Extended Gauss Iterative Map: Bistability and Chimera States

Abstract

We investigate an extended Gauss iterative map by incorporating the q-exponential function, a key component of the Tsallis framework. This extension enables us to investigate the non-linear dynamics of the Gauss iterative map across a broader range of scenarios, encompassing periodic, chaotic, and bistable behaviors. Regular and chaotic phenomena have been observed in coupled systems. In this context, we propose a network of coupled extended Gauss iterative maps. In our network, we found the emergence of chimera states, characterized by the coexistence of coherent and incoherent behaviors. These states are identified within specific parameter regimes using Gopal’s metric. In this work, we show the interplay between chaos and emergent collective dynamics in coupled extended Gauss iterative maps.

1. Introduction

Discrete maps are among the simplest and most fundamental systems that exhibit chaotic behavior, in which even slight variations in initial conditions can lead to significantly different outcomes during simulations. This sensitivity to initial conditions, a hallmark of chaos, introduces high unpredictability, making chaos a key focus in numerous fields, including physics, biology, and mathematics.

In the broader context of dynamical systems, a wide array of phenomena can be observed across various fields of physics. Classical examples include pendulums [1], harmonic oscillators [2], attractors [3], and fluid dynamics, each exhibiting unique and intricate behaviors that highlight the depth of dynamical systems theory. Moreover, many systems exhibit co-dependencies through networks that emerge from long-range [4] and short-range [5] interactions. The dynamics of such interconnected systems are of significant interest, particularly in the study of chaos and its underlying mechanisms.

Due to their simplicity and versatility, chaotic maps are often employed as model systems to investigate the properties of emergent networks [6,7]. They provide a tractable framework for exploring complex behaviors, such as clustering and synchronization [8], pattern formation [9], and phase transitions [10], among others. Such properties are central to understanding the collective dynamics of interconnected systems.

Networks of coupled non-linear systems can exhibit a spatiotemporal coexistence of chaotic and regular structures [11], known as chimera states. Chimeras challenge the conventional expectation of uniformity in coupled systems and have been observed in a diverse range of networked systems. Examples include coupled chaotic oscillators, neuronal networks [12,13], and chaotic map networks [14]. Their discovery has significantly advanced our understanding of the interplay between coherence and incoherence in complex dynamical systems. From a biological perspective, chimera states have been associated with functional asymmetries in neural activity and other crucial physiological processes. The relevance of chimera states underscores the importance of understanding hybrid dynamical regimes in real-world systems, where localized synchronization coexists with desynchronized behavior.

The description of complex systems presents a challenge, motivating the development of many formal aspects. One is the Tsallis framework, which generalizes the Boltzmann–Gibbs entropy by introducing a parameter q that quantifies the degree of nonextensivity. This formulation is especially suitable for systems with long-range interactions [15,16,17], multifractal structures [18] or memory effects, features commonly encountered in chaotic [19,20,21,22,23] and networked dynamical systems [24]. Applications of Tsallis statistics have shed light on anomalous diffusion, fractal basins, and transitions to chaos in low-dimensional maps. It provides a framework for modeling the emergence of order and complexity in systems that deviate from the classical assumptions of extensivity and ergodicity.

In this work, we demonstrate that the parameter q plays a crucial role in the dynamics of the extended Gauss iterative map and in its dynamics when coupled through nonlocal interactions. The map exhibits rich dynamical behavior, including periodic and chaotic attractors and period-doubling routes to chaos. Depending on the parameters and initial conditions, we identify that the map solution can switch between periodic and chaotic attractors, i.e., bistability. Regarding the network, we observe the emergence of chimera states as we vary the coupling strength and the number of coupled neighbors.

The paper is structured as follows. In Section 2, we provide an overview of the key characteristics of the Tsallis framework, which serves as the foundation for our study. We extend the classical Gauss map (often referred to as the mouse map) by incorporating the q-exponential derived from the Tsallis formalism. In this section, we also calculate the Lyapunov exponent and present our results in relation to the map. In Section 3, we present the coupling scheme and its results, and in Section 4, we discuss the implications of our results and present our concluding remarks.

