A New Feigenbaum-Like Chaotic 3 D 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.

Recently, there has been an interest in finding and studying rare examples of simple chaotic systems with fewer terms.For example, in [5], 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 [7], Jafari and Sprott proposed some simple chaotic flows with a line equilibrium.In [8], Munmuangsaen and Srisuchinwong proposed a new five-term simple chaotic system.In [9], 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 [5].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.

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 ∑  =1    ⩾ 0 and ∑ +1 =1    < 0. Therefore, Lyapunov dimension of system (1) is   = 2.1992.Furthermore, the Poincaré image and power spectrum also show that the system (1) is chaotic, as shown in Figures 1(a)-1(d).

Some Basic Properties of the New System
3.1.Dissipativity.For system (1), we can obtain the following divergence: This means that system (1) is dissipative system when  < 0. 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.

Equilibrium and Stability
. First, we discuss equilibrium of this nonlinear system.Let We have the following.
Remark 1. From [5], 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  1,2 .From Routh-Hurwitz criterion, all of the roots of ( 8) have negative real parts if and only if  < 0,  < 2 * < 0 holds.Thus, we have the following.

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,  2 is always unstable, so we only discuss  1 .From (8), we get the characteristic equation of () at  1 as follows: Assume that  0 is a pure imaginary root of (9); then, we have It follows that  =  0 = 0 is a bifurcation value.Submitting  = () into ( 9) and taking the derivation of , we have And then, if  > 0, we have (Re ())/| =0 > 0; the transversality condition holds.

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  < 0, we always suppose that  = −1 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 (−0.201, −0.001, 0.41).As an example, Figure 3 also demonstrates the gradual evolving dynamical process as  varying continuously.

Bifurcation Analysis.
All the above numerical results are summarized in Table 1, which indicate that equilibrium  1 is changed from a stable node-focus to an unstable saddle-focus and equilibrium  2 is always an unstable saddle-focus.

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 2 2 to period 2 2(+1) attractor, and the values of the parameter which makes a transition to a regime of period 2  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 [15]: where   is the value of the changing parameter which makes a transition to a regime of period 2  .Feigenbaum noticed that the ratio   converges rapidly to a constant value  ≈ 4.6692 as  increases.
From the bifurcating values listed in and  3 is a mere 2.65% 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.

Conclusion
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.

Figure 2 (
a) shows a bifurcation diagram versus the parameter  and fix  = 1,  = 1.68, demonstrating a period-doubling route to chaos.

Figure 2 (
b) shows a bifurcation diagram versus the parameter  and fix  = −20,  = 1.68, demonstrating also a period-doubling route to chaos.

Figure 2 (
c) shows a bifurcation diagram versus the parameter  and fix  = −20,  = 1.But in this case, we can see that there is a period 3 window with  ∈ [1.3489, 1.3815].And when  > 1.3815, Figure 2(c) still demonstrates a perioddoubling route to chaos.

Table 1 :
Numerical results for some values of the parameter  with  = −20,  = 1.

Table 2 :
The period-doubling cascade of system (1) versus parameters  and .