A Novel Megastable Hamiltonian System with Infinite Hyperbolic and Nonhyperbolic Equilibria
The dimension of the conservative chaotic systems is an integer and equals the system dimension, which brings about a better ergodic property and thus have potentials in engineering application than the dissipative systems. This paper investigates the phenomenon of megastability in a unique and simple conservative oscillator with infinite of hyperbolic and nonhyperbolic equilibria. Using traditional nonlinear analysis tools, we found that the introduced oscillator possesses an invariable energy and displays either self-excited or hidden dynamics depending on the stability of its equilibria. Besides, the conservative nature of the new system is validated using theoretical measurement. Furthermore, an analog simulator of the oscillator is built and simulated in the PSpice environment to confirm that the previous results were not artifacts.
Considerable efforts have been made recently by several authors in designing chaotic systems with striking features linked to their equilibria [1–7]. These dynamical systems include some with no equilibrium point , stable equilibria [1, 9–11], infinite equilibria [12–17], and unstable equilibria [2, 18, 19]. The latter belongs to self-excited nonlinear dynamical systems, while the formers are new and classified as systems with hidden attractors [20–22]. Several systems with hidden attractors are linked to multistability which refers to the coexistence of multiple attractors for the same group of parameters [23–25]. Indeed, a system may approach different stable states starting from different initial values. When the system basin of initial conditions is associated with unstable equilibrium points, one may find countable coexisting attractors. However, when the system has a great deal of the coexisting attractors, the phenomenon is defined as extreme multistability. Despite the intensive efforts in reporting systems with hidden attractors, most of the attention so far was oriented only on structure and characteristic of their equilibria , while other important specificities including shape and topology of the strange attractors were rarely considered. Few examples are chaotic systems with varying symmetry , chaotic systems with conditional symmetry [28–31], multiscroll attractors [32–34], attractor growing [35, 36], algebraically simple equations , and nested megastable attractors [38–45].
Now in forefront of investigations, megastable systems are a special class of multistable systems with nested attractors . First reported by Sprott and coauthors in , such systems are now widely investigated [38–45]. Although the phenomenon of megastability in chaotic systems was up to date linked to additional external excitation [46, 47], the recent work by Kathikeyan and colleagues demonstrates that the above condition is not mandatory . However, very few megastable systems reported so far are conservative. Conservative chaotic flow is a category of incompressible chaos usually generated from dynamical systems without dissipation. Based on a long history and formalism from Euler, Lagrange Hamilton, and Jacobi , the main characteristic of such systems is that they do not have attractors because their phase space volumes are conserved . However, such systems can exhibit complicated dynamical behaviors including chaos . The dimension of the conservative chaotic systems is an integer and equals the system dimension, which brings about a better ergodic property than the dissipative system. Consequently, the probability distribution associated with conservative chaos is relatively flat like the uniform distribution of white noise, whereas dissipative chaos has either a single peak or multiple peaks, from which the main frequency characteristics can be identified. Clearly, the information encrypted based on the former can be decrypted or attacked based on frequency reconstruction  while the latter (conservative system) is adapted . Therefore, with both having the same bandwidths, the conservative chaotic system is more suitable as a pseudorandom number generator than the dissipative chaotic system. Owing to the importance of conservative/Hamiltonian systems in the mechanical domain, we propose in this work an extremely simple megastable Hamiltonian system with its circuitry implementation.
It is observed from Table 1 that no 2D conservative system with infinite hyperbolic and nonhyperbolic equilibria and few terms (i.e., three) was reported in the literature. Also, Table 2 highlights the megastability (i.e., coexistence of infinite countable attractors) property of some simple conservative systems proposed within this work. To the best of the authors’ knowledge, no such simple conservative 2D system with both megastability and multistability was yet reported in the relevant literature so far and thus deserves dissemination.
The organization of this paper is as follows. Section 2 deals with the introduction and description of the new conservative megastable system. Preliminary analysis including investigations of steady states and their nature is presented. Then, Section 3 is dedicated to the conservative feature/property of the new system with two different methods, namely, divergence and Hamiltonian. In Section 4, numerical investigations of the forced derivative of the newly introduced system are performed based on well-known nonlinear dynamical tools including bifurcation diagrams, Lyapunov exponents, and phase portraits. In Section 5, an experimental study is carried out. The results of the corresponding electronic circuit implemented in the PSpice simulation environment show a good agreement with obtained numerical investigations. The last section concludes the paper.
