Basin of Attraction Analysis in Piecewise-Linear Systems with Big-Bang Bifurcation for the Period-Increment Phenomenon

Abstract

This paper investigates the basins of attraction of periodic orbits arising in one-dimensional piecewise-linear discrete dynamical systems as the system parameters vary in a neighborhood of a Big-Bang bifurcation point associated with the period-increment phenomenon. In this setting, the Big-Bang point corresponds to a parameter value through which infinitely many bifurcation curves pass, leading to the successive emergence of periodic orbits whose periods increase incrementally. The analysis is carried out using a fully analytical approach, exploiting the one-dimensional nature of the system and the occurrence of border-collision bifurcations. Within this framework, we construct analytical sequences that characterize the convergence of any initial condition on the real line toward a periodic point belonging to a periodic orbit, either isolated or coexisting with another periodic orbit. As the main results, we explicitly characterize the basins of attraction of periodic orbits generated in the period-increment Big-Bang scenario and provide explicit analytical conditions on the system parameters for the existence of these periodic orbits. Moreover, we show that, in certain regions of the parameter plane, at most two periodic orbits can coexist, and we describe explicitly the structure of their corresponding basins of attraction. This work provides a new analytical perspective on basin organization in piecewise-linear systems exhibiting the period-increment phenomenon.

1. Introduction

Big-Bang bifurcations constitute a particular class of organizing centers in dynamical systems characterized by the intersection of infinitely many bifurcation curves in parameter space [1,2,3]. In the neighborhood of such points, a wide variety of dynamical behaviors may arise, including the successive emergence of periodic orbits with increasing periods. Owing to this rich structure, Big-Bang bifurcations play a central role in the qualitative analysis of multiparametric dynamical systems.

Big-Bang bifurcations have been studied primarily in the context of piecewise-smooth dynamical systems, whose relevance is well established from both a theoretical and an applied perspective [1,4]. These systems arise naturally in a wide range of applications, including power electronics—particularly DC–DC converters [5,6]—as well as mechanical models such as impact systems and stick–slip oscillators [7,8]. Moreover, certain piecewise-smooth and discontinuous maps appear naturally as Poincaré return maps of continuous-time dynamical systems exhibiting chaotic behavior [9,10,11]. Comprehensive overviews of piecewise-smooth systems and their associated bifurcation phenomena can be found in [1,4,9,12,13,14].

In one-dimensional maps, the specific bifurcation scenario depends fundamentally on local properties near the switching boundary, such as the distances between fixed points—often virtual—and the border, as well as the signs of the eigenvalues determining the local configuration. Under the assumption of contraction on both sides of the switching boundary, the increasing–increasing and increasing–decreasing configurations can be distinguished, while the decreasing–increasing and increasing–decreasing cases are equivalent. In contrast, the decreasing–decreasing configuration admits only two possible scenarios: the existence of a periodic orbit or the presence of two fixed points. These latter cases correspond to first-return maps associated with homoclinic connections in butterfly-type configurations [15,16]. Within this framework, several bifurcation scenarios have been identified, including period-doubling, period-incrementing, and period-adding bifurcations [2,17,18,19], together with more complex behaviors such as coexistence of attractors, period-increment and period-adding combinations, and even chaotic inclusions [15,20].

The increasing–increasing configuration was first analyzed for piecewise-linear maps in [21] through explicit computations. Approximately two decades later, analogous scenarios were confirmed for quadratic maps in [9,22] and subsequently investigated in greater depth using renormalization techniques and low-period computations [23,24]. These studies revealed the emergence of infinitely many periodic orbits from the origin of parameter space, giving rise to a saturation phenomenon and a progressive addition of periods. Later works [20,25,26] demonstrated that similar phenomena may occur in other piecewise-defined maps without requiring a double homoclinic bifurcation. This scenario, termed the period-adding Big-Bang bifurcation, was systematically analyzed in [15].

From a broader perspective, Big-Bang bifurcations have been investigated extensively in engineering and applied contexts [4,27,28]. Despite significant progress, a complete analytical understanding of these phenomena—particularly in multiparameter settings—has historically been difficult to achieve [29] and many open questions remain [13,25]. To address these challenges, piecewise-linear maps are often employed as analytically tractable models that capture essential features of more general piecewise-smooth systems [25,30]. Indeed, such maps are known to act as normal forms for certain piecewise-smooth systems undergoing border-collision bifurcations [31]. Nevertheless, even in the one-dimensional setting, a systematic analytical study of the associated basins of attraction has largely remained absent.

It is well known that Big-Bang bifurcations in piecewise-smooth systems are typically induced by border-collision bifurcations. However, it was shown in [32] that Big-Bang bifurcations may also arise from simple local bifurcations. Related phenomena have further been observed in higher-dimensional and continuous-time dynamical systems [1]. In the one-dimensional case, sufficient conditions for the occurrence of a period-increment Big-Bang bifurcation include contraction near the switching boundary and opposite signs on each side of the discontinuity [15]. The problem of deriving analytical conditions for the existence of Big-Bang bifurcations was explicitly raised at the ENOC 2005 [33]. Subsequent works [15,34] established partial results, showing that, under suitable conditions, two border-collision bifurcation curves in codimension-two piecewise-smooth maps possess an organizing center corresponding to a Big-Bang bifurcation, from which infinitely many periodic orbits emerge.

Numerical investigations of Big-Bang bifurcations in control systems were reported in [35], particularly for first- and second-order systems under sliding-mode control. In parallel, ref. [36] analyzed basins of attraction associated with Big-Bang bifurcations in a Boost power converter governed by a ZAD control strategy. The system was modeled by a two-dimensional piecewise-smooth map, and basin structures in the period-adding scenario were characterized numerically using the Dynamic Bandcounter method [37,38]. These studies highlighted important mechanisms underlying basin organization during transitions involving the coexistence of periodic orbits and further emphasized the relevance of planar systems for qualitative basin analysis [3,35,36].

Despite this extensive body of work, most existing studies focus primarily on the existence, stability, and bifurcation structure of periodic orbits [20,25,26]. Analytical investigations of basins of attraction remain comparatively scarce, especially in one-dimensional piecewise-linear systems. When basin structures are addressed, they are typically studied numerically and mainly in higher-dimensional settings [36,37,38,39,40,41].

Understanding basins of attraction is nevertheless essential for a complete description of long-term system behavior. While bifurcation diagrams describe how periodic solutions are created and organized in parameter space, they do not determine how initial conditions are distributed among coexisting attractors. This limitation is particularly relevant in piecewise-smooth systems, where small variations in initial conditions may lead to abrupt transitions between qualitatively different dynamical regimes [4,12]. Moreover, as emphasized by Datseris et al. [42], basins of attraction in nonlinear systems are often analytically intractable, leading most studies to rely on numerical approximations that necessarily involve finite resolution and may fail to capture hidden attractors [43,44].

Motivated by this gap, the present work focuses on the global analytical study of basins of attraction associated with period-increment Big-Bang bifurcations in one-dimensional piecewise-linear maps. Exploiting the linearity of each branch and the local structure near the switching boundary, we derive explicit analytical constructions that characterize the basins of attraction of all emerging periodic orbits, including regions of coexistence. This one-dimensional framework provides a natural and tractable starting point for the systematic development of analytical tools that may later be extended to higher-dimensional piecewise-smooth systems.

Figure 1 illustrates a representative biparametric diagram in the $(k_{\ell },k_r)$ plane. The origin $(0,0)$ corresponds to a Big-Bang bifurcation point from which infinitely many bifurcation curves emanate and accumulate toward the line $k_{\ell }=0$, revealing the hierarchical organization of the period-increment scenario [15]. Adjacent curves delimit regions of coexistence in which at most two periodic orbits coexist, corresponding to consecutive symbolic itineraries. Such coexistence regions constitute a fundamental structural feature of period-increment Big-Bang bifurcations.

The motivation for studying basins of attraction in this setting goes beyond the identification of periodic orbits. By determining how initial conditions are partitioned among coexisting attractors, basin analysis provides essential insight into attractor selection and long-term predictability. In this work, cobweb diagrams are employed to visualize the iterative dynamics and to support the construction of a fully global analytical description of basin structures on $\mathbb{R}$.

The main contributions of this paper can be summarized as follows: First, we introduce an explicit analytical sequence, defined in terms of the system parameters and obtained from successive preimages of the map near the Big-Bang bifurcation, which encodes the iterative structure governing convergence to periodic points. Second, we develop a complete analytical characterization of the basins of attraction associated with all periodic orbits arising in the considered system, including explicit interval decompositions and convergence criteria. Third, in a biparametric setting, we show that at most two periodic orbits can coexist and provide explicit descriptions of their corresponding attraction domains. Finally, adopting a pointwise perspective, we characterize basins at the level of individual periodic points, leading to a detailed global description of basin organization near period-increment Big-Bang bifurcations.

The results obtained here lay the groundwork for extensions to higher-dimensional piecewise-smooth systems, such as power electronic converters or PWM-controlled oscillators, where Big-Bang bifurcations have been reported numerically and experimentally [34,35,45]. Although the present analysis is restricted to one-dimensional maps, the insights gained from the global organization of basins of attraction are expected to inform future studies in more complex settings.

The remainder of the paper is organized as follows: Section 2 introduces the one-dimensional piecewise-linear map and basic definitions. Section 3 establishes sufficient conditions for the existence of periodic orbits. Section 4 analyzes the period-increment regime and the associated Big-Bang bifurcation structure. Section 5 provides an analytical characterization of basins of attraction, while Section 6 focuses on regions of coexistence. An illustrative example is presented in Section 7 to visualize and consolidate the analytical results developed throughout the paper. Finally, Section 8 summarizes the main conclusions and discusses the scope of the obtained results.

2. Map Definition and Preliminary Concepts

We begin by considering a one-dimensional piecewise-linear map that captures the essential return dynamics of a class of piecewise-smooth systems in a neighborhood of a Big-Bang bifurcation. Such maps arise naturally as reduced or normal-form descriptions of switching systems and border-collision phenomena, where the long-term behavior can be effectively characterized by the iteration of a scalar map.

This reduction allows us to analytically investigate the existence, stability, and coexistence of periodic orbits through the fixed points of the iterates of the map, as well as to describe the organization of basins of attraction in a mathematically tractable setting. This approach is standard in the literature on piecewise-smooth and piecewise-linear dynamical systems, particularly in the study of Big-Bang and period-increment bifurcations; see, for example, [4,12,34,46].

More precisely, we focus on the study of the function $f:\mathbb{R}\to \mathbb{R}$, defined as

$$f\left(x\right)=\left\{\begin{array}{cc}{m}_{1}x+k_{\ell }\hfill & ;\mathrm{if}x\le 0\hfill \\ m_{2}x-k_r\hfill & ;\mathrm{if}x>0\hfill \end{array}\right.,$$

(1)

subject to the conditions $0<m_1<1$ and $-1<m_2<0$, where the parameters satisfy $k_r>0$ and $k_{\ell }>0$, as illustrated in Figure 2.

