Research Article | Open Access
Dynamic Analysis and Circuit Implementation of a New 4D Lorenz-Type Hyperchaotic System
This paper attempts to further extend the results of dynamical analysis carried out on a recent 4D Lorenz-type hyperchaotic system while exploring new analytical results concerns its local and global dynamics. In particular, the equilibrium points of the system along with solution’s continuous dependence on initial conditions are examined. Then, a detailed symmetrical Bogdanov-Takens bifurcation analysis of the hyperchaotic system is performed. Also, the possible first integrals and global invariant surfaces which exist in system’s phase space are analytically found. Theoretical results reveal the rich dynamics and the complexity of system behavior. Finally, numerical simulations and a proposed circuit implementation of the hyperchaotic system are provided to validate the present analytical study of the system.
Nonlinear dynamics analysis of various phenomena and systems of physics, engineering, biology, chemistry, economy, and industry has attracted a great interest among scientists and considered a very active area of research from 1960s in the last century till now [1–4]. There are two key reasons which interpret this great interest. The first one is that the dynamical systems tools help scientists better comprehend and analyze the varieties of nonlinear characteristics and new phenomena exhibited by systems from different disciplines. In particular, the tools of dynamical systems such as the applied bifurcation theories are successfully employed to investigate the qualitative behaviors of nonlinear systems [5–7]. This includes investigation of equilibrium points and their stability, creation, destruction and stability of periodic orbits, quasiperiodic behavior, homoclinic orbits, creation or destruction of chaotic attractors, and chaos control and synchronization. The second reason is that engineers and scientists can utilize some of the fascinating features of nonlinear dynamical systems in wide range of interesting applications.
The spellbinding chaotic dynamics, as an example, is recognized by high sensitivity to initial conditions and positive Lyapunov exponents. The generation of chaos for practical applications can be achieved by exploiting nonlinear electronic circuits, nonlinearities of laser systems with feedbacks, or via digital platforms such as FPGAs and DSPs. The noise-like behavior of chaotic systems, their wide spectrum, and the possibility of attaining chaos synchronization between two chaotic systems render them essential for cutting edge applications related to cryptography and robust physical-layer secure communication systems and ultra-fast physical random bits generation [4–11]. The other applications of dynamical systems methods and chaos theory include financial systems, mathematical biology, nonlinear circuits, nonlinear mechanical systems, plasma physics, chaos control, efficient image encryption, neuroscience research, and geophysics [12–33].
The more complex hyperchaotic system possesses at least two positive Lyapunov exponents and it has a phase space of dimension at least four. Clearly, the hyperchaotic systems have more randomness and higher unpredictability than simple chaotic systems. Therefore, hyperchaos is more preferred than simple chaos and its applications have recently become a central topic in research including chaos-based secure communications, image encryption, and cryptography
In the last two decades, some interesting high-dimensional hyperchaotic systems, in science and engineering, have been explored and their dynamics have been extensively investigated [18, 24, 27, 28, 34, 35]. In fact, it is of great importance from theoretical and practical aspects to explain complicated phenomena and internal structural characteristics of hyperchaotic systems. This research line focuses on applying codimension two or three bifurcation analysis to system under investigation. For example, analysis of Bogdanov-Takens bifurcation, degenerate Hopf bifurcation, and Heteroclinic and Homoclinic bifurcations can be undertaken. Also, it is crucial to determine whether there exists any set of parameters values for which the considered system is integrable and find the corresponding invariant surfaces if exist. As it is practically unattainable goal to apply exhaustive numerical investigation to acquire values of parameters where the hyperchaotic system is integrable, one needs a powerful analytical method that enables achieving this goal easily.
