# A New Feigenbaum-Like Chaotic 3D System

Authors: Huitao Zhao; Yiping Lin; Yunxian Dai
Journal: Discrete Dynamics in Nature and Society
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1] has motivated a great deal of interest and investigation of 3D autonomous chaotic systems with simple nonlinearities, such as Rösser system [2], Chen system [3], and Lü system [4]. 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 [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 Section2, 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(1)x˙=ay,y˙=x+z2,z˙=b+cx+dz,
where a,b,c, and d 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
(2)a=-20,b=1,c=1.58,d=-1,
then we can get that the Lyapunov exponents are
(3)λL1=0.2420,λL2=0.0067,λL1=-1.2488,
and the Lyapunov dimension is defined by
(4)DL=j+1|λLj+1|∑i=1j‍λli=2.1992,
where j is the largest integer satisfying ∑i=1jλli⩾0 and ∑i=1j+1λli<0. Therefore, Lyapunov dimension of system (1) is DL=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).Chaotic attractor, Poincaré map, and frequency spectrum of system (1).
(a)
(b)
(c)
(d)


## 3. Some Basic Properties of the New System

### 3.1. Dissipativity

For system (1), we can obtain the following divergence:
(5)∇·V=∂x˙∂x+∂y˙∂y+∂z˙∂z=d.
This means that system (1) is dissipative system when d<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.


### 3.2.  Equilibrium and Stability

First, we discuss equilibrium of this nonlinear system.Let(6)ay=0,x+z2=0,b+cx+dz=0.We have the following.(i)
Ifd2+4bc>0, system (1) has two equilibria E1,2(x1,2,0,z1,2), where x1,2=-z1,22, z1,2=(d±d2+4bc)/2c.(ii)
Ifd2+4bc=0, system (1) has only one equilibrium E0(x0,0,z0), where x0=-z02,z0=d/2c.(iii)
Ifd2+4bc<0, system (1) has no equilibrium.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
(7)J(z)=(0a0102zc0d).
Thus, the characteristic equation of J(z) at z* is
(8)λ3-dλ2-aλ+ad-2acz*=0,
where z* denotes z1,2. From Routh-Hurwitz criterion, all of the roots of (8) have negative real parts if and only if a<0,d<2cz*<0 holds. Thus, we have the following.Theorem 2.
Assume thata<0,d<0 and d2+4bc>0; we have the following:(i)
ifbc<0, then equilibrium E1 is stable;(ii)
equilibriumE2 is always unstable.


### 3.3.  Existence of Hopf Bifurcation

In this section, we first chooseb as a bifurcation parameter and investigate the conditions for bifurcating periodic solutions. Under the conditions of Theorem 2, E2 is always unstable, so we only discuss E1.From (8), we get the characteristic equation of J(z) at E1 as follows:
(9)λ3-dλ2-aλ-ad2+4bc=0.
Assume that iω0 is a pure imaginary root of (9); then, we have
(10)ω02=a,dω02=ad2+4bc.
It follows that b=b0=0 is a bifurcation value.Submittingλ=λ(b) into (9) and taking the derivation of b, we have
(11)d(Reλ(b))db|b=0=2a2c-d(a2-ad2).
And then, if c>0, we have d(Reλ(b))/db|b=0>0; the transversality condition holds.In the same way, if we chosec as a bifurcation parameter, then c=c0=0 is a bifurcation value. Submitting λ=λ(c) into (9) and taking the derivation of c, we have
(12)d(Reλ(c))dc|c=0=2a2b-d(a2-ad2).
And then, if b>0, we have d(Reλ(c))/dc|c=0>0; the transversality condition holds too. Hence, we have the following.Theorem 3.
Assume thata<0,d<0 and d2+4bc>0; we have the following:(i)
ifc>0, then when b pass through b0=0, and system (1) undergoes Hopf bifurcation at equilibrium E1;(ii)
ifb>0, then when c pass through c0=0, system (1) undergoes Hopf bifurcation at equilibrium E1.


