Oscillator with Line of Equilibiria and Nonlinear Function Terms: Stability Analysis, Chaos, and Application for Secure Communications

Abstract

We explore an oscillator with nonlinear functions and equilibrium lines that displays chaos. The equilibrium stability and complexity of the oscillator have been analysed and investigated. The presence of multiple equilibrium lines sets it apart from previously reported oscillators. The synchronization of the oscillator is considered as an application for secure communications. An observer is designed by considering a transmitted signal as a state, in other words, by injecting a linear function satisfying Lipschitz’s condition to the proposed oscillator. Moreover, the adaptive control of the new oscillator is obtained.

1. Introduction

Chaos was formally defined by Edward Lorenz as a nonlinear dynamical system sensitive to initial conditions. H. Poincaré realized that the motion of a three-body has a very special behaviour such as that similar to how drastically different orbits would result from even a small initial condition change. The unusual and strange geometry of the chaotic attractor received the first look, while the usual attractors were periodic attractors, quasi-periodic, and stable equilibrium attractors. In the 1970s, the Belousov–Zhabotinsky reaction was presented by Anatoliy Zhabotinsky, where the basin of attraction included an unstable equilibrium point. The attractor was called “self-excited”. Simply, by the usual computation, the self-excited attractors can be easily found. Another class of attractor was observed by Gennady Leonov and Nikolay Kuznetsov, which they called the “hidden attractor” in multistable systems [1,2,3].

Due to the complicated and unexpected behaviour of chaotic systems, their applications can be found in different scientific and engineering fields, including communications, image encryption [4], robotics [5], anti-jamming systems [6], chemical reactors [7], modelling and prediction, time-delayed systems [8], and sensors [9].

In addition, the investigations of chaotic systems with special characteristics have attracted a lot of attention from scientists, engineers, and mathematicians over the last two decades. After the discovery of the three-dimensional Lorenz system [10], a large number of chaotic systems were presented. Some typical systems are chaotic oscillators without equilibrium [11], with stable equilibrium [12], and with equilibrium point located on a segmented straight line [13,14]. In [15], authors investigated chaotic oscillators with balanced squares, while a chaotic oscillator with circular equilibrium was studied in [16].

The present challenge is to provide a new chaotic oscillator with special properties. Chaotic oscillators can be divided into conservative and dissipative. A dissipative chaotic system is one in which energy or other stored quantities are continually lost and, for instance, often converted to heat by processes such as friction; see [17]. The behaviour of the system may be unpredictable and limited to the boundaries of these attractors, and vice versa, conservative chaotic oscillators do not lose energy over time and the appearance of their orbits on the surface exhibits constant energy in the corresponding space. In this case, the orbits of these oscillators remain within conserved subspace. Conservative and dissipative chaotic oscillators are structurally stable. Recently, peculiar chaotic oscillators with no linear terms have been presented. The boundedness of this class of chaotic oscillators shows a difficulty to be determined, see [18]. A complicated class of chaotic oscillators generated by only nonlinear terms is shown in

Table 1. Chaotic oscillators generated by only nonlinear terms.

System Eq.	Lyapunov Exponents	No. Equilibria and Their States	Ref.
$\left\{\begin{array}{cc}\dot{x}=yx\hfill & \\ \dot{y}=x\left|x\right|-y\left|y\right|\hfill & \\ \dot{z}=\left|x\right|-1.5xy\hfill \end{array}\right.$	$L_1=0.145$	(0, 0, z)-one line	[19]
	$L_2=0$
	$L_3=-0.203$
$\left\{\begin{array}{cc}\dot{x}=-x^2-y^2+z^2\hfill & \\ \dot{y}=x^2-1\hfill & \\ \dot{z}=-x^2+y^2+9\hfill \end{array}\right.$	$L_1=0.215$	8-unstable	[20]
	$L_2=0$
	$L_3=-4.799$
$\left\{\begin{array}{cc}\dot{x}=yz\hfill & \\ \dot{y}=1-z^2\hfill & \\ \dot{z}=0.05x^3+yz\hfill \end{array}\right.$	$L_1=0.0271$	2-unstable	[21]
	$L_2=0$
	$L_3=-1.799$

.

On the other hand, the study of when and how it is feasible to govern systems showing irregular, chaotic behaviour is known as control of chaos, or control of chaotic oscillators. It is the boundary field that exists between control theory and nonlinear oscillators theory. Nonlinear control and control of chaos are closely related, and chaotic oscillators can benefit from various nonlinear control techniques. Due to the complexity of chaotic oscillators generated by a large number of nonlinear terms, the control process became more complicated. Two of the most known methods of chaos control will be studied. The first is the globally stable adaptive controller based on the second method of Lyapunov, which has been introduced to control the chaotic motion. The second method is by constructing an observer for the corresponding nonlinear system, which depends on the solvability of a linear matrix related to a matrix system.