Recently, theoretical analysis, based on Fishing Principle, is applied to study some global features of a new four-dimensional Lorenz-type hyperchaotic system . More specifically, conditions for existence of homoclinic orbits are obtained. This work aims at extending the aforementioned work and exploring other aspects of complicated dynamical behaviors of the 4D Lorenz-type hyperchaotic system. Analytical bifurcation structure of the model which includes symmetrical Bogdanov-Takens, Pitchfork, Andronov-Hopf bifurcation and homoclinic bifurcation are obtained. Existence, uniqueness, and continuous dependence on initial conditions for the solution of 4D hyperchaotic system are all investigated. It is important to ask whether there is any set of parameter values in which system dynamics are regular and the studied system is integrable. The different integrable cases of the system are studied and the closed-form expressions for invariant surfaces corresponding to first integrals of the system are also obtained. Moreover, the hyperchaotic system has been simulated using a proposed electronic circuit realization and numerical simulations are performed confirming the new results of theoretical analysis. Note that attractors found in dynamical system are classified as being self-excited or hidden attractors [37, 38]. The key difference between them is that the self-excited attractor has a basin of attraction excited from unstable equilibria. The hidden attractor, on the other hand, has a basin of attraction with no intersection with neighborhoods of dynamical systems equilibrium points. In this work, we focus on self-excited attractors exist in the hyperchaotic system.
The rest of the paper is structured as follows: In Section 2, we introduce the 4D Lorenz-type system and discuss the equilibrium points’ existence. A sufficient condition for continuous dependence on initial condition is determined. Phase portraits, bifurcation diagrams, and Lyapunov characteristics spectrum are obtained. In Section 3, the analysis of some possible codimension two bifurcations is performed. It is shown that the 4D Lorenz system undergoes Bogdanov-Takens bifurcation, Andronov-Hopf bifurcation, Pitchfork bifurcation, and homoclinic bifurcation. The integrability analysis of system is investigated in Section 4. A practical application to engineering will be realized by an electronic circuit in Section 5. Finally, Section 6 concludes the paper.
2. The 4D Lorenz-Type Hyperchaotic System
The following 4D hyperchaotic system of Lorenz type was presented in where state variables of the system are denoted by , and , the parameters of the system are represented by , and , and the dot above state variables refers to time derivative of state variables.
The following subsections examine the main properties of system (1) and provide elementary dynamical analysis of the model.
2.1. Equilibrium Points of the System
(I) when , the fixed point is given by
(II) when , the fixed points are
2.2. Existence and Uniqueness of the Solution
The hyperchaotic system (1) can be put in the formwhereand the initial conditions are given by
For the class of continuous functions , we use the following norm in subsequent analysis:while the matrix of continuous functions employs the norm
It is obvious that 4D system (1) is dissipative if Now, solution of the system is examined in specific region where andParameter is utilized to lay a boundary for the phase space region where existence and uniqueness of the solution are investigated.
The solution of (5) and (7) can be represented byThe equivalence of the integral equation (11) and system (5)-(7) is obvious. Now, denoting the right hand side of (11) by , then for and we getand therefore Then, we obtain whereThus, , for as sufficient condition, is a contraction mapping.
Theorem 1. Assume that , then a unique solution of hyperchaotic system (1), starting from initial condition in the region , exists.
2.3. Continuous Dependence on Initial Conditions
The continuous dependence on initial conditions means that solution trajectories of the system which start close to each other still close to each other with evolution of time. This property is contrary to sensitive dependence on initial conditions which specifies chaotic dynamics. The goal of the next analysis is to find the particular parameters and time range where continuous dependence on state variables initial conditions is persevered; i.e., system (1) does not exhibit chaotic dynamics.
Assume that there are two points of initial conditions of system (5)-(7); namely, and satisfy First, suppose that the condition of Theorem 1 holds. Thus and also we obtain and where is defined by (15). Finally, we get where . Thus, we can formulate the following theorem.
Theorem 2. The solution of hyperchaotic system (1) exhibits continuous dependence on initial conditions if system (1) satisfies the condition of Theorem 1. More specifically, such that whenever it implies that .