The bounded ranges of the parameters considered throughout this work were chosen deliberately in order to guarantee the emergence of the period-increment Big-Bang bifurcation scenario that constitutes the focus of our analysis. In particular, the assumptions on the slopes and offsets ensure the existence of periodic orbits whose iterates are strictly contained within the fundamental intervals, allowing for a well-defined interval decomposition and an explicit analytical construction of the associated basins of attraction, as is standard in the analysis of piecewise-smooth and piecewise-linear maps [12,15].

It is worth noting that alternative parameter regimes have been studied in the literature. For instance, when one of the slopes lies in the interval $(0,1)$, the system typically exhibits a period-adding rather than a period-increment scenario, leading to a substantially different dynamical organization and a much more intricate basin structure [15,47]. The analytical treatment of basins in such regimes lies beyond the scope of the present work and will be addressed elsewhere.

Likewise, if both offsets $k_{\ell }$ and $k_r$ share the same sign, no fixed points or periodic orbits arise, which highlights the distinguished role of the origin $(0,0)$ as the organizing center of the Big-Bang bifurcation structure. Without loss of generality, the essential requirement for the validity of the results presented here is that $k_{\ell }$ and $k_r$ have opposite signs, ensuring the persistence of the qualitative dynamical organization and basin structure described in this paper.

Degenerate configurations in which periodic points coincide exactly with switching boundaries or interval endpoints correspond to nongeneric or limiting parameter values. Such cases are excluded by the adopted assumptions and are well known to require separate treatment, as boundary collisions typically mark transitions between distinct dynamical regimes [12]. Under the parameter conditions considered here, all periodic orbits remain strictly interior to the fundamental intervals, and the main results are therefore rigorously established within the intended scope of the framework.

Throughout this section, it is assumed that $m_1$, $m_2$, $k_r$, and $k_{\ell }$ refer to the map f given in (1).

Definition 1 (Orbit). 

The orbit of $x_0$ under the function in (1) is

$$Orb(x_0,f)=\{x_0,f\left(x_0\right),f^2\left(x_0\right),\dots ,f^n\left(x_0\right),\dots \},$$

(2)

where $f^n$ denotes the n-fold composition of f, with $f^0\left(x_0\right)=x_0$ and $n\in {\mathbb{Z}}^{+}$.

The orbit notion introduced above is standard in discrete-time dynamical systems and provides the basic object for describing the evolution of trajectories under iteration; see, for example, [46,48,49]. In the context of piecewise-smooth and piecewise-linear maps, it is often useful to go beyond individual trajectories and introduce relations between orbits in order to capture common dynamical features. In particular, grouping orbits that share the same forward itinerary or switching sequence plays a central role in the analysis of such systems; see, for instance, [4,12].

Definition 2 (Related orbits). 

Let $x_0,y_0$ be two distinct initial conditions. We say that the orbits $Orb(x_0,f)$ and $Orb(y_0,f)$ are related under the relation ∼ if and only if there exists $n\in \mathbb{N}$ such that

$$f^n\left(x_0\right)={\overline{y}}_0,$$

(3)

where ${\overline{y}}_0\in Orb(y_0,f)$.

Since ∼ is an equivalence relation on $Orb\left(f\right)$, we may introduce the corresponding quotient set and characterize the induced orbit structure.

We show that the system in (1) admits periodic orbits for suitable parameter values by identifying fixed points of the iterates $f^n$ on forward-invariant sets (that is, sets $S\subset \mathbb{R}$ satisfying $f\left(S\right)\subseteq S$). Concretely, an n-periodic orbit arises from a solution of

$$f^n\left(x^{*}\right)=x^{*},n\in \mathbb{N},n\ge 1,$$

(4)

where n is the (minimal) period of the orbit generated by $x^{*}$.

It is also immediate that the map has a fixed point when $k_{\ell }=0$ and $k_r=0$. Indeed,

$$k_{\ell }=0,k_r=0⟹f\left(0\right)=0,$$

(5)

so $x=0$ is a fixed point of f.

Definition 3 (Periodic orbit). 

We say that an orbit $Orb(x_0,f)$ is periodic if there exists $n\in \mathbb{N}$ such that $f^n\left(x_0\right)=x_0$. In this case, the period is the smallest positive integer n satisfying this condition. More generally, any periodic orbit of period n satisfies

$$f^q\left(x_0\right)=f^{q+mn}\left(x_0\right),$$

(6)

for all $0\le q<n$ and all $m\in \mathbb{N}$.

Following the approach found in [15], the itinerary of an orbit provides a symbolic representation that facilitates the classification of the types of orbits that may arise in a given phase space. In particular, it is useful for determining the type of periodic orbit to which a given set of initial conditions converges, in which case only the period is relevant.

Definition 4 (Itinerary). 

Let $Orb(x_0,f)$ be the orbit generated by iterating f starting from $x_0$. Its itinerary is defined as

$$\tau (x_0,f)={\vartheta }_1{\vartheta }_2\cdots {\vartheta }_n\cdots ,n\in \mathbb{N},$$

(7)

where

$${\vartheta }_n=\left\{\begin{array}{cc}\mathcal{L}\hfill & ;\mathit{if}{f}^n\left(x_0\right)\le 0\hfill \\ \mathcal{R}\hfill & ;\mathit{if}{f}^n\left(x_0\right)>0\hfill \end{array}\right..$$

(8)

If $Orb(x_0,f)$ is a periodic orbit, its itinerary is denoted by ${\mathcal{O}}_{\tau }$, where τ is a finite sequence of $\mathcal{L}$s and $\mathcal{R}$s.

The symbolic notation of itineraries is especially relevant when studying periodic orbits, as illustrated in

Table 1. Symbolic representation of periodic orbits.

Periodic Orbit	Symbolic Itinerary Notation
$\mathcal{O}=\{-0.5,0.5\}$	${\mathcal{O}}_{\tau }={\mathcal{O}}_{\mathcal{LR}}$
$\mathcal{O}=\{-0.1,-0.2,-0.3,0.5\}$	${\mathcal{O}}_{\tau }={\mathcal{O}}_{\mathcal{LCLR}}={\mathcal{O}}_{{\mathcal{L}}^3\mathcal{R}}$
$\mathcal{O}=\{0.1,0.2,0.3,0.5\}$	${\mathcal{O}}_{\tau }={\mathcal{O}}_{\mathcal{RRRR}}={\mathcal{O}}_{{\mathcal{R}}^4}$
$\mathcal{O}=\{-0.5,0.5\}=\{0.5,-0.5\}$	${\mathcal{O}}_{\tau }={\mathcal{O}}_{\mathcal{LR}}={\mathcal{O}}_{\mathcal{RL}}$

. Repeated symbols $\mathcal{L}$ or $\mathcal{R}$ are written with an exponent to simplify notation. When working with equivalence classes of orbits, some symbolic itineraries are considered equivalent. Moreover, two identical symbolic itineraries do not necessarily correspond to the same physical orbit.

In the remainder of this article, we denote a periodic orbit $\mathcal{O}$ together with its symbolic itinerary $\tau$ by ${\mathcal{O}}_{\tau }$. This notation follows the convention that each point of the orbit is assigned a symbol according to the region of the domain it occupies, as detailed in Table 1 and the accompanying discussion.

As an example, consider the periodic orbit $\mathcal{O}=\{-0.5,0.5\}$ in the first row of Table 1. Under iteration of f, the point $-0.5$ maps to $0.5$, and $0.5$ maps back to $-0.5$, forming a 2-cycle. Since $-0.5$ lies in the left region $\mathcal{L}$ and $0.5$ lies in the right region $\mathcal{R}$, the symbolic itinerary is $\mathcal{LR}$. Therefore, the symbolic representation ${\mathcal{O}}_{\tau }={\mathcal{O}}_{\mathcal{LR}}$ encodes the fact that the orbit alternates between the left and right regions of the domain (see Figure 3).

3. Sufficient Conditions for Obtaining Periodic Orbits

In systems such as (1), where $k_{\ell }$ and $k_r$ are both positive, one may observe convergence to a single periodic orbit or, at most, the coexistence of two periodic orbits. In [15], the following result is stated without proof.

Proposition 1. 

Let f be a map of the form (1), and let $Orb(x_0,f)$ be an orbit with symbolic configuration ${\mathcal{O}}_{\tau }$. For $k_r>0$ and $k_{\ell }>0$, it always holds that, if the itinerary satisfies ${\vartheta }_n=\mathcal{R}$, then ${\vartheta }_{n+1}=\mathcal{L}$ for all $n\in \mathbb{N}$.

Proof. 

Suppose that ${\vartheta }_n=\mathcal{R}$, that is, $f^n\left(x_0\right)>0$. From (1), this implies

$$f^{n+1}\left(x_0\right)=m_2{f}^n\left(x_0\right)-k_r.$$

(9)

Since $m_2<0$, the conditions $f^n\left(x_0\right)>0$ and $k_r>0$ guarantee that $f^{n+1}\left(x_0\right)<0$. Therefore, ${\vartheta }_{n+1}=\mathcal{L}$. □

When aiming to determine the convergence of orbits graphically in $\mathbb{R}$, the cobweb diagram can be employed for discontinuous systems, as illustrated in Figure 4. In this example, the initial condition $x_0$ converges, through successive iterations (in green), towards a period-3 orbit. It is worth noting the role of the line $y=x$ in achieving convergence.

The existence of periodic orbits depends explicitly on the interval $\left(b,-k_r\right)$, as will be shown later in Theorem 2, while coexistence depends on the presence of a point $a_n$ within this interval. The structure of periodic orbits and their corresponding evaluations, depending on the intervals, is illustrated in Figure 5, Figure 6 and Figure 7. In particular, one can observe that every periodic orbit contains at least one point lying in the interval $\left(0,b_0\right)$.

The result of Proposition 1 guarantees a strict alternation between $\mathcal{R}$ and $\mathcal{L}$ symbols in the itineraries when $k_{\ell }>0$ and $k_r>0$. This alternance plays a key role in the construction of the periodic orbits of an arbitrary period.

To construct periodic orbits of an arbitrary period by varying the parameters $k_{\ell }$ and $k_r$ in (1), it is necessary to introduce a quantity that naturally emerges from the iterative structure of the system and allows us to formulate precise parameter conditions for the existence of periodic solutions. To this end, we examine the successive preimages generated by iterating the left branch of the map. This iterative process reveals a distinguished sequence of values that plays a fundamental role in organizing the dynamics and delimiting the parameter regions associated with periodic behavior.

Definition 5. 

Let $k_{\ell }>0$. For every $n\in \mathbb{N}$, we define

$$a_n={\displaystyle \frac{k_{\ell }\left(1-m_1^n\right)}{m_1^n\left(m_1-1\right)}}.$$

(10)

The sequence ${\left\{a_n\right\}}_{n\in \mathbb{N}}$ provides a convenient and effective framework for characterizing parameter intervals associated with the existence and coexistence of periodic orbits. In particular, these values will serve as the main building blocks for establishing the criteria required to generate periodic solutions of a prescribed period in the forthcoming sections.

From Definition 5, it is possible to derive certain results that are fundamental for establishing the existence of the orbits of a given period n. This is feasible because map (1) has been conveniently set up to allow variation of the parameters $k_{\ell }$ and $k_r$, as illustrated in Figure 2.

Lemma 1. 