In this paper, we present an oscillator with seven pure nonlinear terms and its stability in Section 2. Section 3 provides a dynamic analysis. We illustrate a controller and an application of the proposed oscillator in secure communications in Section 4 and Section 5.

2. Proposed Oscillator

We consider the nonlinear function

$$F\left(y\right)=a_1{y}^3-a_2{y}^2-a_{3}y,$$

(1)

where $a_1$, $a_2$, $a_3$ are positive parameters. Based on $F\left(y\right)$, the oscillator is

$$\left\{\begin{array}{cc}\dot{x}=F\left(y\right)+a_4{x}^2;\hfill & \\ \dot{y}=-a_{5}xz;\hfill & \\ \dot{z}=a_{6}xy+a_7{x}^2.\hfill \end{array}\right.$$

(2)

where $a_{i}i=1,\dots ,7$ are systems parameters.

2.1. Stability Analysis

By coordinate transformation, the oscillator (2) possesses the invariance as

$$(x,y,z,a_2,a_4)\to (-x,-y,-z,-a_2,-a_4).$$

(3)

From (3), one can see that the symmetrical feature is attractive when exploring oscillators as illustrated in Figure 1.

We obtain the equilibria of the system (2) by solving the algebraic equations below:

$$F\left(y\right)+a_4{x}^2=0,$$

(4)

$$-a_{5}xz=0,$$

(5)

$$a_{6}xy+a_7{x}^2=0.$$

(6)

One can find the equilibrium points of the oscillator (2) by starting from Equation (6), which yields the following three cases:

• The first equilibirum point is the origin coordinate, i.e., $E_1=(0,0,0).$
• When $x=0$ and $z\ne 0,$ then we have a line of equilibria with following conditions
  • if $a_1,a_2,a_3$ are not equal to zero, then we have

    $$E_2=(0,\frac{a_2-\sqrt{4a_1{a}_3-a_2^2}}{2a_1},z),$$

    $$E_3=(0,\frac{a_2+\sqrt{4a_1{a}_3-a_2^2}}{2a_1},z);$$
  • if $a_1=0$ and $a_2=a_3$ are not equal to zero, then we have

    $$E_4=(0,\frac{-a_2}{a_3},z).$$
• When $x\ne 0$ and $z=0,$ then we have the following

  $$E_5=\left\{\begin{array}{cc}-\frac{a_6}{a_7}\frac{a_2\sqrt{a_7}-\sqrt{4a_1{a}_3{a}_7+4a_1{a}_4{a}_6+a_2^2{a}_7}}{2a_1\sqrt{a_7}},\hfill & \\ \\ \frac{a_2\sqrt{a_7}-\sqrt{4a_1{a}_3{a}_7+4a_1{a}_4{a}_6+a_2^2{a}_7}}{2a_1\sqrt{a_7}},\hfill & \\ \\ 0,\hfill \end{array}\right.$$

  $$E_6=\left\{\begin{array}{cc}-\frac{a_6}{a_7}\frac{a_2\sqrt{a_7}+\sqrt{4a_1{a}_3{a}_7+4a_1{a}_4{a}_6+a_2^2{a}_7}}{2a_1\sqrt{a_7}},\hfill & \\ \\ \frac{a_2\sqrt{a_7}+\sqrt{4a_1{a}_3{a}_7+4a_1{a}_4{a}_6+a_2^2{a}_7}}{2a_1\sqrt{a_7}},\hfill & \\ \\ 0,\hfill \end{array}\right.$$

Consequently, to check the stability of $E_p(x,y,z)$, $(p=1,\dots ,6),$ the given equilibrium points, we calculate the Jacobian matrix as:

$${|J-\lambda I|}_{(x,y,z)}=\left(\begin{array}{ccc}2a_{4}x& F^{\prime }\left(y\right)& 0\\ \\ -a_{5}z& 0& -a_{5}x\\ \\ a_{6}y+2a_{7}x& a_{6}x& 0\end{array}\right)$$

(7)

where $F^{\prime }\left(y\right)=3a_1{y}^2-2a_{2}y-a_3.$ The characteristic polynomial of J is given by

$$\Pi (J,\lambda )={\alpha }_3{\lambda }^3+{\alpha }_2{\lambda }^2+{\alpha }_1\lambda +{\alpha }_0,$$

where

$$\left\{\begin{array}{cc}{\alpha }_3=-1,\hfill & \\ {\alpha }_2=a_4,\hfill & \\ {\alpha }_1=a_{5}z{F}^{\prime }\left(y\right)+a_5{a}_6{x}^2,\hfill & \\ {\alpha }_0=2F^{\prime }\left(y\right)a_7{a}_5{x}^2-F^{\prime }\left(y\right)a_6{a}_{5}xy+a_4{a}_6{a}_5{x}^2\hfill \end{array}\right..$$

According to the Routh–Hurwitz stability criterion and ${\alpha }_3,$ the equilibrium points of the oscillator (2) are unstable when all the corresponding parameters are either positive or negative. Therefore, the attractor of the oscillator (2) will pass all possible equilibrium points.