## 3.1. Dissipativity

For system (1), we can obtain the following divergence:
(5)∇·V=∂x˙∂x+∂y˙∂y+∂z˙∂z=d.
This means that system (1) is dissipative system when d<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.


## 3.2.  Equilibrium and Stability

First, we discuss equilibrium of this nonlinear system.Let(6)ay=0,x+z2=0,b+cx+dz=0.We have the following.(i)
Ifd2+4bc>0, system (1) has two equilibria E1,2(x1,2,0,z1,2), where x1,2=-z1,22, z1,2=(d±d2+4bc)/2c.(ii)
Ifd2+4bc=0, system (1) has only one equilibrium E0(x0,0,z0), where x0=-z02,z0=d/2c.(iii)
Ifd2+4bc<0, system (1) has no equilibrium.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
(7)J(z)=(0a0102zc0d).
Thus, the characteristic equation of J(z) at z* is
(8)λ3-dλ2-aλ+ad-2acz*=0,
where z* denotes z1,2. From Routh-Hurwitz criterion, all of the roots of (8) have negative real parts if and only if a<0,d<2cz*<0 holds. Thus, we have the following.Theorem 2.
Assume thata<0,d<0 and d2+4bc>0; we have the following:(i)
ifbc<0, then equilibrium E1 is stable;(ii)
equilibriumE2 is always unstable.


## 3.3.  Existence of Hopf Bifurcation

In this section, we first chooseb as a bifurcation parameter and investigate the conditions for bifurcating periodic solutions. Under the conditions of Theorem 2, E2 is always unstable, so we only discuss E1.From (8), we get the characteristic equation of J(z) at E1 as follows:
(9)λ3-dλ2-aλ-ad2+4bc=0.
Assume that iω0 is a pure imaginary root of (9); then, we have
(10)ω02=a,dω02=ad2+4bc.
It follows that b=b0=0 is a bifurcation value.Submittingλ=λ(b) into (9) and taking the derivation of b, we have
(11)d(Reλ(b))db|b=0=2a2c-d(a2-ad2).
And then, if c>0, we have d(Reλ(b))/db|b=0>0; the transversality condition holds.In the same way, if we chosec as a bifurcation parameter, then c=c0=0 is a bifurcation value. Submitting λ=λ(c) into (9) and taking the derivation of c, we have
(12)d(Reλ(c))dc|c=0=2a2b-d(a2-ad2).
And then, if b>0, we have d(Reλ(c))/dc|c=0>0; the transversality condition holds too. Hence, we have the following.Theorem 3.
Assume thata<0,d<0 and d2+4bc>0; we have the following:(i)
ifc>0, then when b pass through b0=0, and system (1) undergoes Hopf bifurcation at equilibrium E1;(ii)
ifb>0, then when c pass through c0=0, system (1) undergoes Hopf bifurcation at equilibrium E1.


## 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 d<0, we always suppose that d=-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).