3. Symmetry Bogdanov-Takens Bifurcation
Now, we study the case where and that implies Jacobian matrix has two real zero eigenvalues, , and two negative eigenvalues if either or evaluated at the origin. The following coordinates’ transformation is applied to (1) in order to put the system in standard formwhich yieldswheresuch that
The center manifold is assumed in the form of second-order polynomial for sufficiently small and asthen substituting from ((25a)-(25b)) and the first two equations of (23) into the last two equation of (23). By comparing the coefficient of , in both sides of last two equations of (23) after substitution in ((25a)-(25b)), the values of center manifold coefficients are obtained as follows: The dynamics of the system on the center manifold is then described by where
Conditional normal form at bifurcation values of parameters can be obtained by following Kuznetsov approach which implies that the simplified system which represents the conditional normal form at bifurcation value is the following system:
The next step in this analysis is to obtain the universal unfolding of the BT bifurcation. In order to put the system in an appropriate form, the bifurcation parameters should be perturbed around bifurcation value, such that and The following system is attained Now, applying transformation (22) to getwhere
The center manifold is assumed as follows: where parameters and can be attained through the same way employed in the first part of this subsection. The following reduced system on center manifold is obtained after some simplifications where such that .
Now, it necessary to define the next vector valued functionswhere The exact expression for these vector valued functions is omitted for brevity. Now, following the procedure described in , we assume that the following relation holds between unfolding parameters , and perturbation values
The two real linearly independent generalized eigenvectors and that satisfy and can be found as Also, for the transposed matrix , there exist eigenvectors and that have the properties and they are computed as
After some calculations, the linear part coefficients of (39) can be evaluated as Then, the transversality condition is verified by evaluating which implies that transversality condition is fulfilled. The other coefficients of (39) can be evaluated by solving a set of systems of linear equations described in  which results in By substituting from (39) into (40) and comparing the coefficients of , , on both sides of (40), the unfolding parameters and can be obtained in the following expression
The system representing the universal unfolding of BT bifurcation is thus written asLet and , then apply the following rescaling of coordinates and time to further simplify system (48) After simplification, we can obtainwhere derivative is considered with respect to
Theorem 3. The bifurcation diagram of (50) in plane involves the following bifurcation curves:
(v) , where and where represents a pitchfork bifurcation of system (50), denotes Hopf bifurcation of the origin equilibrium point, refers to the Hopf bifurcation of the nontrivial equilibrium points, corresponds to Homoclinic connection to the origin, and finally denotes the saddle-node bifurcation of symmetric periodic orbits.
The phase portraits of system (1) are obtained via numerical simulations and illustrate examples of different dynamics that can be exhibited by the hyperchaotic system. In particular, Figure 1 verifies the occurrences of Hopf and pitchfork bifurcations of system (2) whereas Figure 2 shows 2D and 3D portraits of hyperchaotic attractor of the system.
Bifurcation diagrams are used to provide useful illustrations for variations of a particular system dynamics with changes in its parameters. Figure 3 shows examples of bifurcation diagrams of system (1) with respect to different parameters in the system. Figure 4 depicts the associated Lyapunov exponent plots to the cases presented in Figure 3.
4. First Integrals of the System
It is well known that, if the analyzed system has chaotic dynamics, the behaviors of the system are irregular behaviors. So, it is impossible to predict the behavior of the system in a long time scale. Therefore, it is important to ask whether there is any set of parameter values in which system dynamics are regular and that the studied system is integrable. In this section, we provide the question of global analytic integrability investigated for hyperchaotic system (1). In particular, we provide conditions of analytic integrability in hyperchaotic system (1) and obtain the possible first integrals of it.
Theorem 4. System (1) has the following first integrals:(1) when ,(2) when ,(3) when and ,(4) when ,(5) when and , where and are arbitrary constants.
Proof. The function is said to be the first integral of system (1) if it satisfies the conditionSubstituting (1) into (51), we obtainAssume that the solution of (52) takes the formwhere is a constant. Substituting (53) into (52), we obtainLet the solution of (54) take the formSubstituting (55) into (54), we obtainBy derivative (56) twice with respect to , we obtainEquating the coefficient of