Let map (1) be given, and let $a_n$ be defined as in Definition 5. Then, for every $n\in \mathbb{N}$, the following properties hold:

(i) 

$a_{n+1}<a_n$;

(ii) 

$a_n\le 0$;

(iii) 

$f\left(a_{n+1}\right)=a_n$.

Proof. 

We proceed in several steps:

(i)

Since

$${(m_1-1)}^2>0,$$

we have

$$\begin{array}{cc}\hfill m_1^2-2m_1+1& >0,\hfill \\ \hfill 2m_1& <m_1^2+1.\hfill \end{array}$$

(11)

Since $m_1^n>0$ for all $n\in \mathbb{N}$, it follows that

$$\begin{array}{cc}\hfill 2m_1^n{m}_1& <m_1^n(m_1^2+1),\hfill \\ \hfill m_1^{n+1}+m_1^{n+1}& <m_1^{n+2}+m_1^n,\hfill \\ \hfill -m_1^n+m_1^{n+1}& <-m_1^{n+1}+m_1^{n+2}.\hfill \end{array}$$

(12)

Adding and subtracting the same term on both sides, we obtain

$$\begin{array}{cc}\hfill m_1^{2n+1}-m_1^{2n+2}-m_1^n+m_1^{n+1}& <m_1^{2n+1}-m_1^{2n+2}-m_1^{n+1}+m_1^{n+2}\hfill \\ \hfill (m_1^{n+1}-1)(m_1^n-m_1^{n+1})& <(m_1^n-1)(m_1^{n+1}-m_1^{n+2})\hfill \end{array}$$

(13)

Since $m_1^n-m_1^{n+1}>0$, $m_1^{n+1}-m_1^{n+2}>0$, and $k_{\ell }>0$, we have

$${\displaystyle \frac{k_{\ell }(m_1^{n+1}-1)}{m_1^{n+1}-m_1^{n+2}}}<{\displaystyle \frac{k_{\ell }(m_1^n-1)}{m_1^n-m_1^{n+1}}}.$$

(14)

Therefore,

$$a_{n+1}<a_n.$$

(15)

(ii)

From part (i), we have

$$\cdots <a_n<a_{n-1}<\cdots <a_1<a_0=0,$$

which implies that $a_n\le 0$ for all $n\in \mathbb{N}$.

(iii)

Since $a_n\le 0$ for all $n\in \mathbb{N}$, by part (ii), we have from (1)

$$\begin{array}{cc}\hfill f\left(a_{n+1}\right)& =m_1{a}_{n+1}+k_{\ell }\hfill \\ \hfill & =m_1{\displaystyle \frac{k_{\ell }(1-m_1^{n+1})}{m_1^{n+1}(m_1-1)}}+k_{\ell }\hfill \\ \hfill & ={\displaystyle \frac{k_{\ell }(1-m_1^{n+1})+k_{\ell }{m}_1^n(m_1-1)}{m_1^n(m_1-1)}}\hfill \\ \hfill & ={\displaystyle \frac{k_{\ell }-k_{\ell }{m}_1^{n+1}+k_{\ell }{m}_1^{n+1}-k_{\ell }{m}_1^n}{m_1^n(m_1-1)}}\hfill \\ \hfill & =k_{\ell }{\displaystyle \frac{1-m_1^n}{m_1^n(m_1-1)}}\hfill \\ \hfill & =a_n.\hfill \end{array}$$

(16)

□

From expression (1), it follows that the function is increasing on the interval $(-\infty ,0]$ since $f^{\prime }\left(x\right)=m_1>0$ in this domain and decreasing on $(0,\infty )$ as $f^{\prime }\left(x\right)=m_2<0$ there.

Lemma 2. 

Under the conditions given in (1), and for $a_n$ as defined in Definition 5, the following hold:

(i) 

If $x\in \left(a_1,a_0\right]$, then $f\left(x\right)\in \left(0,k_{\ell }\right]$.

(ii) 

If $x\in \left(a_{n+1},a_n\right]$, then $f\left(x\right)\in \left(a_n,a_{n-1}\right]$ for all $n\in \mathbb{N}$.

Proof. 

(i)

Let $x\in \left(a_1,a_0\right]$. By Lemma 1, we have $f\left(a_1\right)=a_0=0$, and, moreover, $f\left(0\right)=k_{\ell }$. Since f is strictly increasing on the interval $(-\infty ,0]$, it follows that $f\left(x\right)\in \left(0,k_{\ell }\right]$.

(ii)

Let $x\in \left(a_{n+1},a_n\right]$. Then, $a_{n+1}<x\le a_n$. Since f is increasing on $(-\infty ,0]$, we obtain $f\left(a_{n+1}\right)<f\left(x\right)\le f\left(a_n\right)$, that is, $a_n<f\left(x\right)\le a_{n-1}$.

□

Lemma 2 allows us to establish the following proposition, which describes how f acts on the subintervals of $(-\infty ,0]$:

Proposition 2. 

For every $n\in {\mathbb{Z}}^{+}$, if $A_n=(a_n,a_{n-1}]$, then $f\left(A_{n+1}\right)=A_n$.

Proof. 

From Lemmas 1 and 2, together with the monotonicity of f on $(-\infty ,0]$, we have $f\left((a_{n+1},a_n]\right)\subseteq (a_n,a_{n-1}]$, and the endpoints are mapped accordingly, which yields $f\left(A_{n+1}\right)=A_n$. □

The following result will be used to prove Proposition 3, which ensures that the sequence $\{a_0,a_1,\dots ,a_n,\dots \}$ generates a partition of the interval $(-\infty ,0]$. This, in turn, allows for a global analysis of the basins of attraction of periodic orbits.

Lemma 3. 

For every $n\in {\mathbb{Z}}^{+}$, the inequality $a_n<-n{k}_{\ell }$ holds.

Proof. 

Since $0<m_1<1$ and $k_{\ell }>0$, we have

$$-{\displaystyle \frac{k_{\ell }}{m_1}}<-k_{\ell }.$$

Similarly, since $0<m_1^2<m_1<1$,

$$-{\displaystyle \frac{k_{\ell }}{m_1^2}}<-k_{\ell }.$$

By induction, for any $n\in {\mathbb{Z}}^{+}$,

$$-{\displaystyle \frac{k_{\ell }}{m_1^n}}<-k_{\ell }.$$

(17)

Inequality (17) implies

$$-k_{\ell }\sum _{i=1}^n{\displaystyle \frac{1}{m_1^i}}<-n{k}_{\ell }.$$

Now, we explicitly compute

$$-k_{\ell }\sum _{i=1}^n{\displaystyle \frac{1}{m_1^i}}=-{\displaystyle \frac{k_{\ell }}{m_1}}\sum _{k=0}^{n-1}{\left({\displaystyle \frac{1}{m_1}}\right)}^k=-{\displaystyle \frac{k_{\ell }}{m_1}}·{\displaystyle \frac{1-{\left(\frac{1}{m_1}\right)}^n}{1-\frac{1}{m_1}}},$$

where $m_1\ne 1$. This simplifies to

$$-k_{\ell }\sum _{i=1}^n{\displaystyle \frac{1}{m_1^i}}={\displaystyle \frac{k_{\ell }\left(m_1^n-1\right)}{m_1^n\left(m_1-1\right)}}=a_n.$$

(18)

Therefore, combining (18) with the previous inequality, we obtain

$$a_n<-n{k}_{\ell },$$

(19)

which completes the proof. □

The following result characterizes the set covered by the sequence of intervals $\left(a_{n+1},a_n\right]$ and shows that their union exactly coincides with the negative half-line up to zero.

Proposition 3. 

${\displaystyle \bigcup _{n\in \mathbb{N}}\left(a_{n+1},a_n\right]=(-\infty ,0]}$.

Proof. 

Suppose that $x\in {\bigcup }_{n\in \mathbb{N}}\left(a_{n+1},a_n\right]$. Then, there exists $n\in \mathbb{N}$ such that $x\in \left(a_{n+1},a_n\right]$. Since $a_n\le 0$ by Lemma 1, it follows that $x\in (-\infty ,0]$. Conversely, suppose that $x\in (-\infty ,0]$. There are two possible cases, namely, $-x\le k_{\ell }$ and $-x>k_{\ell }$. If $-x\le k_{\ell }$, then $a_1<-k_{\ell }\le x$, and, since $x\le a_0=0$, we conclude that $x\in \left(a_1,a_0\right]$. If $-x>k_{\ell }$, there exists $m\in \mathbb{N}$ such that $-x<m{k}_{\ell }$. Therefore, $-m{k}_{\ell }<x$.

By Lemma 1, we have

$$a_m<-m{k}_{\ell }<x<a_0.$$

Since $a_m<a_{m-1}<\cdots <a_0$, there exists $n\in \{0,1,\dots ,m\}$ such that $x\in \left(a_{n+1},a_n\right]$. □

All intervals $\left(a_{n+1},a_n\right]$ are mapped to the intervals $\left(a_n,a_{n-1}\right]$, as illustrated in Figure 8. This fact is established in the following proposition:

Proposition 4. 

If $x\in \left(a_{n+1},a_n\right]$, then for every $0\le m\le n$,

$$f^m\left(x\right)\in \left(a_{n-m+1},a_{n-m}\right].$$

(20)

Proof. 

Suppose $x\in \left(a_{n+1},a_n\right]$. We show by induction on m that, for every $0\le m\le n$, the iterate $f^m\left(x\right)$ belongs to $\left(a_{n-m+1},a_{n-m}\right]$. For the base case, since $f^0\left(x\right)=x$, it follows that $x\in \left(a_{n+1},a_n\right]$. Now, assume that, for some m with $0\le m<n$, Equation (20) holds. Then, by Lemma 2,

$$f^{m+1}\left(x\right)\in \left(a_{n-(m+1)+1},a_{n-(m+1)}\right],$$

(21)

which completes the induction. □

The following result is mentioned in [15]; however, we provide here a proof in the context of the piecewise-linear system (1).

Theorem 1. 

Let $b=m_2{k}_{\ell }-k_r$. If there exists $a_n\in \left(b,-k_r\right)$, then it is unique.

Proof. 

Suppose there exist $a_n$ and $a_m$ in the interval $\left(b,-k_r\right)$. Without loss of generality, assume that $a_m<a_n$. Then,

$$\left|a_n-a_m\right|<\left|-k_r-b\right|,$$

(22)

and consequently

$$0<a_n-a_m<\left|m_2\right|k_{\ell }.$$

(23)

From Definition 5, $a_n$ can be written as

$$a_n=-k_{\ell }\sum _{i=1}^n{\displaystyle \frac{1}{m_1^i}},$$

(24)

and similarly

$$a_m=-k_{\ell }\sum _{i=1}^m{\displaystyle \frac{1}{m_1^i}}.$$

(25)

Therefore,

$$-k_{\ell }\sum _{i=1}^n{\displaystyle \frac{1}{m_1^i}}+k_{\ell }\sum _{i=1}^m{\displaystyle \frac{1}{m_1^i}}<\left|m_2\right|k_{\ell },$$

(26)

which implies

$$\sum _{i=n}^m{\displaystyle \frac{1}{m_1^i}}<\left|m_2\right|.$$

(27)

Since

$$\sum _{i=n}^m{\displaystyle \frac{1}{m_1^i}}>1,$$

(28)

it follows that $\left|m_2\right|>1$, which is impossible because $-1<m_2<0$. Hence, $a_n$ must be unique. □

4. Period-Increment Regime

Given that the negative real axis has been partitioned in a particular way to obtain certain results, it is also necessary to determine a partition of the positive side in order to obtain the basins of attraction.