Due to the positive values of all system’s parameters, in particular, we select the parameters $a_1=0.1,$ $a_7=0.2$ and $a_i=1$ $(i=2,\dots ,6)$, to calculate the eigenvalues of J for the corresponding points in

Table 2. The stability for equilibrium points in oscillator (2).

Equilibrium Point	Eigenvalues	Stability
$E_1=(0,0,0)$	${\lambda }_1=1$	Unstable
	${\lambda }_2=0$
	${\lambda }_3=0$
$E_2=E_3=(0,5\pm 3.8i,q),q\in \mathbb{R}$	${\lambda }_1=0$	Unstable saddle focus
	${\lambda }_2=\frac{1/2-\sqrt{1-q(50\pm 34.96i)}}{2}$
	${\lambda }_3=\frac{1/2+\sqrt{1-q(50\pm 34.96i)}}{2}$
$E_4=(21.097,-4.219,0)$	${\lambda }_1=-3.2328$	Unstable saddle focus
	${\lambda }_2=-2.1164+21.3071i$
	${\lambda }_3=-2.1164-21.3071i$
$E_5=E_6=(\pm 71.097,$ $\pm 14.219,0)$	${\lambda }_1=-5.1903$	Unstable saddle focus
	${\lambda }_2=3.0951+71.2484i$
	${\lambda }_3=3.0951-71.2484i$

.

According to Table 2, the proposed oscillator is self-excited.

In addition, we will consider one more important feature of chaotic oscillators which is the “dissipativity”. In Lorenze’s system, the largest number of those oscillators is dissipative. In fact, the oscillator we proposed in (2) is one dissipative oscillator. The dissipativity is calculated by:

$$V=\frac{\partial \dot{x}}{\partial x}+\frac{\partial \dot{y}}{\partial y}+\frac{\partial \dot{z}}{\partial z}=2a_{4}x.$$

(8)

From (8), one can see that the proposed oscillator is dissipative if $a_4<0$ and $x\left(t\right)>0$ as $t\to \infty$ or $a_4>0$ and $x\left(t\right)<0$ as $t\to \infty .$

2.2. Zero-Hopf Bifurcation

A particular kind of stability is the bifurcation at a point known as a zero-Hopf singular point. Note that such a class of equilibrium points occurs when the Jacobian matrix of a three-dimensional autonomous differential system has two pure imaginary eigenvalues and zero. For certain differential systems, averaging techniques are helpful in examining and figuring out how many periodic orbits there are. Specifically, it is a crucial technique for examining the cyclicity of the limit cycle that splits off from the zero-Hopf point. The first-order averaging theory is commonly used to derive the amplitude equation describing the limit cycle’s dynamics near the bifurcation point. This amplitude equation can be used to determine the stability and bifurcation properties of the limit cycle. Applying such a method, various researchers have concentrated on investigating the existence of periodic orbits. Some typical work has been presented by Marsden and McCracken [22]. In the following, the basic results are presented. Such results are vital to illustrate a possible bifurcation of limit cycles via a selected zero-Hopf equilibrium point in system (2) using the first-order averaging method.

The following differential systems are considered by us:

$$\dot{X}=F_0(t,X)+ϵF_1(t,X)+{ϵ}^2{F}_2(t,X),$$

(9)

where ${ϵ}_0>0$ is a small integer number and $ϵ\in (-{ϵ}_0,{ϵ}_0)$, the maps $F_0,F_1:\mathbb{R}\times H\to {\mathbb{R}}^n$ and $F_2:\mathbb{R}\times H\times (-{ϵ}_0,{ϵ}_0)$ are continuous maps belong to space ${\mathbb{C}}^2$. T-periodic is in the first variable. One of the most important dynamical features is the bifurcation of T-periodic solutions in system (2), which will be studied by a well-known method called averaging theory. Consider

$$\dot{X}=F_0(t,X)$$

(10)

to be an unperturbed system which may have periodic solutions in a sub-manifold. Assume that $X(t,X_0)$ is the periodic solution of system (10) such that $X(0,X_0)=X_0=X(T,X_0)$. Thus, system (10) can be linearized along its periodic solution $X(t,X_0)$ as:

$$\dot{Y}=D_X{F}_0(t,X(t,X_0))Y.$$

(11)

By $M_{X_0}\left(t\right),$ we denote the so-called fundamental matrix of the above equation. Let V be an open set and $Cl\left(V\right)$ be a set formed only by periodic solutions. Suppose there exists V where $Cl\left(V\right)\subset H.$ In particular, for each $X_0\in Cl\left(V\right)$, one has that $X(t,X_0)$ is a T-periodic solution of (10). Then, $Cl\left(V\right)$ may contain a vision about the bifurcation of T-periodic solutions starting from $X(t,X_0).$ For more detailed information about averaging theory and zero-Hopf bifurcation, see [23].