2. The New Simple Conservative Megastable Oscillator
Consider the following two-dimensional (2D) dissipative nonlinear oscillator which is investigated in :
Inspired by system (1), we introduce the following 2D new and simple conservative nonlinear system:
In this paper, the most following elegant  set of parameters (, , and ) are used in the detail investigation of the new conservative megastable system in order to uncover exciting and interesting dynamical behaviors. The equilibrium point or fixed points of system (2) can be calculated by equating to zero the right hand side of system (2). Therefore, we obtained the following transcendental equation:
The graphical representation of function in the interval [−140, 140] is shown in Figure 1. According to this graph, one can see that there are infinite values of which satisfy equation (3) when the range of the graphical representation is increased. Therefore, the new conservative system (2) has infinite fixed points, that is, with .
The Jacobian matrix for this system in any equilibrium points is
Then, the eigenvalues can be calculated accordingly as
For , system (2) has infinite hyperbolic equilibria:
For , system (2) has infinite nonhyperbolic equilibria:
One can observe that the eigenvalues of the characteristic equation strongly depend on the fixed points . Table 3 shows an example of the two eigenvalues of equation (5) for some equilibrium points, that is, , , , , , and . Based on the analysis provided in Table 3, we concluded that the points , and are examples of hyperbolic fixed points, while , and are examples of nonhyperbolic fixed points. Using the previous parameter set, we carried out the numerical calculation of system (2). Some interesting and rare trajectories in this system are detected (see Figure 2). In particular, Figure 2 represents a plot of fourteen coexisting nested limit cycles in system (2). Remember that they are just an example from the infinite set of nested limit cycles around the x-axis and can be obtained under other proper initial conditions.
3. Conservative Nature of the New Oscillator
3.1. Conservative Using Divergence Property
The nonlinear dynamical system (2) can be expressed in vector notation aswhere is a state vector and is the smooth function given by
The divergence of the vector field for nonlinear system (6) can be calculated as
As a result, the megastable system (2) has a conservative nature.
Next, suppose that system (2) is expressed as . Let be any space in with a smooth boundary/surface and also , where is the flow of . Also, consider the volume element of in the phase space. The volume element into a new smooth boundary in an infinitesimal time is given by and can be obtained accordingly as where represents the area of the smooth boundary and is the outward normal apply on the surface . It is clear from (10) that
Thus, by using the Ostrogradsky theorem (i.e., divergence theorem) in (11), we getwhere is the divergence of .
From (13), we get
3.2. Conservative Nature Using Hamiltonian-Based Analysis
The electrical activity of the systems is related to their energy modification . Recently, the Hamiltonian energy function-based method is employed for the study of chaotic systems. Some technics to find the Hamilton energy function in chaotic systems have been presented . In particular, a general procedure for investigation of generalized Hamiltonian realization of an n-dimensional autonomous dynamical system is reported .
Consider the following dynamical system:where and is a smooth function so that . The dynamical system (15) can be written in a generalized Hamiltonian form [63, 64] aswhere is the local structure matrix and is the gradient vector of a smooth energy function . For Hamiltonian system, is a skew-symmetric matrix which satisfies the Jacobian identity. Otherwise, where is not skew-symmetric, the system is generalized Hamiltonian. However, for a generalized Hamiltonian system, can be divided as a sum of one skew-symmetric matrix plus one symmetric matrix . Therefore, equation (16) can be written accordingly as
As a result, and can be obtained as follows:
According to Helmholtz’s theorem, system (15) can be regarded as the vector field to discuss the energy problems and can be decomposed into the rotational tensor and gradient vector , that is,thus
For to be a skew-symmetric matrix,where is the Hamiltonian function and is the gradient vector of defined by
The new 2D conservative autonomous system (2) can be rewritten aswhere
Since , it means that there is no dissipation term in system (2) and trajectories are corresponding to the unique conservative field . That is, when the trajectories begin from an isosurface of the constant Hamilton energy , they cannot escape from the isosurface [57, 58].
According to (22), the Hamilton energy function will satisfy the partial differential equation given by
So, a general solution of (25) can be approached by
The time derivative of (26) can be calculated by
Since the time derivative of Hamiltonian energy of system (2) is zero, this Hamilton energy function is invariable. This justifies the conservative nature of the new system (2). Thus, the conservative property of system (2) is validated using theoretical measures.
4. The Forced Chaotic Oscillator
In the following sections, the dynamical analysis of system (28) is discussed using various numerical tools including two-parameter Lyapunov exponents, Lyapunov spectrum plots, and phase trajectories. Here, the main objective is finding possible domains of chaotic attractors in (28). In this regard, several combinations of can be used to obtain chaos. For more simplicity, we choose , and we consider the same set of system parameter as in the previous section. Therefore, the dynamics of the new system (28) is investigated in terms of .