Definition 6. 

Let

$$a_q=max\left\{\dots ,a_n,\dots ,a_2,a_1\right\}$$

(29)

such that $a_q<-k_r$, with $k_r$ and $m_2$, as in Definition 5. Then, for each $n\in {\mathbb{Z}}^{+}$, we define $b_m$ as

$$b_m={\displaystyle \frac{a_{q+m}+k_r}{m_2}}$$

(30)

for all $m\in \mathbb{N}$.

Lemma 4. 

Let $a_q$ and $b_m$ be as in Definition 6. If

$$\left(b,-k_r\right)\subset \left(a_q,a_{q-1}\right),$$

(31)

with b as in Theorem 1, then the following apply:

(i) 

For all $m\in \mathbb{N}$,

$$f\left(b_m\right)=a_{q+m}.$$

(32)

(ii) 

Let $A=\left(a_1,a_0\right]$. Then,

$$f\left(A\right)=\left(0,k_{\ell }\right]\subset \left(0,b_0\right).$$

(33)

(iii) 

Let $A=\left(0,b_0\right)$. Then,

$$f\left(A\right)=\left(a_q,-k_r\right)\subset \left(a_q,a_{q-1}\right).$$

(34)

(iv) 

If $A_k=\left(b_k,b_{k+1}\right)$, then, for all $k\in \mathbb{N}$,

$$f\left(A_k\right)=\left(a_{q+k+1},a_{q+k}\right).$$

(35)

Proof. 

(i)

We have

$$f\left(b_m\right)=f\left({\displaystyle \frac{a_{q+m}+k_r}{m_2}}\right)=m_2{\displaystyle \frac{a_{q+m}+k_r}{m_2}}-k_r=a_{q+m}.$$

(36)

(ii)

If $x\in \left(a_1,a_0\right]$, since f is increasing on the interval $(-\infty ,0]$, it follows that

$$f\left(x\right)\in \left(f\left(a_1\right),f\left(a_0\right)\right]=\left(0,k_{\ell }\right].$$

(37)

Given that $b>a_q$ by the stated hypotheses, we have

$$m_2{k}_{\ell }-k_r>a_q,$$

(38)

which implies

$$m_2{k}_{\ell }>a_q+k_r,$$

(39)

and therefore

$$k_{\ell }<{\displaystyle \frac{a_q+k_r}{m_2}}.$$

(40)

Since ${\displaystyle \frac{a_q+k_r}{m_2}}=b_0$, we conclude that

$$k_{\ell }<b_0,$$

(41)

and thus $\left(0,k_{\ell }\right]\subset \left(0,b_0\right)$.

(iii)

Let $x\in \left(0,b_0\right)$. Then, $0<x<b_0$. Since f is decreasing on $(0,\infty )$, we have

$$f\left(b_0\right)<f\left(x\right)<f\left(0^{+}\right),$$

(42)

where

$$f\left(0^{+}\right)=\underset{x\to 0^{+}}{lim}f\left(x\right)=-k_r.$$

(43)

Using $f\left(b_0\right)=a_q$, we conclude that

$$f\left(x\right)\in \left(a_q,-k_r\right).$$

(44)

(iv)

Let $x\in \left(b_k,b_{k+1}\right)$ for some $k\in \mathbb{N}$. As in the previous item, since f is decreasing, we obtain

$$f\left(x\right)\in \left(f\left(b_{k+1}\right),f\left(b_k\right)\right)=\left(a_{q+k+1},a_{q+k}\right).$$

(45)

□

Proposition 5. 

The set

$$\left\{0,b_0,b_1,\dots ,b_n,\dots \right\}$$

(46)

generates a partition of the interval $(0,\infty )$.

Proof. 

First, we show that $0<b_m<b_{m+1}$ for all $m\in \mathbb{N}$. Indeed, from Lemma 1, we have $a_{n+m+1}<a_{n+m}$, and, since $k_r,m_2<0$, it follows that $b_m<b_{m+1}$.

Next, we prove that

$$\bigcup _{m\in \mathbb{N}}\left[b_m,b_{m+1}\right)\cup \left(0,b_0\right)=(0,\infty ).$$

(47)

If $x\in {\bigcup }_{m\in \mathbb{N}}\left[b_m,b_{m+1}\right)\cup \left(0,b_0\right)$, then there exists $m\in \mathbb{N}$ such that $x\in \left[b_m,b_{m+1}\right)$. Since $b_m>0$ for all $m\in \mathbb{N}$, it follows that $x\in (0,\infty )$. Similarly, if $x\in (0,b_0)$, it is clear that $x\in (0,\infty )$.

Conversely, let $x\in (0,\infty )$. Then, by the definition of f and Lemma 4, we have

$$f\left(x\right)\in (-\infty ,0],$$

(48)

which means that there exists $p\in \mathbb{N}$ such that

$$a_{p+1}<m_{2}x-k_r\le a_p.$$

(49)

There are three possible cases:

1.

Case $p=q$. In this case,

$${\displaystyle \frac{a_q+k_r}{m_2}}\le x<{\displaystyle \frac{a_{q+1}+k_r}{m_2}},$$

(50)

that is,

$$b_0\le x<b_1,$$

(51)

so $x\in \left[b_0,b_1\right)$.

2.

Case $p<q$. Here, $a_q<a_p<0$ and $a_q\le a_{p+1}<a_p$; thus,

$$a_q\le m_{2}x-k_r<0,$$

(52)

which gives

$$0<x\le {\displaystyle \frac{a_q+k_r}{m_2}}=b_0.$$

(53)

Therefore, $x\in \left(0,b_0\right]$.

3.

Case $p>q$. There exists $m\in \mathbb{N}$ such that $p=q+m$. Then,

$$a_{q+m+1}<m_{2}x-k_r\le a_{q+m},$$

(54)

or equivalently

$$b_m\le x<b_{m+1}.$$

(55)

Hence, $x\in \left[b_m,b_{m+1}\right)$.

In all cases, x belongs to ${\bigcup }_{m\in \mathbb{N}}\left[b_m,b_{m+1}\right)\cup \left(0,b_0\right)$, completing the proof of (47). □

The contraction property will be used to establish the existence of the periodic orbit identified in Theorem 2. Since any iterate of f that acts as a strict contraction on a forward-invariant interval possesses a unique fixed point on that interval, it suffices to verify that $f^n$ contracts distances on the corresponding domain. This standard consequence of the Banach fixed-point framework allows us to locate the required periodic point.

Theorem 2. 

Let b be as defined in Theorem 1. If $\left(b,-k_r\right)\subset \left(a_n,a_{n-1}\right)$, then, for every $x_0\in \mathbb{R}$, the orbit $Orb(x_0,f)$ converges to a periodic orbit of period $n+1$.

Proof. 

Let

$$A_1=\left(a_n,a_{n-1}\right),A_2=\left(a_{n-1},a_{n-2}\right),\dots ,A_n=\left(a_1,a_0\right),A_{n+1}=\left(0,b_0\right)$$

(56)

for all $n\in \mathbb{N}$.

By Lemma 4, we have $f\left(A_k\right)=A_{k+1}$ for $k=1,2,\dots ,n-1$ and $f\left(A_n\right)\subset A_{n+1}$ and $f\left(A_{n+1}\right)\subset A_1$. This means that

$$f_{A_k}^{n+1}:A_k⟶A_k,$$

(57)

where

$$f_{A_k}^{n+1}\left(x\right)=k_{\ell }\sum _{i=0}^{k-2}{m}_1^i-m_1^{k-1}{k}_r+k_{\ell }{m}_2\sum _{i=k-1}^{n-1}{m}_1^i+m_2{m}_1^{n}x,$$

(58)

with $1\le k\le n+1$.

Recall that ${\sum }_{i=0}^{-1}{m}_1^i$ denotes the empty sum, which is taken to be zero.

We now show that $f_{A_k}^{n+1}\left(x\right)$ is a contraction for each $1\le k\le n+1$. We compute

$$\left|f_{A_k}^{n+1}\left(x\right)-f_{A_k}^{n+1}\left(y\right)\right|=\left|m_2{m}_1^{n}x-m_2{m}_1^{n}y\right|,$$

(59)

Hence,

$$\left|f_{A_k}^{n+1}\left(x\right)-f_{A_k}^{n+1}\left(y\right)\right|=\left|m_2{m}_1^n\right||x-y|.$$

(60)

Since $\delta =\left|m_2{m}_1^n\right|<1$, it follows that $f_{A_k}^{n+1}\left(x\right)$ is a contraction for each $1\le k\le n+1$. By the Banach Fixed-Point Theorem, each $f_{A_k}^{n+1}$ is continuous and has a unique fixed point $x_k^{*}$ in $A_k$. Therefore, the set $\{x_1^{*},x_2^{*},\dots ,x_{n+1}^{*}\}$ forms a periodic orbit of period $n+1$. □

The previous theorem implies that the parameters $k_{\ell }$ and $k_r$ can be varied in such a way that, for any n, it is possible to obtain the periodic orbits of any desired period.

5. Characterization of Attraction Basins

Definition 7 (Base of attraction for a periodic orbit). 

Let $Orb\left(x_1,f\right)=\left\{x_1,x_2,\dots ,x_n\right\}$ be a periodic orbit of period n. A base of the domain of attraction for $x_k$, with $1\le k\le n$, is given by

$$B{A}_{x_k}=\left\{x_0:\underset{m\to \infty }{lim}{f}^{nm}\left(x_0\right)=x_k\right\}.$$

(61)

The attraction basin for the periodic orbit $Orb\left(x_1,f\right)$ is given by

$$B{A}_{Orb\left(x_1,f\right)}=\bigcup _{1\le k\le n}B{A}_{x_k}.$$

(62)

Corollary 1. 

Let

$$A_1=\left(0,b_0\right),A_2=\left(a_1,0\right],A_3=\left(a_2,a_1\right],\dots ,A_{n+1}=\left(a_n,a_{n-1}\right]$$

(63)

and assume that $\left(b,-k_r\right)\subset \left(a_n,a_{n-1}\right)$. Then, a periodic orbit

$$\left\{x_1,x_2,\dots ,x_n,x_{n+1}\right\}$$

(64)

is generated with $x_k\in A_k$. If $x\in A_k$, $1\le k\le n+1$, then

$$\underset{m\to \infty }{lim}{f}^{(n+1)m}\left(x\right)=x_k.$$

(65)

Proof. 

By Theorem 2, if for $x\in A_k$ we define $g\left(x\right)=f^{n+1}\left(x\right)$, then, since $x_k$ is the fixed point in $A_k$, we have

$$\underset{m\to \infty }{lim}{g}^m\left(x\right)=x_k.$$

(66)

Moreover, since $g^m\left(x\right)={\left(f^{n+1}\right)}^m\left(x\right)$, it follows that

$$\underset{m\to \infty }{lim}{f}^{(n+1)m}\left(x\right)=x_k.$$

(67)

This holds for all $1\le k\le n+1$. □

The following definition allows us to determine the fixed points of the subfunctions derived from the main function, in order to identify, within each of them, fixed points that will form part of the periodic orbit of the main function.