The next classical result brings clear information:

Theorem 1. 

Let V be an open and bounded set where $Cl\left(V\right)\in H.$ Let for each $X_0\in Cl\left(V\right)$, if the solution $X(t,X_0)$ is a T-periodic, then we have map $F:Cl\left(V\right)\to {\mathbb{R}}^n$ such that:

$$F\left(X_0\right)=\frac{1}{T}{\int }_0^T{M}_{X_0}^{-1}(t,X_0)F(t,X(t,X_0))dt.$$

Then, the following statements hold:

(I) If there exist $X^{*}\in V$ with $F\left(X^{*}\right)=0$ and $det\left(\frac{\partial F}{\partial X_0}\left(X^{*}\right)\right)\ne 0$, then there exist a T-periodic solution $X(t,ϵ)$ of system (10) such that $X(0,ϵ)\to X^{*}$ as $ϵ\to 0$.

(II) The type of stability of the periodic solution $X(t,ϵ)$ is given by the eigenvalues of the Jacobian matrix $\frac{\partial F}{\partial X_0}\left(X^{*}\right).$

Proposition 1. 

For system (2), the point $(0,0,z)$ is a zero-Hopf singular point if and only if the following conditions are satisfied

1. 

$a_1=a_2=a_4=0;$

2. 

$a_3,a_5,a_6,a_7>0$ and $z\ne 0.$

Proof. 

It is clear to show that the characteristic equation $\Pi (J,\lambda )$ has three roots $0,\pm i\omega ;\omega =\sqrt{a_3}\sqrt{a_5}$ the following conditions are satisfied

• $a_1=a_2=a_4=0;$
• $a_3,a_5,a_6,a_7>0$ and $z\ne 0.$ □

Theorem 2. 

Consider system (2), provided that $a_3,a_5>0$. Let $a_3={\omega }^2+ϵ\tau$ where $\tau \ne 0.$ Let ϵ be a sufficiently small positive parameter. Using first order averaging theory, it cannot find periodic orbits bifurcating from the zero-Hopf singular point at the origin.

Proof. 

When $a={\omega }^2+ϵ\tau$ with using the scale of variables $(x,y,z)=(ϵX,ϵY,ϵZ).$ Denote $(X,Y,Z)$ by $(x,y,z)$ and $a_1=a_2=a_4=0,$ we have

$$\left(\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\end{array}\right)=\left(\begin{array}{c}-({\omega }^2+\tau ϵ)x_2\\ 0\\ 0\end{array}\right)+ϵ\left(\begin{array}{c}0\\ -a_{5}xy\\ a_{6}xy+a_7{x}^2\end{array}\right)$$

(12)

Now, the following transformation transforms the unperturbed terms of system (12) into its Jordan normal form

$$\begin{array}{c}\hfill \left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& -\omega & 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right).\end{array}$$

(13)

In the new variables $(u_1,u_2,u_3)$, system (12) becomes

$$\begin{array}{cc}\hfill \dot{u_1}=& -(\omega +{\displaystyle \frac{\tau ϵ}{\omega }})\omega u_2,\hfill \\ \hfill \dot{u_2}=& -a_5{u}_1{u}_3,\hfill \\ \hfill \dot{u_3}=& a_6{u}_1{u}_2+a_7{u}_1^2.\hfill \end{array}$$

(14)

Its clear that (14) is written as (9).

Now, let us consider the following:

$$\dot{X}=F_0(t,X),X_0=(u_0,v_0,w_0).$$

(15)

The solution of (15) is $X(t,X_0)=(u_1\left(t\right), u_2\left(t\right), u_3\left(t\right))$ where

$$\left\{\begin{array}{c}{u}_1\left(t\right)=v_0,\hfill \\ u_2\left(t\right)=u_0+\omega t{v}_0,\hfill \\ u_3\left(t\right)=w_0.\hfill \end{array}\right.$$

Obviously, all solutions $X(t,X_0)$ involved in $u_2\left(t\right)=u_3\left(t\right)=0$ are not periodic. The given system does not have any nontrivial solutions that satisfy $F\left(X_0\right)=0$ provided that the determinant of the Jacobian matrix at that point is not zero. As a result, there cannot be any periodic orbits that bifurcate from the associated zero-Hopf point. This indicates that averaging theory does not help to localize the periodic solutions of a complicated structure of chaotic oscillator such as having no linear (or at least) terms.  □

3. Dynamics of the Oscillator

Important tools to indicate the complex dynamics of nonlinear oscillators are the Lyapunov exponents and bifurcation diagram. Bifurcation analysis studies the evolution of a dynamic system’s basic properties as a parameter changes. Its main goal is to identify precise points in the parameter space where notable qualitative changes in the behaviour of the system appear. It is possible to determine possible multistable zones in the parameter space by using the forward and backward methods to generate bifurcation diagrams. It is standard procedure to conduct bifurcation analysis in the case of flow-based dynamical systems by locating the local maxima in the system’s time evolution for each unique parameter configuration. Conversely, the Lyapunov exponents analysis is a technique used to evaluate numerically the speed at which neighbouring trajectories in a dynamical oscillator either converge or spread out exponentially. This method provides insightful information about the system’s vulnerability to initial conditions. In the following, we provide the Lyapunov exponents and bifurcation analysis of the proposed oscillator (2).