2. Extended Gauss Iterative Map

In Section 2.1, we present key aspects of the Tsallis formalism and extend the standard mouse map by incorporating the q-exponential. The results of this analysis are discussed in Section 2.2.

2.1. Map Dynamics

One of the primary challenges in statistical mechanics is describing macroscopic behavior in terms of microscopic dynamics. This point has always promoted intense discussion in many aspects, connecting statistical mechanics with thermodynamics, which resulted in the well-known Boltzmann–Gibbs statistical mechanics based on the entropy:

$$S_{BG}=-k_B\sum _i{p}_{i}ln{p}_i,$$

(1)

where $p_i$ represents the probability of a microstate and $k_B$ is the Boltzmann constant. It has been successfully applied to various phenomena, including both classical and quantum contexts. On the other hand, in the case of long-range interactions, such as self-gravitating systems and anomalous correlated diffusion, which constitute a large class of systems, difficulties have been found with the standard approach. To overcome the difficulties inherent to the nature of the system, Tsallis’ formalisms introduced the Tsallis entropy [25,26]:

$$S_q=k{\displaystyle \frac{1-{\sum }_i{p}_i^q}{q-1}},$$

(2)

where q is the entropic index that characterizes the degree of nonextensivity. For instance, for two independent systems $\mathcal{A}$ and $\mathcal{B}$, the entropy satisfies the pseudo-additivity rule [27]: $S_q^{\mathcal{A}\cup \mathcal{B}}=S_q^{\mathcal{A}}+S_q^{\mathcal{B}}+\left[(1-q)/k\right]S_q^{\mathcal{A}}{S}_q^{\mathcal{B}}$ which accounts for correlations between subsystems. In the limit $q\to 1$, the Tsallis entropy is reduced to the Boltzmann–Gibbs entropy, recovering classical statistical mechanics. By applying the maximum principle in Equation (2) subjected to the constraints

$$\begin{array}{c}\hfill U_q={\displaystyle \frac{{\sum }_i{E}_i{p}_i^q}{{\sum }_i{p}_i^q}}\mathrm{and}\sum _i{p}_i=1,\end{array}$$

(3)

yields

$$\begin{array}{c}\hfill p_i={\displaystyle \frac{1}{Z_q}}{exp}_q\left[-\tilde{\beta }\left(E_i-U_q\right)\right],\end{array}$$

(4)

with $\tilde{\beta }=\beta /{\sum }_i{p}_i^q$ and the partition function

$$\begin{array}{c}\hfill Z_q=\sum _i{exp}_q\left[-\tilde{\beta }\left(E_i-U_q\right)\right].\end{array}$$

(5)

In the previous equation, we have the function ${exp}_q\left(x\right)$, which is defined as follows:

$$\begin{array}{c}\hfill {exp}_q\left(x\right)=\left\{\begin{array}{cc}{(1+(1-q)x)}^{\frac{1}{1-q}}& x\ge 1/(q-1)\\ 0& x<1/(q-1).\end{array}\right.\end{array}$$

(6)

Note that we have a cut-off point to retain the probabilistic interpretation of the distributions that emerge from the maximum principle of entropy. The Tsallis framework has been successfully applied in diverse fields, demonstrating its versatility in modeling complex and correlated systems. For instance, it has been successfully applied in econophysics [28], plasma physics [29], biological systems [30], astrophysics [31], system with long-range interactions [32]. For discrete maps, such as the logistic and Henon maps and their unimodal generalizations, transitions from regular to chaotic regimes occur, as well as critical regions where Tsallis entropy may offer a more suitable characterization of the dynamics than Shannon entropy. At the transition point to chaos, known as the Feigenbaum edges, the Tsallis entropy with an index $q\ne 1$ can effectively capture the anomalous sensitivity to initial conditions and the non-Gaussian distribution of the states visited, revealing connections between discrete chaotic dynamics and the theoretical framework of non-extensive statistical mechanics. Additionally, the Gauss iterative map,

$$\begin{array}{c}\hfill x_{n+1}=exp(-\alpha x_n^2)+\beta ,\end{array}$$

(7)

can be used as an alternative to the logistic map to study transitions to chaos (period doubling, Feigenbaum route), investigate sensitivity to initial conditions, and fractal structures in time series. It can also generate non-Gaussian probability distributions connected to the Tsallis approach, providing a suitable context for analysis.