Definition 8 (Fundamental domain of attraction). 

A fundamental domain of attraction for a point $x_k$ of a periodic orbit of period n is the largest set $A\subseteq \mathbb{R}$ such that

$$f^n\left(A\right)\subseteq A,$$

(68)

where $f^n\left(x_k\right)=x_k$.

Each fundamental interval of attraction serves to determine the point of the periodic orbit to which the initial conditions inside that interval converge, as illustrated in Figure 9, Figure 10 and Figure 11. In these figures, starting from two different initial conditions, one can observe their convergence to a periodic orbit of period 3; in each of its fundamental intervals, the point of the periodic orbit to which they belong can be identified.

Similarly, for example, in the case of a periodic orbit of period 4 without coexistence, Figure 12, Figure 13 and Figure 14 show three fundamental intervals that make convergence possible. This occurs when

$$\left(b,-k_r\right)\subset \left(a_2,a_1\right).$$

(69)

Theorem 3. 

If

$$\left(b,-k_r\right)\subset \left(a_n,a_{n-1}\right),$$

(70)

for some $n\in \mathbb{N}$, then the fundamental domain of attraction for the periodic orbit of period $n+1$,

$$\left\{x_1,x_2,\dots ,x_n,x_{n+1}\right\},$$

(71)

is given by

$$\bigcup _{1\le k\le n+1}{A}_k,$$

(72)

where

$$A_1=\left(a_n,a_{n-1}\right],A_2=\left(a_{n-1},a_{n-2}\right],\dots ,A_n=\left(a_1,a_0\right],A_{n+1}=\left(0,b_0\right).$$

(73)

Proof. 

As established in Theorem 2, for open intervals we define

$$f_{A_k}^{n+1}:A_k⟶A_k.$$

(74)

In our case, we must examine the right endpoints of the intervals for this property to hold. Specifically,

$$\begin{array}{cc}\hfill & f\left(a_{n-1}\right)\in A_2,f^2\left(a_{n-1}\right)\in A_3,\dots ,f^{n-1}\left(a_{n-1}\right)\in A_n,\hfill \\ \hfill & f^n\left(a_{n-1}\right)\in A_{n+1},f^{n+1}\left(a_{n-1}\right)\in A_1,\hfill \end{array}$$

(75)

and similarly

$$\begin{array}{cc}\hfill & f\left(a_{n-2}\right)\in A_3,f^2\left(a_{n-2}\right)\in A_4,\dots ,f^{n-1}\left(a_{n-2}\right)\in A_{n+1},\hfill \\ \hfill & f^n\left(a_{n-2}\right)\in A_1,f^{n+1}\left(a_{n-2}\right)\in A_2.\hfill \end{array}$$

(76)

The same reasoning applies for all points $a_{n-3},\dots ,a_0,b_0$.

If a value smaller than $a_n$ is chosen, more iterations are required according to Lemma 2. Likewise, if values larger than $b_0$ are chosen, additional iterations are also necessary. □

The following theorem characterizes part of the attraction basins corresponding to nonfundamental intervals in ${\mathbb{R}}^{-}$.

Theorem 4. 

Let

$$A_1=\left(0,b_0\right),A_2=\left(a_1,0\right],A_3=\left(a_2,a_1\right],\dots ,A_{n+1}=\left(a_n,a_{n-1}\right].$$

(77)

Let

$$p_q=n+1+q+nt,$$

(78)

for $0\le q\le n$ and $t\in {\mathbb{Z}}^{+}$. If

$$x\in \left(a_{p_q+1},a_{p_q}\right],$$

(79)

then x belongs to the same attraction basin to which the interval $A_{q+1}$ belongs.

Proof. 

Let $x\in \left(a_{p_q+1},a_{p_q}\right]$. By Proposition 4, we have

$$f^{(n+1)t}\left(x\right)\in \left(a_{q+1},a_q\right].$$

(80)

If $q\ge 1$,

$$f^{(n+1)t}\left(x\right)\in \left(0,b_0\right).$$

(81)

Since $\left(a_{q+1},a_q\right]$ is a fundamental interval for every $1\le q\le n$, by Corollary 1 it follows that x belongs to the same attraction basin as $A_{q+1}$. □

Theorem 5. 

If $x\in \left[b_k,b_{k+1}\right)$, then x belongs to the same attraction basin to which the interval $\left(a_{k+1},a_k\right]$ belongs.

Proof. 

Let $x\in \left[b_k,b_{k+1}\right)$. By Lemma 4,

$$f\left(x\right)\in \left(a_{q+k+1},a_{q+k}\right].$$

(82)

By Proposition 4,

$$f^{q+1}\left(x\right)\in \left(a_{k+1},a_k\right].$$

(83)

By Theorem 4, $f^{q+1}\left(x\right)$ belongs to the same domain of attraction as $\left(a_{k+1},a_k\right]$, and therefore the same holds for x. □

6. Attraction Basins for the Coexistence of Periodic Orbits

To determine the attraction basins of two coexisting periodic orbits, certain basic parameter criteria must be met which generally depend on the existence of some $a_n$ within the interval $\left(b,-k_r\right)$. For instance, in Figure 15, two periodic orbits of periods 2 and 3 coexist because $a_1\in \left(b,-k_r\right)$. In Figure 16, one can observe the coexistence of two periodic orbits of periods 3 and 4, which arises from having $a_2$ within the interval $\left(b,-k_r\right)$. The same occurs in Figure 17, where $a_3\in \left(b,-k_r\right)$ leads to the coexistence of orbits of periods 4 and 5. Similarly, in Figure 18, $a_8\in \left(b,-k_r\right)$ yields the coexistence of periodic orbits of periods 9 and 10.

To compute the attraction basins in cases of coexisting orbits, we define the variables $c_p$ for the negative real axis and ${\overline{c}}_p$ for the positive real axis as follows:

Definition 9. 

Let $a_q$ be as in Equation (5) with $k_{\ell },k_r>0$, $0<m_1<1$, and $-1<m_2<0$. For $p\in \mathbb{N}$, we define

$$c_p={\displaystyle \frac{a_q+k_r}{m_1^p{m}_2}}+a_p,$$

(84)

and, for $p>0$,

$${\overline{c}}_p={\displaystyle \frac{c_{q+p}+k_r}{m_2}}.$$

(85)

The following lemma will be used to prove all subsequent theorems regarding the attraction basins:

Lemma 5. 

Let $a_q\in \left(b,-k_r\right)$ for some $q\in \mathbb{N}$, and let $c_p$ and ${\overline{c}}_p$ satisfy the conditions given in Definition 9, as defined in (84) and (85). Then, for all $j\in \mathbb{N}$, the following hold:

(i) 

$c_{j+1}<a_j$;

(ii) 

$c_{j+1}<c_j$;

(iii) 

${\overline{c}}_j<{\overline{c}}_{j+1}$, $j>0$;

(iv) 

$b_j<{\overline{c}}_{j+1}$;

(v) 

$f\left(c_{j+1}\right)=c_j$, $j>0$;

(vi) 

$f\left({\overline{c}}_j\right)=c_{n+j}$, $j>0$;

(vii) 

$b_0<k_{\ell }<{\overline{c}}_1$.

Proof. 

We prove each item separately.

(i)

Since $b<a_n$, we have

$$-k_r+m_2{k}_{\ell }<a_n,$$

(86)

which implies

$${\displaystyle \frac{a_n+k_r}{m_2}}<k_{\ell }.$$

(87)

Adding and subtracting convenient terms, for $j\in \mathbb{N}$ we can write

$$\begin{array}{}\mathrm{(88)}& \hfill {\displaystyle \frac{a_n+k_r}{m_2}}& <-k_{\ell }{m}_1^j+k_{\ell }+m_1^j{k}_{\ell }\hfill \mathrm{(89)}& \hfill & =-k_{\ell }(m_1^j-1)+m_1^j{k}_{\ell }.\hfill \end{array}$$

Since $m_1^j>0$ for all $j\in \mathbb{N}$, dividing by $m_1^j$ yields

$${\displaystyle \frac{a_n+k_r}{m_2{m}_1^j}}<{\displaystyle \frac{-k_{\ell }(m_1^j-1)+m_1^j{k}_{\ell }}{m_1^j}}.$$

(90)

Rewriting the second term in terms of $a_j$,

$$\begin{array}{}\mathrm{(91)}& \hfill {\displaystyle \frac{a_n+k_r}{m_2{m}_1^j}}& <{\displaystyle \frac{k_{\ell }(m_1^j-1)(m_1-1)}{m_1^j(1-m_1)}}+k_{\ell }\hfill \mathrm{(92)}& \hfill & =a_j(m_1-1)+k_{\ell }\hfill \mathrm{(93)}& \hfill & =m_1{a}_j-a_j+k_{\ell }.\hfill \end{array}$$

Adding $a_{j+1}$ to both sides and rearranging gives

$${\displaystyle \frac{a_n+k_r}{m_2{m}_1^{j+1}}}+a_{j+1}<a_j.$$

(94)

By the definition of $c_{j+1}$, this is exactly $c_{j+1}<a_j$.

(ii)

Since $0<m_1<1$, we have $m_1^j>m_1^{j+1}$ for all j; hence,

$${\displaystyle \frac{a_n+k_r}{m_1^{j+1}{m}_2}}<{\displaystyle \frac{a_n+k_r}{m_1^j{m}_2}},$$

(95)

using the fact that ${\displaystyle \frac{1}{m_2}}<0$ and $a_n<-k_r$. Adding $a_p$ to both sides yields $c_{j+1}<c_j$.

(iii)

From (ii), $c_{n+j+1}<c_{n+j}$, and applying the definition of ${\overline{c}}_p$ immediately gives ${\overline{c}}_j<{\overline{c}}_{j+1}$ for $j>0$.

(iv)

From (i), $c_{n+j+1}<a_{n+j}$; hence,

$${\displaystyle \frac{a_{n+j}+k_r}{m_2}}<{\displaystyle \frac{c_{n+j+1}+k_r}{m_2}}.$$

(96)

Recognizing the left-hand side as $b_j$ and the right-hand side as ${\overline{c}}_{j+1}$, we obtain $b_j<{\overline{c}}_{j+1}$.