The proposed oscillator (2) displays chaotic behaviour when the parameters $a_1=0.1,$ $a_7=0.2,$ $a_i=1$ $(i=2,\dots ,6)$ and initial condition (−1.3, 1.5, 1.4). To gain a better understanding of the complexity and behaviour of the proposed oscillator, we will investigate the Lyapunov exponents. Due to the corresponding parameter values and initial conditions, the LEs are found to be $L_1=0.313, L_2=0, L_3=-0.744.$ Moreover, based on the Lyapunov exponents, we will determine the Kaplan–Yorke fractional dimension ($D_{KY}$), which is given by:

$$D_{KY}=j+\frac{{\sum }_{i=1}^j{L}_i}{|L_{j+1}|},$$

(16)

where $L_j$ is the Lyapunov exponent. Therefore, the Kaplan–Yorke fractional dimension is found to be $D_{KY}=2.42$, which shows the geometric complexity of the attractor (see Figure 1).

To explore the dynamics of the introduced oscillator along a wide range of its two parameters $a_1$ and $a_7$, the bifurcation diagram for the parameter $a_1$ on the interval 0 < $a_1$≤ 1 with keeping $a_7=0.2,$ and bifurcation diagram for the parameter $a_7$ on the interval 0 < $a_7$≤ 1 with keeping $a_1=0.1$ have been plotted as shown in Figure 2. By using the oscillator initial conditions, $x_0=y_0=z_0=1$. In all bifurcation diagrams, the oscillator has simulated along the time interval $t\in [0,1000],$ where the Poincare section has been captured on $x-z$ plane after the steady state reached (i.e., t > 300). The bifurcation diagram and the corresponding Lyapunov exponents have been plotted with the step-size $\Delta a_1=0.02$ as shown in Figure 3, while in Figure 2 the bifurcation parameter $a_7$ is discretized with the step-size $\Delta a_7=0.01.$ It is clear from Figure 2 that the system performs chaotic motion as long as $a_1$ > 0.1 when keeping $a_7=0.2.$ On the other hand, Figure 2 demonstrates that the system oscillates chaotically for $a_7$ > 0.2 when keeping $a_1=0.1.$ The phase portraits and time response with parameters of the system are illustrated in Figure 2 and Figure 4, respectively.

4. Adaptive Control of System (2)

Adaptive control, an advanced methodology adapted for dynamic systems, stands as a beacon of innovation in the realm of control theory. Providing unparalleled versatility and efficacy, adaptive control mechanisms facilitate seamless customization and optimization of control processes in response to fluctuations in system dynamics. This adaptive capability is particularly invaluable in scenarios characterized by the inherent unpredictability and complexity of chaotic systems. Given the propensity of chaotic systems to exhibit a sensitive dependence on initial conditions and nonlinear dynamics, conventional control strategies often fail to adequately regulate such systems. Researchers have directed considerable efforts towards the development and refinement of adaptive controllers specifically tailored to address the unique challenges posed by chaotic systems. By harnessing the intrinsic adaptability of adaptive control methodologies, these activities aim to imbue control systems with the resilience and responsiveness requisite for navigating the intricacies of chaotic dynamics.

In this section, we present a control study of oscillator (2). Briefly, the control strategy is the earliest control strategy proposed to resolve the synchronization issue. The first work in this direction [24] shows that the active control theory may synchronize the connected Lorenz system which is also confirmed by the simulation. However, Perez-Cruz et al. [25] looked into the synchronization of a new three-dimensional chaotic system by using Lyapunov analysis to design a nonlinear controller that guaranteed the exponential convergence of the synchronization error, and the results of their numerical simulation confirmed the controller’s excellent performance.

Now, let fix some the parameters of oscillator (2) as $a_j=1,j=2,\dots ,6$ and let us define the following driver system as follows

$$\left\{\begin{array}{cc}\dot{x_1}=a_1{x}_2^3-x_2^2-x_2+x_1^2;\hfill & \\ \dot{x_2}=-x_1{x}_3;\hfill & \\ \dot{x_3}=x_1{x}_2+a_7{x}_1^2\hfill \end{array}\right.$$

(17)

Next, let us provide the adaptive synchronization of an identical novel chaotic oscillator with parameters which are not valued. The response system is presented as

$$\left\{\begin{array}{cc}\dot{x_1}=a_1{x}_2^3-x_2^2-x_2+x_1^2+u_1;\hfill & \\ \dot{x_2}=-x_1{x}_3+u_2;\hfill & \\ \dot{x_3}=x_1{x}_2+a_7{x}_1^2+u_3\hfill \end{array}\right.$$