Motivated by the rich class of behaviors manifested by the q-Gaussian, i.e., short and long tailed behaviors (see Figure 1), which can be related to different distributions such as the Lévy distributions in the asymptotic limit [33], we extend the exponential of Equation (7) to the q-exponential, resulting in a q-Gaussian in Equation (8), i.e.,

$$\begin{array}{c}\hfill x_{n+1}={exp}_q\left(-\alpha x_n^2\right)+\beta =f\left(x_n\right),\end{array}$$

(8)

which opens up new possibilities for the application of the Gauss iterative map by incorporating the class of behaviors present in the q-Gaussian. In fact, the extended Gauss iterative map can provide a unifying discrete framework for studying how nonextensive effects, encoded by the parameter q, control the transition between periodic, chaotic, and chimera-like regimes. Since the parameter q can modulate the degree of correlation and non-Gaussianity, this approach can also enable parallels with systems exhibiting long-range interactions, memory effects, or collective synchronization. Specifically, in the study of chaos, the Lyapunov exponent is a fundamental concept for analyzing its behavior, measuring how much an infinitesimal increment in position affects future measurements. For a time discrete equation, it is defined as follows:

$$\begin{array}{c}\hfill \lambda =\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{n=N-1}ln\left(f^{\prime }\left(x_n\right)\right).\end{array}$$

(9)

For Equation (8), in our simulations, $\lambda$ becomes:

$$\begin{array}{c}\hfill \lambda =\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{n=N-1}ln\left\{\left|2\alpha x_n{\left[{exp}_q\left(-\alpha x_n^2\right)\right]}^{-q}\right|\right\}.\end{array}$$

(10)

By using logarithm properties, Equation (10) can be written as

$$\begin{array}{c}\hfill \lambda =ln\left\{2\alpha \right\}+\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{n=N-1}ln\left\{\left|x_n\right|\right\}+{\displaystyle \frac{q}{q-1}}ln\left\{\left(1+(q-1)\alpha x_n^2\right)\right\}.\end{array}$$

(11)

We may develop Equation (11) further considering the case where $q\to 1$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F\left(q\right)& ={\displaystyle \frac{q}{q-1}}ln\left\{\left(1+(q-1)\alpha x_n^2\right)\right\},\mathrm{let}\gamma =\alpha x_n^2,\mathrm{then}\hfill \\ \hfill F\left(1\right)& =\underset{q\to 1}{lim}{\displaystyle \frac{q}{q-1}}ln\left\{\left(1+(q-1)\gamma \right)\right\}.\hfill \end{array}\end{array}$$

(12)

Since both the numerator and denominator go to zero as q goes to one, we may use L’Hôpital’s rule:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F\left(1\right)& =\underset{q\to 1}{lim}{\displaystyle \frac{\frac{d}{dq}\left[qln\left\{\left(1+(q-1)\gamma \right)\right\}\right]}{\frac{d}{dq}\left[q-1\right]}},\hfill \\ & =\underset{q\to 1}{lim}ln\left\{\left(1+(q-1)\gamma \right)\right\}+q{\displaystyle \frac{\gamma }{\gamma (q-1)+1}},\hfill \\ & =ln\left\{1\right\}+\gamma =\alpha x_n^2.\hfill \end{array}\end{array}$$

(13)

Therefore, $\lambda (q=1)$ can be written as:

$$\begin{array}{c}\hfill \lambda (q=1)=ln\left\{2\alpha \right\}+\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{n=N-1}ln\left\{\left|x_n\right|\right\}+\alpha x_n^2.\end{array}$$

(14)

It is worth mentioning that transient steps must not be included in the averaging process for the Lyapunov exponent. In this work, the first $2\times {10}^3$ steps were discarded, and ${10}^4$ were used to compute $\lambda$.