(v)

Direct substitution into f yields

$$\begin{array}{}\mathrm{(97)}& \hfill f\left(c_{j+1}\right)& =m_1\left({\displaystyle \frac{a_n+k_r}{m_1^{j+1}{m}_2}}+a_{j+1}\right)+k_{\ell }\hfill \mathrm{(98)}& \hfill & ={\displaystyle \frac{a_n+k_r}{m_1^j{m}_2}}+a_j\hfill \mathrm{(99)}& \hfill & =c_j.\hfill \end{array}$$

(vi)

Again, by direct substitution,

$$f\left({\overline{c}}_j\right)=m_2\left({\displaystyle \frac{c_{n+j}+k_r}{m_2}}\right)-k_r=c_{n+j}.$$

(100)

(vii)

From $b<a_q$, we deduce

$$m_2{k}_{\ell }-k_r<a_q\Rightarrow {\displaystyle \frac{a_q+k_r}{m_2}}<k_{\ell }.$$

(101)

Thus, $b_0<k_{\ell }$. Suppose ${\overline{c}}_1<k_{\ell }$. Then, there exists $x\in (a_1,a_0)$ with

$${\overline{c}}_1=m_{1}x+k_{\ell }.$$

(102)

Substituting ${\overline{c}}_1={\displaystyle \frac{c_{q+1}+k_r}{m_2}}$ and solving for x leads to

$$x={\displaystyle \frac{b_0}{m_1^{q+2}{m}_2}}+{\displaystyle \frac{k_r}{m_1{m}_2}}-{\displaystyle \frac{k_{\ell }}{m_1}}.$$

(103)

Since ${\displaystyle \frac{b_0}{m_1^{q+2}{m}_2}}<0$ and ${\displaystyle \frac{k_r}{m_1{m}_2}}<0$, we obtain $x<a_1$, a contradiction. Therefore, $k_{\ell }<{\overline{c}}_1$.

□

Lemma 5 states that the set $\{\dots ,c_n,\dots ,c_2,c_1,0\}$ forms a partition of the interval $(-\infty ,0]$, while the set $\{0,{\overline{c}}_1,{\overline{c}}_2,\dots ,{\overline{c}}_n,\dots \}$ forms a partition of the interval $(0,\infty )$.

It can also be observed that, since the map f is increasing on $(-\infty ,0]$, if $C_k=(c_{k+1},c_k)$, then

$$f\left(C_k\right)=C_{k-1}.$$

(104)

Similarly, because f is decreasing on $(0,\infty )$, if ${\overline{C}}_k=({\overline{c}}_k,{\overline{c}}_{k+1})$, then

$$f\left({\overline{C}}_k\right)=(c_{n+k+1},c_{n+k}),$$

(105)

for all $k\in {\mathbb{Z}}^{+}$.

This local behavior allows us to extend the analysis to the image of the intervals involved in the transition between the left and right branches of f. The following proposition formalizes these relations:

Proposition 6. 

Let $a_q\in (b,-k_r)$ for some $q\in \mathbb{N}$, and let $c_p,{\overline{c}}_p$ satisfy the conditions stated in Definition (9), as defined in Equations (84) and (85). Then, for every $j\in \mathbb{N}$, the following relations hold:

(i) 

$f\left((c_{j+2},a_{j+1}]\right)=(c_{j+1},a_j]$;

(ii) 

$f\left((a_{j+2},c_{j+2}]\right)=(a_{j+1},c_{j+1}]$;

(iii) 

$f\left((0,b_0)\right)=(a_q,-k_r)$;

(iv) 

$f\left([b_j,{\overline{c}}_{j+1})\right)=(c_{q+j+1},a_{q+j}]$;

(v) 

$f\left([{\overline{c}}_{j+1},b_{j+1})\right)=(a_{q+j+1},c_{q+j+1}]$.

Proof. 

Items (i) and (ii) follow directly from Lemma 5(i), Lemma 5(ii), and Lemma 5(v), taking into account that the function f is increasing on the interval $(-\infty ,0)$.

Similarly, items (iv) and (v) are derived from Lemma 5(iii), Lemma 5(iv), and Lemma 5(vi), together with the fact that f is decreasing on the interval $(0,\infty )$.

Finally, for item (iii), since $f\left(b_0\right)=a_q$ and ${\displaystyle \underset{x\to 0^{+}}{lim}f\left(x\right)=-k_r}$, and given that f is decreasing on $(0,\infty )$, the result follows immediately. □

The fundamental domains for the coexistence of periodic orbits strongly depend on the values of $c_n$ and ${\overline{c}}_1$ for the positive side of the real line. For instance, Figure 19 illustrates the fundamental intervals associated with the coexistence of periodic orbits of periods 3 and 4, while Figure 20 shows the corresponding intervals for the coexistence of orbits with periods 5 and 6.

Theorem 6. 

If there exists n such that $a_n\in (b,-k_r)$, then two periodic orbits of periods $n+2$ and $n+1$ coexist, whose fundamental domains of attraction are respectively given by

$$\begin{array}{cc}\hfill & A_1=(c_{n+1},a_n],A_2=(c_n,a_{n-1}],A_3=(c_{n-1},a_{n-2}],\dots ,A_n=(c_2,a_1],\hfill \\ \hfill & A_{n+1}=(c_1,a_0],A_{n+2}=(b_0,{\overline{c}}_1],\hfill \\ \hfill & B_1=(a_n,c_n],B_2=(a_{n-1},c_{n-1}],B_3=(a_{n-2},c_{n-2}],\dots ,B_n=(a_1,c_1],\hfill \\ \hfill & B_{n+1}=(a_0,b_0].\hfill \end{array}$$

(106)

Proof. 

Since $b<a_n<-k_r$, it follows that $(b,a_n)\subset (c_{n+1},a_n]$ and $(a_n,-k_r)\subset (a_n,c_n]$. This means that, for the interval $(b,a_n)$, $n+2$ compositions of the function f are required to return to the same interval, while, for $(a_n,-k_r)$, only $n+1$ compositions of f are needed to return to it.

Now, $f\left(A_1\right)=A_2$, $f\left(A_2\right)=A_3$, …, $f\left(A_n\right)=A_{n+1}$, and $f\left(A_{n+1}\right)\subset A_{n+2}$. By iterating the function $n+2$ times over each interval $A_k$ with $1\le k\le n+2$, we obtain

$$f_{A_k}^{n+2}:A_k⟶A_k$$

(107)

defined as

$$f_{A_k}^{n+2}\left(x\right)=k_{\ell }\sum _{i=0}^{k-2}{m}_1^i-m_1^{k-1}{k}_r+k_{\ell }{m}_2\sum _{i=k-1}^n{m}_1^i+m_2{m}_1^{n+1}x.$$

(108)

Next, we show that $f_{A_k}^{n+2}\left(x\right)$ is a contraction for each $1\le k\le n+2$. Indeed,

$$\begin{array}{cc}\hfill \left|f_{A_k}^{n+2}\left(x\right)-f_{A_k}^{n+2}\left(y\right)\right|& =\left|m_2{m}_1^{n}x-m_2{m}_1^{n}y\right|\hfill \\ \hfill & =\left|m_2{m}_1^n\right||x-y|.\hfill \end{array}$$

(109)

Since $\delta =|m_2{m}_1^n|<1$, the mapping $f_{A_k}^{n+2}\left(x\right)$ is a contraction for every $1\le k\le n+2$.

Therefore, $f_{A_k}^{n+2}\left(x\right)$ is continuous and possesses a fixed point $x_k^{*}$ for each $1\le k\le n+2$, thus constructing a periodic orbit of period $n+2$, denoted by

$$\left\{x_1^{*},x_2^{*},\dots ,x_{n+2}^{*}\right\}.$$

(110)

Similarly, since $f\left(B_1\right)=B_2$, $f\left(B_2\right)=B_3$, …, and $f\left(B_n\right)\subset B_{n+1}$, this implies the existence of a periodic orbit of period $n+1$. □

Corollary 2. 

Let $A_1=[b_0,{\overline{c}}_1)$, $A_2=(c_1,a_0]$, $A_3=(c_2,a_1]$, …, $A_n=(c_{n-1},a_{n-2}]$, $A_{n+1}=(c_n,a_{n-1}]$, $A_{n+2}=(c_{n+1},a_n]$, and $B_1=(a_0,b_0)$, $B_2=(a_n,c_n]$, $B_3=(a_{n-1},c_{n-1}]$, $B_4=(a_{n-2},c_{n-2}]$, …, $B_{n+1}=(a_1,c_1]$. Assume that $a_n\in (b,-k_r)$. Then, two periodic orbits are generated:

$$\{x_1,x_2,\dots ,x_{n+1},x_{n+2}\},\mathit{with}{x}_k\in A_k,$$

(111)

and

$$\{y_1,y_2,\dots ,y_n,y_{n+1}\},\mathit{with}{y}_k\in B_k.$$

(112)

Moreover, the following apply:

1.

If $x\in A_k$ for $1\le k\le n+2$, then

$$\underset{m\to \infty }{lim}{f}^{(n+2)m}\left(x\right)=x_k.$$

(113)

2.

If $y\in B_k$ for $1\le k\le n+1$, then

$$\underset{m\to \infty }{lim}{f}^{(n+1)m}\left(y\right)=y_k.$$

(114)

Proof. 

1.

If $x\in A_k$, with $1\le k\le n+2$, define $g_1\left(x\right)=f^{n+2}\left(x\right)$. By Theorem 6, $x_k$ is a fixed point of $g_1$ within each $A_k$; hence,

$$\underset{m\to \infty }{lim}{g}_1^m\left(x\right)=x_k.$$

(115)

Consequently,

$$\underset{m\to \infty }{lim}{f}^{(n+2)m}\left(x\right)=x_k.$$

(116)

2.

If $x\in B_k$, with $1\le k\le n+1$, define $g_2\left(x\right)=f^{n+1}\left(x\right)$. By Theorem 6, $y_k$ is a fixed point of $g_2$ within each $B_k$; thus,

$$\underset{m\to \infty }{lim}{g}_2^m\left(x\right)=y_k.$$

(117)

Therefore,

$$\underset{m\to \infty }{lim}{f}^{(n+1)m}\left(x\right)=y_k.$$

(118)

□

The following results show that all points lying outside the fundamental intervals belong to one and only one basin of attraction. However, this membership depends on whether $(n+1)$ or $(n+2)$ evaluations of the function f are required for each initial condition $x_0$.

Theorem 7. 

Let $A_1=[b_0,{\overline{c}}_1)$, $A_2=(c_1,a_0]$, $A_3=(c_2,a_1]$, …, $A_n=(c_{n-1},a_{n-2}]$, $A_{n+1}=(c_n,a_{n-1}]$, and $A_{n+2}=(c_{n+1},a_n]$. Define $p_q=n+q+nm$, with $0\le q\le n+1$ and $m\in \mathbb{N}$. Then, if

$$x\in (c_{p_q+2},a_{p_q+1}],$$

(119)

it follows that x belongs to the same basin of attraction as the interval $A_{q+1}$.

Proof. 

Let $x\in (c_{p_q+2},a_{p_q+1}]$. By Proposition 6, there exists $m\in \mathbb{N}$ such that

$$f^{(n+2)m}\left(x\right)\in (c_{q+1},a_q],$$

(120)

for some $q\ge 1$, or equivalently

$$f^{(n+2)t}\left(x\right)\in (b_0,{\overline{c}}_1),$$

(121)

for some $t\in \mathbb{N}$.

Since both $(c_{q+1},a_q]$ and $(b_0,{\overline{c}}_1)$ are fundamental intervals for every $0\le q\le n+1$, it follows from Corollary 2 that x belongs to the same basin of attraction as $A_{q+1}$. □

Theorem 8. 

If $x\in [b_k,{\overline{c}}_{k+1})$, then x belongs to the same basin of attraction as the interval $(c_k,a_{k-1}]$.

Proof. 