### 4.1. Bifurcation Analysis

Figure2(a) shows a bifurcation diagram versus the parameter a and fix b=1,c=1.68, demonstrating a period-doubling route to chaos.Bifurcation diagram of system (1): (a) b=1 and c=1.68, (b) a=-20 and c=1.68, and (c) a=-20 and b=1.
(a)
(b)
(c)Figure2(b) shows a bifurcation diagram versus the parameter b and fix a=-20,c=1.68, demonstrating also a period-doubling route to chaos.Figure2(c) shows a bifurcation diagram versus the parameter c and fix a=-20,b=1. But in this case, we can see that there is a period 3 window with c∈[1.3489,1.3815]. And when c>1.3815, Figure 2(c) still demonstrates a period-doubling route to chaos.As an example, Figure3 also demonstrates the gradual evolving dynamical process as c varying continuously.Phase portraits of system (1): (a) period 1,  c=0.1, (b) period 2,  c=1.3, (c) period 3,  c=1.36, (d) period 4,  c=1.5, (e) period 8,  c=1.52, and (f) chaotic,  c=1.6.
(a)
(b)
(c)
(d)
(e)
(f)All the above numerical results are summarized in Table1, which indicate that equilibrium E1 is changed from a stable node-focus to an unstable saddle-focus and equilibrium E2 is always an unstable saddle-focus.Table 1
Numerical results for some values of the parameterc with a=-20, b=1.
Parameterc
Eigenvalues ofE1
Eigenvalues ofE2
Lyapunov exponents
D
L
-
0.1
-
0.7813
,
-
0.1094
±
4.4516
i
0.7287
,
-
0.8643
±
4.5291
i
-
0.1091
,
-
0.1096
,
-
0.7813
0
0.1
-
1.1715,0.0857
±
4.4937
i
1.0659
,
-
1.0329
±
4.5973
i
0
,
-
0.2602
,
-
0.7416
1
1.3
-
2.1997,0.5999
±
4.7201
i
1.9381
,
-
1.4691
±
4.8514
i
0
,
-
0.3568
,
-
0.6428
1
1.36
-
2.2312,0.6156
±
4.7295
i
1.9652
,
-
1.4826
±
4.8610
i
0.0005
,
-
0.4998
,
-
0.5007
1.001
1.4
-
2.2517,0.6259
±
4.7357
i
1.9827
,
-
1.4914
±
4.8672
i
0
,
-
0.4990
,
-
0.5009
1
1.5
-
2.3012,0.6506
±
4.7509
i
2.0253
,
-
1.5126
±
4.8825
i
0
,
-
0.1167
,
-
0.8830
1
1.52
-
2.3108,0.6554
±
4.7539
i
2.0336
,
-
1.5168
±
4.8855
i
0
,
-
0.0081
,
-
0.9916
1
1.6
-
2.3484,0.6742
±
4.7657
i
2.0660
,
-
1.5330
±
4.8974
i
0.2821,0.0002
,
-
1.2823
2.2201


### 4.2. Feigenbaum’s Constant

Figure2 shows when a or b is gradually increased; then, the attractors of system (1) undergo a period-doubling bifurcation which converts period 22k to period 22(k+1) attractor, and the values of the parameter which makes a transition to a regime of period 2k are listed in Table 2.Table 2
The period-doubling cascade of system (1) versus parameters a and b.
Parametera
Parameterb
Bifurcation
-
28.5583
0.7908
Period-1 to Period-2
-
23.6810
0.9243
Period-2 to Period-4
-
22.8528
0.9545
Period-4 to Period-8
-
22.6984
0.9595
Period-8 to Period-16
-
22.6652
0.9606
Period-16 to Period-32As 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]:
(13)δk=Pk-Pk-1Pk+1-Pk,
where Pk is the value of the changing parameter which makes a transition to a regime of period 2k. Feigenbaum noticed that the ratio δn converges rapidly to a constant value δ≈4.6692 as n increases.From the bifurcating values listed in Table2, we can compute that, for the parameter a,
(14)δ3=-22.6984+22.8528-22.6652+22.6984≈4.7950,
and δ3 is a mere 2.69% increase in comparison to Feigenbaum's constant. For the parameter b, we have
(15)δ3=0.9595-0.95450.9606-0.9595≈4.5455,
and δ3 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.