Regarding Equation (8), we can also compute k-periodic orbits by considering the equations:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f\left(x\right)& ={exp}_q\left(-\alpha x_n^2\right);\hfill \\ \hfill f^2\left(x\right)& =f\left(f\right(x\left)\right),\hfill \\ \hfill f^k\left(x\right)& =f\left(f\right(...\left)\right).\hfill \end{array}\end{array}$$

(15)

And for $f^k\left(x\right)=x$, the solution is a fixed point for $k=1$ or a k-periodic orbit for $k\ne 1$. For some q, we can find fixed points analytically, for example, let us consider $q=2$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f\left(x\right)& =x\hfill \\ \hfill x& ={(1+(q-1)\alpha x^2)}^{{\displaystyle \frac{1}{q-1}}}+\beta \hfill \\ & ={(1+\alpha x^2)}^{-1}+\beta \hfill \\ \hfill x(1+\alpha x^2)& =1+\beta (1+\alpha x^2)\hfill \\ \hfill 0& =\alpha x^3-\beta \alpha x^2+x-(\beta +1)\hfill \end{array}\end{array}$$

(16)

Fixed points can be tested for stability by considering it’s derivative, let $\widehat{x}$ be a fixed point, if $|f^{\prime }\left(\widehat{x}\right)|<1$ then $\widehat{x}$ is a sink (stable), if $|f^{\prime }\left(\widehat{x}\right)|>1$ then it is a source (unstable) and if $|f^{\prime }\left(\widehat{x}\right)|=1$ then the result is inconclusive and must be tested.

2.2. Extended Gauss Iterative Map Dynamics

A complex dynamical behavior arises in both standard and extended Gauss iterative maps, characterized by the presence of chaotic and stable regions. As illustrated in Figure 2, the interplay between stable and chaotic regimes exhibits significant diversity as a function of q. In Figure 3 we compute the Lyapunov exponent $\lambda$ by varying q and $\beta$ for $\alpha$ equal to 5, 10, 15, and 20, as shown in panels (a), (b), (c), and (d), respectively. We observe that an increase in q requires higher values of $\alpha$ for the emergence of a chaotic regime. Faster tail decay is an important factor in the existence of chaos.

In Figure 4, we compute the bifurcation diagram by varying $\beta$ for $\alpha =12$, (a) $q=1.01$, (b) $q=1.25$, (c) $q=1.50$, and (d) $q=1.75$. The diagram shows a period-doubling bifurcation cascade, periodic orbits, and chaotic attractors. It is possible to observe the presence of bistability, which is the existence of two distinct attractors. Due to this fact, there are two distinct Lyapunov exponents for small values of $\beta$. We find a threshold or critical point (black dashed line), known as the tipping point, in which the system switches between a fixed point and a chaotic attractor with small changes in $\beta$.

3. Network of Coupled Maps and Chimera States

We study a periodic network of coupled extended Gauss iterative maps, as displayed in Figure 5. The coupled map network equation is given by:

$$x_{i,n+1}=(1-ϵ)f_{i,n}+{\displaystyle \frac{ϵ}{2R}}\sum _{j=-R}^R\left(f_{i+j,n}\right),$$

(17)

where

$$f_{i,n}={[1+(q-1)\alpha x_{i,n}^2]}^{{\displaystyle \frac{1}{1-q}}}+\beta ,$$

(18)

$x_{i,n}$ is the variable at time n, i is the map index ($j\ne 0$) and $ϵ$ is the coupling strength between the map and its R neighbors.