4.1. Two-Parameter Lyapunov Exponent Analysis and Multistability
The dynamical regions of the simple Hamiltonian megastable system are drawn in this part by means of two-parameter Lyapunov exponent analysis when varying the control parameter and in the appropriate domain. However, the global dynamic behaviors of the system can be well quantified exploiting two-parameter Lyapunov exponents by adjusting simultaneously these two last parameters of the 2D system (28) through the establishment of appropriate colorful diagrams. The obtained colorful diagrams are produced by the Lyapunov exponent spectrum on a grid of values of the two control space parameters (i.e., and ). The main advantage of the two-parameter Lyapunov exponent is that the complexity of the system can be directly analyzed on the variation of two parameters, which offers a major advantage over the traditional exponent which is defined on a single parameter. To properly obtain this result, system (28) is numerically solved using the standard fourth-order Runge–Kutta integration method with a fixed step size equal to . We also used steps to compute the Lyapunov exponent (LE). Figure 3 shows the two-parameter Lyapunov exponents in the parameter space using two different initial conditions. Periodic and quasiperiodic regions are indicated by blue and cyan colors, respectively, while chaotic oscillations are materialized by yellowish-reddish color. The graphs are obtained by increasing both control parameters. From these graphs, one can observe the regions of coexisting behaviors (i.e., hysteretic dynamics) characterized by the presence of different colors in the map. To better highlight the presence of multiple coexisting attractors, we have plotted the trajectories showing different coexisting strange attractors of system (28) for and as shown in Figure 4.
4.2. Infinite Number of Coexisting Nested Strange Attractors
For more simplicity, we set and consider as a control parameter of system (28). Figure 5 shows the Lyapunov spectrum plots when parameter is varied from to with initial conditions . One can note from this diagram that the forced version of new system (2) presents a very large range of chaotic dynamic provided that in the region , , , and . However, the conservative nature of the new two-dimensional megastable system is also appreciated because [55, 56, 65]. That is, the maximum value of LE and the minimum value are symmetric with respect to . When ICs are and , the corresponding LEs are , , and .
However, and . The Kaplan–Yorke dimension (Lyapunov dimension) is defined aswhere is the largest integer satisfying and . Thus, provided that .
Furthermore, using four different initial conditions, the nested coexisting bifurcation diagrams of system (28) are plotted and superimposed as shown in Figure 6. The graphs represent the plot of local maximal values of in terms of the control parameter that is varied (i.e., increased) in the range . The presence of four different coexisting data in Figure 6 further confirms the complicated dynamical behaviors observed in this work. As a result, the presence of infinite number of nested attractors for is depicted in Figure 7(a) and the zoom (i.e., attractors in the red box of Figure 7(a)) is shown in Figure 7(b). Note that these plots are just examples in an infinite number of stable states around the x-axis, which could be found under other proper initial conditions.
5. Circuit Implementation
In this part of our work, an analog circuit is constructed and used to make a comparison between the theoretical/numerical results obtained previously and the experimental results. The circuit implementation defines the analog circuit of our model, which confirms a possibility of laboratory measurement. Unlike other simulation software, PSpice remains the best software by its capacity to integrate the transient time. The circuit diagram that allows us to perform the various simulations in the PSpice software is presented in Figure 8. This electronic implementation of the model is very singular because the model is built based only on linear and trigonometric terms. The circuit of the new system is designed using two capacitors and ten resistors including five op-amps TL082CD. In addition, it uses two sine blocs, one AC source, and a symmetric power supply. The circuit equation using Kirchhoff’s electrical circuit laws can be obtained as
Set , , and except , , , , and ; adopt the rescale of time and variables and .
, , , , , and with .
The dynamics of the circuit provided in Figure 8 is simulated in PSpice as provided in Figure 9. This obtained result agrees with the one obtained based on Pascal simulation. From this figure, it can be seen that some of the coexisting attractors of the new megastable oscillator have been captured and support the fact that the obtained results were not artifacts.
6. Concluding Remark
This paper was focused on the dynamical analysis of the simple 2D autonomous system with infinite of hyperbolic and nonhyperbolic equilibria reported to date. The investigations of the Hamiltonian and dissipation properties of the model reveal that it is conservative and displays the coexistence of a large number of stable states. Equally, it is demonstrated under the presence of a sinusoidal excitation that the proposed model exhibits the striking phenomenon of megastability characterized by the coexistence of an infinite number of countable stable states. Finally, the electronic implementation of the introduced model has been provided using PSPice simulations to further support our results. In the future, it would be interesting to carry out the analytical methods which can predict megastability behaviors in a nonlinear system and address the megastable system-based secure communication.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
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