## 4.1. Bifurcation Analysis

Figure2(a) shows a bifurcation diagram versus the parameter a and fix b=1,c=1.68, demonstrating a period-doubling route to chaos.Bifurcation diagram of system (1): (a) b=1 and c=1.68, (b) a=-20 and c=1.68, and (c) a=-20 and b=1.
(a)
(b)
(c)Figure2(b) shows a bifurcation diagram versus the parameter b and fix a=-20,c=1.68, demonstrating also a period-doubling route to chaos.Figure2(c) shows a bifurcation diagram versus the parameter c and fix a=-20,b=1. But in this case, we can see that there is a period 3 window with c∈[1.3489,1.3815]. And when c>1.3815, Figure 2(c) still demonstrates a period-doubling route to chaos.As an example, Figure3 also demonstrates the gradual evolving dynamical process as c varying continuously.Phase portraits of system (1): (a) period 1,  c=0.1, (b) period 2,  c=1.3, (c) period 3,  c=1.36, (d) period 4,  c=1.5, (e) period 8,  c=1.52, and (f) chaotic,  c=1.6.
(a)
(b)
(c)
(d)
(e)
(f)All the above numerical results are summarized in Table1, which indicate that equilibrium E1 is changed from a stable node-focus to an unstable saddle-focus and equilibrium E2 is always an unstable saddle-focus.Table 1
Numerical results for some values of the parameterc with a=-20, b=1.
Parameterc
Eigenvalues ofE1
Eigenvalues ofE2
Lyapunov exponents
D
L
-
0.1
-
0.7813
,
-
0.1094
±
4.4516
i
0.7287
,
-
0.8643
±
4.5291
i
-
0.1091
,
-
0.1096
,
-
0.7813
0
0.1
-
1.1715,0.0857
±
4.4937
i
1.0659
,
-
1.0329
±
4.5973
i
0
,
-
0.2602
,
-
0.7416
1
1.3
-
2.1997,0.5999
±
4.7201
i
1.9381
,
-
1.4691
±
4.8514
i
0
,
-
0.3568
,
-
0.6428
1
1.36
-
2.2312,0.6156
±
4.7295
i
1.9652
,
-
1.4826
±
4.8610
i
0.0005
,
-
0.4998
,
-
0.5007
1.001
1.4
-
2.2517,0.6259
±
4.7357
i
1.9827
,
-
1.4914
±
4.8672
i
0
,
-
0.4990
,
-
0.5009
1
1.5
-
2.3012,0.6506
±
4.7509
i
2.0253
,
-
1.5126
±
4.8825
i
0
,
-
0.1167
,
-
0.8830
1
1.52
-
2.3108,0.6554
±
4.7539
i
2.0336
,
-
1.5168
±
4.8855
i
0
,
-
0.0081
,
-
0.9916
1
1.6
-
2.3484,0.6742
±
4.7657
i
2.0660
,
-
1.5330
±
4.8974
i
0.2821,0.0002
,
-
1.2823
2.2201


## 4.2. Feigenbaum’s Constant

Figure2 shows when a or b is gradually increased; then, the attractors of system (1) undergo a period-doubling bifurcation which converts period 22k to period 22(k+1) attractor, and the values of the parameter which makes a transition to a regime of period 2k are listed in Table 2.Table 2
The period-doubling cascade of system (1) versus parameters a and b.
Parametera
Parameterb
Bifurcation
-
28.5583
0.7908
Period-1 to Period-2
-
23.6810
0.9243
Period-2 to Period-4
-
22.8528
0.9545
Period-4 to Period-8
-
22.6984
0.9595
Period-8 to Period-16
-
22.6652
0.9606
Period-16 to Period-32As 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]:
(13)δk=Pk-Pk-1Pk+1-Pk,
where Pk is the value of the changing parameter which makes a transition to a regime of period 2k. Feigenbaum noticed that the ratio δn converges rapidly to a constant value δ≈4.6692 as n increases.From the bifurcating values listed in Table2, we can compute that, for the parameter a,
(14)δ3=-22.6984+22.8528-22.6652+22.6984≈4.7950,
and δ3 is a mere 2.69% increase in comparison to Feigenbaum's constant. For the parameter b, we have
(15)δ3=0.9595-0.95450.9606-0.9595≈4.5455,
and δ3 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.


## 5. 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.



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