We investigate the coexistence of incoherent and coherent states using the incoherence metric introduced by Gopal [34,35]. It involves the partition of the network with $G_s$ nodes into M slices containing $w=G_s/M$ elements. By partitioning the system into slices, we see the interplay of its elements, a necessary step in evaluating the existence of chimera states. To do that, we compute the local standard deviation ${\chi }_{m,n}$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\chi }_{m,n}& =\sqrt{{\displaystyle \frac{1}{w}}\sum _{j=mw}^{(m+1)w-1}{\left[z_{j,n}-\langle z_m\rangle \right]}^2},\hfill \\ \hfill {\chi }_m& ={\langle {\chi }_{m,n}\rangle }_t.\hfill \end{array}\end{array}$$

(19)

where m is the slice index and $z_{j,n}=x_{i,n}-x_{i+1,n}$ and $\langle z_m\rangle$ is the space average of $z_{j,n}\in m$. We continue with the local coherence state, which is given by:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill S_m=\mathcal{H}\{\delta -{\chi }_m\},\end{array}\end{array}$$

(20)

where $\delta$ is a cutoff and $\mathcal{H}$ is the Heaviside step function:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathcal{H}\{\delta -{\chi }_m\}=\left\{\begin{array}{c}1,\delta -{\chi }_m>0\hfill \\ 0,\delta -{\chi }_m\le 0\hfill \end{array}\right.\end{array}.\end{array}$$

(21)

Then, the incoherence metric is calculated as follows:

$$\begin{array}{c}\hfill S_I=1-{\displaystyle \frac{1}{M}}\sum _{m=0}^{m<M}{S}_m.\end{array}$$

(22)

The $S_I$ metric provides a quantitative measure of the coherence or incoherence of coupled oscillator systems. When $S_I=0$, the system achieves perfect synchronization, indicating that all oscillators share the same phase or state, resulting in fully coherent and uniform dynamics. On the other hand, when $S_I=1$, the system is in a state of complete incoherence, where each oscillator operates independently. For values of $S_I$ between 0 and 1, coherence coexists with incoherence. The oscillators with identical coupling conditions spontaneously divide into coherent and incoherent subpopulations.

Chimera states are inherently unstable phenomena that often emerge as transient dynamics over time. We investigate the diversity of chimera states considering the $S_I$ metric. Figure 6 displays the time series of two simulations for identical parameters and different initial conditions, resulting in $S_I=0.30$ and $S_I=0.70$. These simulations reveal different dynamical behaviors. For $S_I=0.30$, the network exhibits a prolonged transient, taking approximately 225 steps to reach equilibrium, while for $S_I=0.70$, it achieves equilibrium significantly faster, requiring only 60 steps. A lower value $S_I$ (closer to synchronization) corresponds to a slower convergence to equilibrium, reflecting the persistence of coherent structures. In contrast, a higher $S_I$ value (closer to incoherence) leads to a faster equilibration, indicative of weaker coherence and stronger disorder. Other simulations with different parameters can be seen in Appendix A.

Chimera states are observed most readily in simulations where $S_I$ is approximately 0.5, indicating a balanced coexistence of coherence and incoherence. However, our findings reveal that no universal combination of parameters R (coupling distance) and $ϵ$ (coupling strength) consistently produces such states. This is due to a variation in the distribution of $S_I$, which is highly sensitive to other parameters such as q, $\alpha$, and $\beta$.

We observe two distinct patterns in the transition between chaotic and stable regions for the parameter space of ($ϵ,R$). The first pattern is characterized by a single interface between coherent and incoherent domains, which is evident for $q=1.5$ (Figure 7). The second pattern, featuring a double interface structure, emerges for $q=1.01$ (Figure 7). Another important observation is the role of the coupling distance R. Although R has a lower influence on the mean $S_I$ when it is sufficiently large, it can significantly affect the quadratic deviation of $S_I$. Specifically, larger values of R tend to produce a broader range of $S_I$ values, indicating greater variability in the balance between coherence and incoherence in our simulations.

Our analysis of chimera states in the parameter space $\alpha$, $\beta$, and q reveals that the most complex patterns are generated in $\beta \times \alpha$ (Figure 8) and $\beta \times q$ (Figure 9). In contrast, the simplest patterns are found in $\alpha \times q$, as shown in Figure 10. These results suggest that the mechanisms governing the coexistence of chaotic and stable domains are highly sensitive, with minor parameter variations capable of triggering transitions between fully stable and fully chaotic regimes. It is worth mentioning that stable, fully synchronized domains are represented in blue, chaotic, fully desynchronized domains are shown in red, and intermediate values, indicating the coexistence of coherent and incoherent elements (chimera states), are depicted in yellow.