(18)

where $x_1,x_2,x_3$ are the states, $a_1,a_7$ are unknown system parameters and

$$U={[u_1,u_2,u_3]}^T$$

is the adaptive controller to be determined. We consider the adaptive controller defined by

$$\left\{\begin{array}{cc}{u}_1=-{\epsilon }_{a_1}{x}_2^3+x_2^2+x_2-x_1^2-k_1{x}_1;\hfill & \\ u_2=x_1{x}_3-k_2{x}_2;\hfill & \\ u_3=-x_1{x}_2-{\epsilon }_{a_7}{x}_1^2-k_3{x}_3.\hfill \end{array}\right.$$

(19)

where ${\epsilon }_{a_1},{\epsilon }_{a_7}$ denote the estimated parameters of the system coefficients $a_1,a_7$, respectively, and $k_1,k_2,k_3>0.$

Substituting (19) into (18), we obtain the closed-loop system:

$$\left\{\begin{array}{cc}\dot{x_1}=[a_1-{\epsilon }_{a_1}\left(t\right)]x_2^3-k_1{x}_1;\hfill & \\ \dot{x_2}=-k_2{x}_2;\hfill & \\ \dot{x_3}=[a_7-{\epsilon }_{a_7}\left(t\right)]x_1^2-k_3{x}_3\hfill \end{array}\right.$$

(20)

Let us denote the error estimation of the parameters as follows:

$$\left\{\begin{array}{c}{\epsilon }_1\left(t\right)=[a_1-{\epsilon }_{a_1}\left(t\right)];\hfill \\ {\epsilon }_2\left(t\right)=[a_7-{\epsilon }_{a_7}\left(t\right)];\end{array}\right.$$

(21)

Due to (21), the derivatives of the parameter estimation errors can be expressed as:

$$\left\{\begin{array}{cc}\dot{{\epsilon }_1}=-\dot{{\epsilon }_{a_1}},\hfill & \\ \dot{{\epsilon }_2}=-\dot{{\epsilon }_{a_7}}.\hfill \end{array}\right.$$

(22)

In the following, we reduced (20) to

$$\left\{\begin{array}{cc}\dot{x_1}={\epsilon }_1-k_1{x}_1;\hfill & \\ \dot{x_2}=-k_2{x}_2;\hfill & \\ \dot{x_3}={\epsilon }_2{x}_1^2-k_3{x}_3.\hfill \end{array}\right.$$

(23)

Theorem 3. 

If the controllers are chosen as (19) and let the parameter’s update laws is

$$\left\{\begin{array}{cc}\dot{{\epsilon }_1}\left(t\right)=x_1{x}_2^3-\eta (a_1-{\epsilon }_{a_1});\hfill & \\ \dot{{\epsilon }_2}\left(t\right)=x_3{x}_1^2-\eta (a_7-{\epsilon }_{a_7}).\hfill \end{array}\right.$$

(24)

then the synchronization between the driver system (19) and the response system (18) is approached if $k_1,k_2,k_3$ are positive constants.

Proof. 

We consider the Lyapunov function defined by

$$V(x_1,x_2,x_3,{\epsilon }_1,{\epsilon }_2)=\frac{1}{2}\left(x_1^2+x_2^2+x_3^2+{\epsilon }_1^2+{\epsilon }_2^2\right)$$

Differentiating the above function, we have

$$\dot{V}(x_1,x_2,x_3,{\epsilon }_1,{\epsilon }_2)=$$

$$=\left(x_1\dot{x_1}+x_2\dot{x_2}+x_3\dot{x_3}+{\epsilon }_1\dot{{\epsilon }_1}+{\epsilon }_2\dot{{\epsilon }_2}\right)$$

Taking time derivative of the above function along the trajectories of (24), we have

$$\dot{V}=-(k_1{x}_1^2+k_2{x}_2^2+k_3{x}_3^2+\eta {\epsilon }_1^2+\eta {\epsilon }_2^2)$$

(25)

which is a negative function for $k_1,k_2,k_3>0$. Thus, due to the Lyapunov stability theory, we find that ${\epsilon }_1\left(t\right)\to 0,{\epsilon }_2\left(t\right)\to 0$ exponentially when $t\to \infty$.  □

Numerical Simulation

We consider the fourth-order Runge–Kutta method to numerically simulate the adaptive control for system (18) with the adaptive control law (19) and the parameter update law (24). The parameters of system (17) are selected as $a_1=0.1, a_7=0.2.$ In addition, we take the adaptive and update laws as $k_i={\eta }_j=2, (i=1,\dots ,3;j=1,2).$ Suppose that the initial values of the estimated parameters are $(0,0,0)$ and the initial values of system (2) are taken as (1, 1, 1). When the adaptive control law (24) and the parameter update law are used, the controlled system converges to the equilibrium E = (0, 0, 0) exponentially, as shown in Figure 5.

