Networked Dynamcial Systems: Analysis and SynthesisView this Special Issue
A New Feigenbaum-Like Chaotic 3D System
Based on Sprott N system, a new three-dimensional autonomous system is reported. It is demonstrated to be chaotic in the sense of having positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping, and period-doubling route to chaos are analyzed with careful numerical simulations. The obtained results also show that the period-doubling sequence of bifurcations leads to a Feigenbaum-like strange attractor.
A chaotic system is a nonlinear deterministic system that displays a complex and unpredictable behavior. Since Lorenz found the first chaotic attractor in a smooth three-dimensional autonomous system, considerable research interests have been made in searching for the new chaotic attractors [1–14]. The classic Lorenz system  has motivated a great deal of interest and investigation of 3D autonomous chaotic systems with simple nonlinearities, such as Rösser system , Chen system , and Lü system . These cases are characterized by seven terms and either two quadratic nonlinearities or one quadratic nonlinearity.
Recently, there has been an interest in finding and studying rare examples of simple chaotic systems with fewer terms. For example, in , Sprott proposed nineteen simple chaotic systems. Among which, Sprott A-E systems are characterized by five terms with two nonlinearities, and Sprott F-S systems are characterized by six terms with only one nonlinearity. In , Jafari and Sprott proposed some simple chaotic flows with a line equilibrium. In , Munmuangsaen and Srisuchinwong proposed a new five-term simple chaotic system. In , Wang and Chen studied a chaotic system with six terms and only one stable equilibrium.
Stimulated by the above works, in this paper, we propose a new chaotic system based on Sprott N system . This new system is a three-dimensional autonomous system characterized by six terms but equipped with only one nonlinear term. The chaotic attractor obtained from this new system is also one-band attractor similar to Sprott N system but it is not equivalent to Sprott N system. The obtained results show that there is a period-doubling sequence of bifurcations leading to a Feigenbaum-like strange attractor. This paper is devoted to a more detailed analysis of this new chaotic attractor.
The rest of the paper is organized as follows. In Section 2, we propose the new chaotic system. In Section 3, we study some basic properties of the new system including the dissipativity, equilibrium and its stability, and the existence of Hopf bifurcation. In Section 4, we will give some numerical simulations including bifurcation diagram and Feigenbaum's constant. In Section 5, a brief discussion is given.
2. The Proposed System
In this section, a new chaotic system is proposed in this paper: the autonomy differential equations that describe the system are where , and are parameters. System (1) has a different term with Sprott N system in the third equation, and system (1) has different number of equilibria with Sprott N system, so system (1) is not equivalent to Sprott N system.
For system (1), if we let then we can get that the Lyapunov exponents are and the Lyapunov dimension is defined by where is the largest integer satisfying and . Therefore, Lyapunov dimension of system (1) is . Furthermore, the Poincaré image and power spectrum also show that the system (1) is chaotic, as shown in Figures 1(a)–1(d).
3. Some Basic Properties of the New System
For system (1), we can obtain the following divergence: This means that system (1) is dissipative system when . Previous numerical simulations seem to suggest that solutions of the system are bounded. However, we cannot prove that it is bounded, and if we take initial values large enough, numerical simulations show that the solution of this system cannot be in the basin of attraction of any chaotic attractor.
3.2. Equilibrium and Stability
First, we discuss equilibrium of this nonlinear system.
Remark 1. From , Sprott N system has only one equilibrium. However, system (1) has two equilibria or has no equilibrium under different condition, so system (1) is not equivalent to Sprott N system.
Suppose that system (1) has two equilibria. Then, the Jacobian matrix of system (1) is Thus, the characteristic equation of at is where denotes . From Routh-Hurwitz criterion, all of the roots of (8) have negative real parts if and only if holds. Thus, we have the following.
Theorem 2. Assume that and ; we have the following:(i)if , then equilibrium is stable;(ii)equilibrium is always unstable.
3.3. Existence of Hopf Bifurcation
In this section, we first choose as a bifurcation parameter and investigate the conditions for bifurcating periodic solutions. Under the conditions of Theorem 2, is always unstable, so we only discuss .
Submitting into (9) and taking the derivation of , we have And then, if , we have ; the transversality condition holds.
In the same way, if we chose as a bifurcation parameter, then is a bifurcation value. Submitting into (9) and taking the derivation of , we have And then, if , we have ; the transversality condition holds too. Hence, we have the following.
Theorem 3. Assume that and ; we have the following:(i)if , then when pass through , and system (1) undergoes Hopf bifurcation at equilibrium ;(ii)if , then when pass through , system (1) undergoes Hopf bifurcation at equilibrium .
4. Numerical Simulation
In this section, we give some numerical results to show the basic dynamics of the new chaotic system, which can be summarized in the following phase portraits, Lyapunov exponents, Lyapunov dimensions, bifurcation diagrams, and so on. Because system (1) is dissipative when , we always suppose that in this section.
The following results are obtained by using MATLAB program, the phase graphs are drawn by using ode45 codes, and the initial values of system (1) are selected as .
4.1. Bifurcation Analysis
Figure 2(a) shows a bifurcation diagram versus the parameter and fix , demonstrating a period-doubling route to chaos.
Figure 2(b) shows a bifurcation diagram versus the parameter and fix , demonstrating also a period-doubling route to chaos.
Figure 2(c) shows a bifurcation diagram versus the parameter and fix . But in this case, we can see that there is a period 3 window with . And when , Figure 2(c) still demonstrates a period-doubling route to chaos.
As an example, Figure 3 also demonstrates the gradual evolving dynamical process as varying continuously.
All the above numerical results are summarized in Table 1, which indicate that equilibrium is changed from a stable node-focus to an unstable saddle-focus and equilibrium is always an unstable saddle-focus.
4.2. Feigenbaum’s Constant
Figure 2 shows when or is gradually increased; then, the attractors of system (1) undergo a period-doubling bifurcation which converts period to period attractor, and the values of the parameter which makes a transition to a regime of period are listed in Table 2.
As we see, the behaviour is indicative of the onset of chaos. Then, a number now known as Feigenbaum’s constant can be approximatively computed. The Feigenbaum constant is the limiting ratio of each bifurcation interval to the next between every period doubling and can be calculated according to the following expression : where is the value of the changing parameter which makes a transition to a regime of period . Feigenbaum noticed that the ratio converges rapidly to a constant value as increases.
From the bifurcating values listed in Table 2, we can compute that, for the parameter , and is a mere increase in comparison to Feigenbaum's constant. For the parameter , we have and is a mere decline in comparison to Feigenbaum's constant. It is illustrated, in this case, that a period-doubling sequence of bifurcations leads to a Feigenbaum-like strange attractor.
In this paper, a new chaotic system has been proposed from Sprott N system, but with a different term from Sprott N system. Some basic properties of the new system have been investigated in terms of chaotic attractors, equilibria, Hopf bifurcation, Lyapunov exponents, bifurcation diagram, and associated Poincaré map, as well as Feigenbaum’s constant. There are still abundant and complex dynamical behaviors, and the topological structure of the new system should be completely and thoroughly investigated and exploited. It is expected that more detailed theoretical analysis and simulation investigations about this system will be provided in a forthcoming study.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the anonymous referee for the very helpful suggestions and comments which led to improvements of their original paper. This work is supported by the National Natural Science Foundation of China (no. 11061016), the Science and Technology Department of Henan Province (no. 122300410417), and the Education Department of Henan Province (no. 13A110108).
T. Kathleen, D. Tim, and A. James, Chaos: An Introduction to Dynamical Systems, Springer, New York, NY, USA, 1996.