The parameter space of ($\alpha ,\beta$) shows that the distribution of chimera states remains largely consistent with the variability of q. The primary effect is an upward shift as q increases. This consistent behavior reinforces the hypothesis that the system’s properties are governed by the tail-decay rate of the q-exponential. The parameter $\alpha$ can modulate this effect, effectively counteracting the influence of the heavy tail to preserve the stability and structure of the chimera states, as seen in Figure 8:

Figure 9 displays the parameter space ($q,\beta$) for $S_I$ and ${\sigma }^2\left(S_I\right)$ in color bars. We verify that the system dynamics are governed by the tail decay of the q-exponential. Furthermore, the distribution of chimera states exhibits more pronounced and spiky features in Figure 9 compared to the broader structures observed in Figure 8.

In Figure 10, we calculated the parameter space ($q,\alpha$) for different values of $\beta$. The panels demonstrate similar behavior; however, a detailed examination reveals that the system’s properties depend sensitively and often unpredictably on small changes in $\beta$, we also may see an animation in the Supplementary Materials or at https://youtu.be/DgIvOlTjEzA (accessed 22 October 2025) where we cover $\langle S_I\rangle$ in smaller increments of $\beta$. We also note a region of high deviation in panel (b).

4. Conclusions

We have explored the Tsallis entropic framework and utilized it to extend the Gauss iterative map by incorporating the q-exponential function. This extension has enabled us to uncover a richer spectrum of dynamical behaviors within the chaotic map, including the computation of Lyapunov exponents and the analysis of both pristine and coupled states.

For the extended Gauss iterative map, our results revealed the presence of bistability. In particular, the critical value of $\beta$ at which the two stable regions merge into a single stable state, known as the tipping point, varies significantly depending on the parameters of the system. This variability underscores the system’s sensitivity to changes in $\beta$ and highlights the complex interplay between the parameters and their dynamic behavior. Furthermore, we computed the Lyapunov exponents, demonstrating that the Tsallis parameter q plays a crucial role in shaping the map’s dynamics. Specifically, variations in q can lead to different bifurcation patterns and stability regimes. In the context of coupled states, we performed simulations with identical parameter values but different initial conditions, resulting in distinct values of the incoherence metric $S_I$. Our simulations demonstrated that changes in initial conditions can yield a diverse range of dynamical outcomes, underscoring the system’s sensitivity to its initial state. Furthermore, we have shown that chimera states—regions of coexisting coherence and incoherence—are achievable across all the parameters we tested. This finding suggests that the extended Gauss map, enriched by the q-exponential, provides a fertile ground for exploring the emergence and stability of chimera states in non-linear systems.

Our findings contribute to a deeper understanding of the Tsallis entropic framework and its application to chaotic maps. By extending the Gauss iterative map using the q-exponential, we unveiled a diverse range of dynamical behaviors that are highly sensitive to both system parameters and initial conditions. The presence of bistability, tipping points, and the emergence of chimera states underscore the richness of the extended map and its potential as a model for studying complex systems. For example, in neuroscience, chimera states have been associated with functional brain asymmetries and sleep–wake transitions, where localized synchronization coexists with incoherent neural activity [12,13]. The parameter q introduced by the presence of the q-Gaussian can serve as a measure of long-range neuronal correlations, memory effects, or deviations from Gaussian activity patterns. Bistability and partial synchronization can occur in coupled lasers and in plasma turbulence [10,29], where fluctuations are often associated with non-Gaussian distributions and, consequently, with Tsallis statistics. Another scenario can be found in social systems, where clusters of coherent opinions can coexist with incoherent fluctuations, resembling chimera-like dynamics. The parameter q can be used to capture the patterns [27,28].

We hope that these results will be useful for future research into the Tsallis framework and its applications to non-linear dynamics.