Let $x\in [b_k,b_{k+1})$. By Lemma 4,

$$f\left(x\right)\in (a_{q+k+1},a_{q+k}],$$

(122)

and by Proposition 4,

$$f^{q+1}\left(x\right)\in (a_{k+1},a_k].$$

(123)

From Theorem 7, $f^{q+1}\left(x\right)$ belongs to the same domain of attraction as $(a_{k+1},a_k]$, and therefore the same holds for x. □

Theorem 9. 

Let $B_1=(a_0,b_0)$, $B_2=(a_n,c_n]$, $B_3=(a_{n-1},c_{n-1}]$, $B_4=(a_{n-2},c_{n-2}]$, …, $B_{n+1}=(a_1,c_1]$. Define $p_q=n+q+nm$, with $0\le q\le n$ and $m\in \mathbb{N}$. Then, if

$$x\in (a_{p_q+1},c_{p_q+1}],$$

(124)

it follows that x belongs to the same basin of attraction as the interval $B_{n+1-q}$.

Proof. 

Let $x\in (a_{p_q+1},c_{p_q+1}]$. By Proposition 6, there exists $m\in \mathbb{N}$ such that

$$f^{(n+1)m}\left(x\right)\in (a_q,c_q],$$

(125)

and, for $q\ge 1$, in particular, for sufficiently large q,

$$f^{(n+1)t}\left(x\right)\in (0,b_0).$$

(126)

Since both $(a_q,c_q]$ and $(0,b_0)$ are fundamental intervals for all $1\le q\le n$, it follows from Corollary 2 that x belongs to the same basin of attraction as $B_{n+1-q}$. □

Theorem 10. 

If $x\in [{\overline{c}}_k,b_k)$, then x belongs to the same basin of attraction as the interval $(a_k,c_k]$.

Proof. 

Let $x\in [b_k,b_{k+1})$. By Lemma 4,

$$f\left(x\right)\in (a_{q+k+1},a_{q+k}],$$

(127)

and by Proposition 4,

$$f^{q+1}\left(x\right)\in (a_{k+1},a_k].$$

(128)

From Theorem 4, $f^{q+1}\left(x\right)$ belongs to the same domain of attraction as $(a_{k+1},a_k]$, and therefore the same holds for x. □

The bifurcation diagram shown in Figure 1 reveals the geometric structure underlying the period-increment scenario in the biparametric setting. The figure displays a collection of straight bifurcation curves whose accumulation and mutual intersection at the origin organize the transitions between periodic regimes and the coexistence of attractors. The following theorem formalizes these observations by proving that the origin of the parameter plane is a Big-Bang bifurcation point, from which an infinite number of bifurcation curves emanate, each associated with a transition between periodic orbits of successive periods.

Theorem 11. 

Let f be as in (1). Then, the point in the parameter space

$$(k_{\ell },k_r)=(0,0)$$

(129)

is a Big-Bang bifurcation point.

Proof. 

Let $f\left(k_{\ell }\right)=a_n$ for some $n\in \mathbb{N}$, with $k_r$ fixed. If $f\left(k_{\ell }\right)$ is sufficiently close to and smaller than $a_n$, then

$$f\left(k_{\ell }\right)<a_n<-k_r,$$

(130)

which implies, by Theorem 6, that two periodic orbits of periods $n+1$ and $n+2$ coexist.

On the other hand, if $f\left(k_{\ell }\right)>a_n$ and sufficiently close, then

$$(b,-k_r)\subset (a_n,a_{n-1}),$$

(131)

which implies that all orbits converge to a periodic orbit of period $n+1$. Consequently, the value $f\left(k_{\ell }\right)=a_n$ corresponds to a bifurcation point. This occurs when

$$-k_r+m_2{k}_{\ell }=a_n,$$

(132)

or equivalently

$$k_{\ell }={\displaystyle \frac{a_n+k_r}{m_2}}.$$

(133)

This reasoning is valid for every $n\in \mathbb{N}$. Consequently, for each n, one obtains a bifurcation curve in the parameter space $(k_{\ell },k_r)$ given by

$$k_r=m_2{k}_{\ell }-{\displaystyle \frac{k_{\ell }(m_1^n-1)}{m_1^n-m_1^{n+1}}}.$$

(134)

If $k_{\ell }=0$ is substituted into (134), one obtains $k_r=0$, which occurs for all $n\in \mathbb{N}$. Let

$$F=\left\{g\left(k_{\ell }\right)=m_2{k}_{\ell }-{\displaystyle \frac{k_{\ell }(m_1^n-1)}{m_1^n-m_1^{n+1}}}:n\in \mathbb{N}\right\}$$

(135)

be the set of all bifurcation curves thus defined. This set is infinite and all curves satisfy $g\left(0\right)=0$. Therefore, the point $(0,0)$ belongs to infinitely many bifurcation curves, from which it follows that $(0,0)$ is a Big-Bang bifurcation point. □

Remark 1. 

Theorem 11 establishes that the point $(k_{\ell },k_r)=(0,0)$ is a Big-Bang bifurcation point in the precise sense that an infinite family of bifurcation curves, indexed by $n\in \mathbb{N}$, intersects at this point. Each of these curves corresponds to a transition between periodic dynamics of consecutive periods and is explicitly given by (134). Consequently, in any arbitrarily small neighborhood of $(0,0)$ in the parameter plane, infinitely many bifurcation curves are present.

This result provides the geometric framework within which the period-increment scenario must be interpreted. Since the Big-Bang point is the intersection of infinitely many bifurcation curves, the ordering of bifurcations observed under a continuous variation of the parameters $(k_{\ell },k_r)$ necessarily depends on the specific one-dimensional path along which such variation is performed. Different paths may intersect the family of curves (134) in different orders, giving rise to distinct sequences of transitions between periodic orbits.

Therefore, the relevant notion of “bifurcation ordering” in this context is not path-independent, but must be understood as the ordering induced by the sequence of crossings of bifurcation curves along a prescribed path in the parameter space. This interpretation is essential for a correct understanding of the period-increment phenomenon in biparametric settings and ensures a consistent and reproducible interpretation of the results.

7. Illustrative Example

In this section, we present an illustrative example that demonstrates how the analytical results developed in the previous sections can be applied to explicitly construct and characterize the basin of attraction of a periodic orbit. The purpose of this example is not to introduce new theoretical results, but rather to provide a concrete application of the proposed framework and to clarify the structure of the attraction domain in a specific region of the parameter space.

7.1. Explicit Construction of Basins of Attraction for Period-3 Attractor

Consider the piecewise-linear system defined in (1) with the parameter values

$$m_1=0.5,m_2=-0.5,k_{\ell }=1,k_r=2.5.$$

(136)

The fundamental interval is given by

$$(b,-k_r),$$

(137)

where

$$b=m_2{k}_{\ell }+k_r.$$

(138)

For the parameter values under consideration, this yields

$$b=(-0.5)\left(1\right)+(-2.5)=-3.$$

(139)

Consequently, the fundamental interval takes the explicit form

$$(b,-k_r)=(-3,-2.5).$$

(140)

Let $k_{\ell }>0$ and $0<m_1<1$. We define the sequence ${\left\{a_n\right\}}_{n\in {\mathbb{Z}}^{+}}$ by

$$a_n={\displaystyle \frac{k_{\ell }\left(1-m_1^n\right)}{m_1^n\left(m_1-1\right)}}.$$

(141)

In the particular case under study, substituting $m_1=0.5$ and $k_{\ell }=1$ yields the explicit expression

$$a_n=2-2^{n+1}.$$

(142)

The sequence satisfies $a_0=0$, $a_n<0$ for all $n\ge 1$ and is strictly decreasing. Consequently, the negative semi-axis can be partitioned into the generalized intervals

$$(a_n,a_{n-1}],n\ge 1.$$

(143)

In particular, the first intervals are given by

$$(a_3,a_2]=(-14,-6],(a_2,a_1]=(-6,-2],(a_1,a_0]=(-2,0].$$

(144)

From the previous computations, we observe that

$$(b,-k_r)=(-3,-2.5)\subset (a_2,a_1]=(-6,-2].$$

(145)

Therefore, according to Theorem 2, the map f admits an periodic orbit of period 3.

By explicitly solving the equation $f^3\left(x\right)=x$ under the admissible itinerary $\tau =\mathcal{R}{\mathcal{L}}^2$, the period-3 orbit is obtained as

$$\mathcal{O}=\left\{{\displaystyle \frac{7}{9}},-{\displaystyle \frac{26}{9}},-{\displaystyle \frac{4}{9}}\right\},$$

(146)

that is,

$${\displaystyle \frac{7}{9}}\stackrel{\mathcal{R}}{\to }-{\displaystyle \frac{26}{9}}\stackrel{\mathcal{L}}{\to }-{\displaystyle \frac{4}{9}}\stackrel{\mathcal{L}}{\to }{\displaystyle \frac{7}{9}}.$$

(147)

The stability of the orbit is determined from the multiplier

$$\Lambda =f^{\prime }\left(x_2\right)f^{\prime }\left(x_1\right)f^{\prime }\left(x_0\right)=(-0.5)\left(0.5\right)\left(0.5\right)=-{\displaystyle \frac{1}{8}}.$$

(148)

Since $|\Lambda |<1$, the periodic orbit $\mathcal{O}$ is attracting.

7.2. Fundamental Domains and Basin Decomposition for the Period-3 Cycle

The sequence ${\left\{a_n\right\}}_{n\in {\mathbb{Z}}^{+}}$ is given by

$$a_n=2-2^{n+1},$$

(149)

which is strictly decreasing, satisfies $a_0=0$, and verifies $a_n<0$ for all $n\ge 1$.