5. Secure Communication Application

In the contemporary information age, the escalating concern for security within information systems is palpable and ever-growing. This heightened apprehension is principally propelled by the exponential surge in news dissemination and data generation, necessitating the proliferation and refinement of sophisticated information systems. The imperative to exchange and transmit information seamlessly in our interconnected world catalyzes the inception and evolution of advanced information systems, amplifying the importance of fortifying their security measures. Addressing the security challenges confronting both individuals and organizations within information systems has become imperative in contemporary times. Indeed, the imperative of secure information systems has become paramount in today’s digital landscape, wherein the repercussions of breaches and cyber attacks can be profound and far-reaching. As such, the development and implementation of robust security mechanisms within information systems are indispensable. Traditional security techniques have long served as the cornerstone of information security efforts, offering protection against a myriad of threats ranging from unauthorized access to data breaches. However, the burgeoning sophistication of cyber threats necessitates a continuous evolution in security strategies and methodologies.

One such evolutionary stride in the realm of information security involves the advent of secure information systems predicated on the principles of chaotic systems. Leveraging the dynamic characteristics inherent in chaotic systems engenders a novel paradigm for ensuring the security of information systems. Chaotic systems, characterized by sensitive dependence on initial conditions and deterministic unpredictability, offer a fertile ground for devising innovative encryption and authentication mechanisms. The inherent complexity and nonlinearity of chaotic systems furnish a formidable barrier against adversarial intrusion, thereby augmenting the resilience of secure information systems.

The exploration of secure information systems, underpinned by chaotic dynamics, represents a burgeoning frontier in information security research. The attempt to harness the intricate dynamics of chaotic systems for fortifying information systems against cyber threats holds immense promise and potential. Ongoing research activities in this domain are poised to yield novel breakthroughs, bolstering the efficacy and robustness of secure information systems.

The quest for enhancing the security of information systems encompasses a multifaceted approach, encompassing both theoretical investigations and practical implementations. Fundamental research activities delve into elucidating the underlying principles governing the dynamics of chaotic systems, discerning patterns amidst apparent randomness, and devising algorithms capable of harnessing chaotic dynamics for cryptographic purposes. Concurrently, applied research actions focus on integrating chaotic encryption techniques into real-world information systems, evaluating their efficacy, scalability, and compatibility with existing infrastructure.

The burgeoning proliferation of interconnected devices and the advent of the Internet of Things (IoT) further accentuate the exigency of secure information systems. As the fabric of our digital ecosystem becomes increasingly intricate and interconnected, the vulnerability surface susceptible to exploitation by malicious actors commensurately expands. In this context, the integration of chaotic encryption techniques offers a salient avenue for bolstering the security posture of IoT devices and networks, mitigating the risks associated with cyber attacks and unauthorized access.

In this section, an application to the proposed oscillator will be considered, especially to secure communications problems. Before we begin, we need to provide a brief introduction to $X=C[a,b]$- Banach spaces. Let f be a continuous function, the norm of g is given by

$${\parallel g\parallel }_{l_1}=\underset{t}{max}\left|g\left(t\right)\right|.$$

The function $g:X\to X$ is said to satisfy the Lipschitz condition if there exists a positive integer $\xi$ such that for every $\mu ,\nu \in l_1,$ we have

$${\parallel g\left(\mu \right)-g\left(\nu \right)\parallel }_X\le \xi {\parallel \mu -\nu \parallel }_X.$$

Because the oscillator (2) has no linear term, we inject the linear term $q\left(t\right)$ which is the transmitted signal. As a result, we consider the following system:

$$\left\{\begin{array}{cc}\dot{x}=F\left(y\right)+a_4{x}^2+b_{1}q\left(t\right);\hfill & \\ \dot{y}=-a_{5}xz+b_{2}q\left(t\right);\hfill & \\ \dot{z}=a_{6}xy+a_7{x}^2+b_{3}q\left(t\right);\hfill & \\ Y\left(t\right)=C_1{\left(x\left(t\right)y\left(t\right)z\left(t\right)\right)}^T+\Gamma q\left(t\right),\hfill \end{array}\right.$$