We define

$$a_q=max\{a_n:a_n<k_r\}.$$

(150)

Since $-k_r=-2.5$, we have

$$a_1=-2>-2.5,a_2=-6<-2.5.$$

(151)

Therefore,

$$q=2,a_q=a_2=-6.$$

(152)

For $m\in {\mathbb{Z}}^{+}$, we define the sequence

$$b_m={\displaystyle \frac{a_{q+m}-k_r}{m_2}}.$$

(153)

Substituting $q=2$, $m_2=-0.5$, and $-k_r=-2.5$ yields

$$b_m={\displaystyle \frac{(2-2^{m+3})+2.5}{-0.5}}=2^{m+4}-9.$$

(154)

In particular,

$$b_0=7.$$

(155)

Based on the previous constructions, the fundamental attraction domains associated with the points of the period-3 orbit are given by

$$A_3=(a_2,a_1]=(-6,-2],A_2=(a_1,0]=(-2,0],A_1=(0,b_0)=(0,7).$$

(156)

For each periodic point $x_k$ of the orbit $\mathcal{O}=\{x_0,x_1,x_2\}$, we define its basin of attraction as

$$B{A}_{x_k}=\left\{x\in \mathbb{R}:\underset{m\to \infty }{lim}{f}^{3m}\left(x\right)=x_k\right\}.$$

(157)

Case 1: $x\in A_1=(0,7)$. Since $x>0$, the first iteration is governed by the right branch

$$R\left(x\right)=m_{2}x+k_r=-\frac{1}{2}x-\frac{5}{2}.$$

(158)

Because $m_2<0$ and $k_r<0$, it follows that

$$R\left(x\right)\le k_r=-2.5<0,\forall x\ge 0.$$

(159)

Hence, the two subsequent iterations are necessarily performed by the left branch, and we obtain

$$f^3\left(x\right)=L\left(L\left(R\left(x\right)\right)\right)=-{\displaystyle \frac{1}{8}}x+{\displaystyle \frac{7}{8}},x\in A_1.$$

(160)

This map is a contraction with fixed point $7/9$, and therefore

$$\underset{m\to \infty }{lim}{f}^{3m}\left(x\right)={\displaystyle \frac{7}{9}},\forall x\in A_1,$$

(161)

which implies

$$A_1\subset B{A}_{7/9}.$$

(162)

Case 2: $x\in A_2=(-2,0]$. In this case,

$$f^3\left(x\right)=L\left(R\left(L\left(x\right)\right)\right)=-{\displaystyle \frac{1}{8}}x-{\displaystyle \frac{1}{2}},x\in A_2,$$

(163)

which is a contraction with fixed point $-4/9$. Consequently,

$$\underset{m\to \infty }{lim}{f}^{3m}\left(x\right)=-{\displaystyle \frac{4}{9}},\forall x\in A_2,$$

(164)

and hence

$$A_2\subset B{A}_{-4/9}.$$

(165)

Case 3: $x\in A_3=(-6,-2]$. For x in this interval,

$$f^3\left(x\right)=R\left(L\left(L\left(x\right)\right)\right)=-{\displaystyle \frac{1}{8}}x-{\displaystyle \frac{13}{4}},x\in A_3,$$

(166)

which is a contraction with fixed point $-26/9$. Therefore,

$$\underset{m\to \infty }{lim}{f}^{3m}\left(x\right)=-{\displaystyle \frac{26}{9}},\forall x\in A_3,$$

(167)

and we conclude that

$$A_3\subset B{A}_{-26/9}.$$

(168)

We now invoke Theorem 4. In the present example, the period is 3, so that $n+1=3$ and $n=2$. Hence,

$$p_q=3+q+2t,0\le q\le 2,t\in {\mathbb{Z}}^{+}.$$

(169)

Accordingly, the nonfundamental intervals are classified into three families.

(i)

Family $q=0$ (same basin as $A_1$).

$$p_0=3+2t,x\in (a_{p_0+1},a_{p_0}]=(a_{2t+4},a_{2t+3}]⟹x\in B{A}_{A_1}.$$

(170)

(ii)

Family $q=1$ (same basin as $A_2$).

$$p_1=4+2t,x\in (a_{p_1+1},a_{p_1}]=(a_{2t+5},a_{2t+4}]⟹x\in B{A}_{A_2}.$$

(171)

(iii)

Family $q=2$ (same basin as $A_3$).

$$p_2=5+2t,x\in (a_{p_2+1},a_{p_2}]=(a_{2t+6},a_{2t+5}]⟹x\in B{A}_{A_3}.$$

(172)

Since

$$A_1\subset B{A}_{7/9},A_2\subset B{A}_{-4/9},A_3\subset B{A}_{-26/9},$$

(173)

the complete basins can be written as

$$\begin{array}{cc}\hfill B{A}_{7/9}& \supseteq A_1\cup \bigcup _{t\in {\mathbb{Z}}^{+}}(a_{2t+4},a_{2t+3}],\hfill \\ \hfill B{A}_{-4/9}& \supseteq A_2\cup \bigcup _{t\in {\mathbb{Z}}^{+}}(a_{2t+5},a_{2t+4}],\hfill \\ \hfill B{A}_{-26/9}& \supseteq A_3\cup \bigcup _{t\in {\mathbb{Z}}^{+}}(a_{2t+6},a_{2t+5}].\hfill \end{array}$$

(174)

In this example, $a_n=2-2^{n+1}$, and therefore

$$a_3=-14,a_4=-30,a_5=-62,a_6=-126,a_7=-254,a_8=-510,\dots$$

(175)

The first nonfundamental extensions are thus given by

$$\begin{array}{cc}\hfill (a_6,a_5],(a_8,a_7],\dots & \subset B{A}_{7/9},\hfill \\ \hfill (a_7,a_6],(a_9,a_8],\dots & \subset B{A}_{-4/9},\hfill \\ \hfill (a_8,a_7],(a_{10},a_9],\dots & \subset B{A}_{-26/9}.\hfill \end{array}$$

(176)

Recalling (153), the sequence $\left\{b_k\right\}$ in this example is given by

$$b_k=2^{k+4}-9,k\in {\mathbb{Z}}^{+},$$

(177)

so that

$$b_0=7,b_1=23,b_2=55,b_3=119,b_4=247,b_5=503,\dots$$

(178)

By Theorem 5, and meaning that the fundamental intervals satisfy

$$(a_1,a_0]=A_2\subset B{A}_{-4/9},(a_2,a_1]=A_3\subset B{A}_{-26/9},(a_3,a_2]\subset B{A}_{7/9},$$

(179)

one obtains the following initial classification for the intervals induced by the sequence $\left\{b_k\right\}$:

$$[b_0,b_1)=[7,23)\subset B{A}_{-4/9},[b_1,b_2)=[23,55)\subset B{A}_{-26/9},[b_2,b_3)=[55,119)\subset B{A}_{7/9}.$$

(180)

From this point on, the assignment repeats cyclically with period 3. Consequently, the partition of $[0,\infty )$ induced by the sequence $\left\{b_k\right\}$ allows us to explicitly describe the decomposition of the attraction basins along the right branch as

$$\begin{array}{cc}\hfill B{A}_{-4/9}\cap [0,\infty )& =\bigcup _{t\in {\mathbb{Z}}^{+}}[b_{3t},b_{3t+1}),\hfill \\ \hfill B{A}_{-26/9}\cap [0,\infty )& =\bigcup _{t\in {\mathbb{Z}}^{+}}[b_{3t+1},b_{3t+2}),\hfill \\ \hfill B{A}_{7/9}\cap [0,\infty )& =(0,b_0)\cup \bigcup _{t\in {\mathbb{Z}}^{+}}[b_{3t+2},b_{3t+3}).\hfill \end{array}$$

(181)

Moreover, the classification of the first nonfundamental intervals, corresponding to $k=0,1,2$, is given in (180). The same periodic pattern continues, for instance,

$$[b_3,b_4)=[119,247)\subset B{A}_{-4/9},[b_4,b_5)=[247,503)\subset B{A}_{-26/9},\dots$$

(182)

8. Conclusions

In this study, the basins of attraction of a one-dimensional piecewise-smooth linear system defined by (1) have been determined analytically and globally under the parameter conditions $0<m_1<1$, $-1<m_2<0$, and positive variations of $k_{\ell }$ and $k_r$. The analysis is based on a decomposition of the real line into fundamental domains, which establishes a complete correspondence between the intervals $(a_{n+1},a_n]$ and the periodic orbits generated through period-increment Big-Bang bifurcations. This construction allows for the explicit determination of all attraction basins and their coexistence regions, showing that, in the biparametric plane $(k_{\ell },k_r)$, at most two periodic orbits may coexist.

A central role in this characterization is played by the results summarized in Equations (58) and (108), which generalize the recursive formulation of the iterates $f^n$ on each interval $A_k$. These expressions yield a unified analytical criterion for convergence toward periodic points and enable a complete description of the asymptotic evolution of any initial condition $x_0\in \mathbb{R}$ in terms of its membership in a basin of attraction associated with a periodic orbit of period $n+1$ or $n+2$.

The present analysis clarifies the geometric mechanisms governing the organization and coexistence of periodic attractors in discontinuous and piecewise-smooth systems. In particular, it reveals how the parameters $k_{\ell }$ and $k_r$ shape the structure of the attraction domains and control the coexistence of stable branches organized around the Big-Bang bifurcation point. In this sense, the analytical expressions derived here extend and complement previous numerical observations reported in [15,34,50], providing a global and exact description of basin structures rather than local or approximate representations.

The analytical construction of the basins of attraction presented in this work is exact within the assumed piecewise-linear framework and the parameter restrictions considered throughout the manuscript. The results rely on a precise interval decomposition induced by the switching structure of the map and on the explicit analytical characterization of the corresponding periodic points. Consequently, the obtained basin organization is rigorously valid within this idealized setting.

Nevertheless, it should be noted that small perturbations in the system parameters—such as variations in slopes, offsets, or switching conditions—may affect the interval decomposition underlying the recursive construction of the basins. Such perturbations can modify the location of periodic points or alter the fine structure of basin boundaries. A systematic robustness analysis with respect to modeling inaccuracies or numerical perturbations lies beyond the scope of the present work and constitutes a natural direction for future research.

The analytical framework developed here relies essentially on the piecewise-linear structure of the system. In this respect, nonlinear perturbations introduced in any of the system branches may significantly alter the global dynamics and lead to substantial changes in the basin organization. Under such nonlinear perturbations, the basin structure characterized in this manuscript may persist only locally, while the global organization described here may no longer be preserved. By contrast, linear perturbations affecting the branches can be absorbed through suitable affine changes of variables, allowing the analytical results and convergence properties of the periodic orbits to remain valid.

For analytical convenience, the switching point was fixed at $x=0$. This choice does not restrict the generality of the results, since any alternative switching location can be translated to the origin through an appropriate change of variables. Consequently, the basin structure and convergence results obtained in this work are invariant under translations of the switching point.

The robustness of the results with respect to variations in the slopes is ensured as long as the fundamental assumptions of the model are satisfied. Specifically, the left-branch slope must remain within $0<m_1<1$ and the right-branch slope within $-1<m_2<0$. Parameter variations within these ranges do not qualitatively alter the dynamics or the existence of periodic attractors. In addition, the analytical characterization of the basins is preserved provided that the parameters $k_{\ell }$ and $k_r$ have opposite signs, guaranteeing an effective switching mechanism between the two branches. When the switching point is displaced, this condition can be equivalently formulated by requiring that one affine branch lies above the identity line $y=x$ while the other lies below it. This geometric configuration is essential for maintaining the period-increment scenario and the basin structure described in this work.

Under the assumptions established throughout the manuscript, convergence toward all periodic orbits has been analytically demonstrated. A distinctive feature of the period-increment phenomenon analyzed here is that all periodic orbits generated within this framework are attractors. As a result, although the fine structure of basin boundaries may be sensitive to perturbations, the attractor nature of the periodic orbits and the global convergence of trajectories are preserved within the class of systems considered.

As for perspectives for future research, the analytical framework developed in this work opens several directions of interest for the academic community. In particular, the methods introduced here suggest the possibility of extending the analytical characterization of basins of attraction to the period-adding scenario, where symbolic and geometric organization exhibits additional complexity. Moreover, the extension to two-dimensional piecewise-smooth systems constitutes a natural next step, since many application-driven models—especially in power electronics and control systems—are governed by planar maps. In this sense, the present one-dimensional analysis can be regarded as a foundational case that provides the conceptual tools required to address more general analytical constructions. Furthermore, the principles established here may serve as a basis for the study of basins of attraction in higher-dimensional systems, encouraging the development of a broader analytical theory that complements and goes beyond the predominantly numerical approaches available in the literature.