(26)

where $F\left(y\right)=y^3-y^2-y,$ $a_2=a_3=a_4=a_5=a_6=1$, $y\left(t\right)$ the measurable output, $b_1, b_2, b_3$ are the coefficients of the transmitted signal. The matrices $C_1$ and $\Gamma$ are appropriate dimensions. In order to successfully estimate the input signal and proceed with synchronization, we reformulate the system as follows:

$$M\dot{x}\left(t\right)=Ax\left(t\right)+\Phi \left(x\right)$$

(27)

and

$$Y\left(t\right)=Cx\left(t\right)$$

(28)

with

$$\left\{\begin{array}{cc}M=(I_3,0_{3\times 1})\hfill & \\ \\ A=\left(\begin{array}{cccc}0& 0& 0& b_1\\ 0& 0& 0& b_2\\ 0& 0& 0& b_3\end{array}\right)\hfill & \\ \\ C=(C_1,\Gamma )\hfill & \\ \\ x\left(t\right)=\left(\begin{array}{c}x\left(t\right)\\ y\left(t\right)\\ z\left(t\right)\end{array}\right)\hfill & \\ \\ \Phi \left(x\right)=\left(\begin{array}{cc}{y}^3-y^2-y+a_4{x}^2;& \\ -a_{5}xz;& \\ a_{6}xy+a_7{x}^2.\end{array}\right)\hfill \end{array}\right.$$

The state $x\left(t\right)$ is estimated by an observer constructed for the system (27). Thus, we can achieve synchronization and input estimation.

In order to continue the process, we assume the following:

• Let n be the number of columns of matrix M, rank ${(M^T,C^T)}^T=n.$ Therefore, due to the form of matrix M, it is equivalent to $\Gamma$.
• The non-linear parts satisfy the Lipschitz condition

  $${\parallel \Phi \left(x\right)-\Phi \left(y\right)\parallel }_X\le \xi {\parallel x-y\parallel }_X,$$

  where $\xi >0$ is positive integer number.

Note that assumption (B) includes a broad spectrum of systems; even for ones not satisfying Lipschitz’s condition “globally”, the constant $\xi$ can be obtained for systems which are satisfying the Lipschitz condition “locally”. Due to the mentioned assumptions, our aim here is to design an observer for system (27), which is given by:

$$\dot{Z}\left(t\right)=RZ\left(t\right)+JY\left(t\right)+K\Phi \left(\hat{x}\right)$$

(29)

and

$$\hat{x}\left(t\right)=x\left(t\right)+EY\left(t\right)$$

(30)

where $R,J,K,E$ are the matrices computed via $\hat{x}$ such that

$$\parallel \hat{x}{-x\parallel }_X\to 0\mathrm{as}t\to \infty .$$

We apply the method in [26] to obtain the observer matrices:

• The matrix K computation can be carried out by the procedure shown in [26];
• Compute the matrices $KM$ and $KA$ by

  $$KM\dot{x}=KA+K\Phi \left(x\right),Y=Cx$$

  considering the solvability of the LMI

  $$W=\left(\begin{array}{cc}{\left(KA\right)}^{T}P+p\left(KA\right)-C^T{\hat{Q}}^T-\hat{Q}C+{\xi }^{2}I& PN\\ N^{T}P& -I\end{array}\right)<0$$

  for $P>0.$
• If W is solvable, then the matrices of the observer are:

  $$Q=P^{-1}\hat{Q},R=KA-QC$$

  (31)

  $$KM=I-EC,V=Q+RE$$

  (32)

This way, the observer (29) can synchronize with system (26).

Theorem 4. 

Let ${\mathbb{C}}_{+}=\left\{q\right|q\in \mathbb{C}Re\left(q\right)\ge 0\}.$ The necessary condition for the solvability of W is

$$rank\left(\begin{array}{c}qM-A\\ C\end{array}\right)=n\forall q\in {\mathbb{C}}_{+}$$

Proof. 

To satisfy Theorem 4, it easy to check that it is based on the structure of $A,$ we need to counteract the lack of linear terms by adding more outputs.  □

Simulation Study

For implementing the simulation, we take $a_1=0.1,a_7=0.2$ and $b_1=1,b_2=0.5$, $b_3=0$ and

$$C=\left(\begin{array}{cccc}1& 3& 0& 0.2\\ 0& 2& 0.5& 0\\ 2& 0& 1& 0\end{array}\right).$$

Assume that the initial conditions are: $x_0=-1.3,y_0=1.5,z_0=1.4$ and $z_{1,0}=1,z_{2,0}=1,z_{3,0}=1,z_{4,0}=0.$ Again, we assume that $q\left(t\right)=0.4cos\left(\pi t\right)$ is the transmitted signal. Figure 6 shows the simulation results starting from an equilibrium point. It is evident that the observer is able to accurately assess the incoming signal and states. The simulation is carried out by the Matlab 2019 platform codes in [27].

6. Conclusions

A noticeable oscillator with nonlinear continuous functions and chaos has been introduced. The oscillator displayed line equilibria and four equilibrium points. The stability analysis verifies that all the equilibrium points of the proposed oscillator are unstable. The proposed oscillator is conditionally dissipative. Unlike typical oscillators, the mathematical model of this oscillator exclusively incorporates nonlinear terms. The dynamics of the oscillator are reported. An application for secure communications is obtained by injecting a linear function (observer), satisfying Lipschitz’s condition in a Banach space of continuous functions, into the proposed oscillator. In addition, we have provided the adaptive control for the oscillator.