A novel chaotic system and its topological horseshoe

Chunlai Li a1, Lei Wu, Hongmin Li , Yaonan Tong c College of Physics and Electronics Hunan Institute of Science and Technology Yueyang 414006, China hnistlichl@163.com Department No. 6 of Air Force Paratrooper College Guilin 541003, China 5950595@qq.com College of Information and Communication Engineering Hunan Institute of Science and Technology Yueyang 414006, China lihmin171@sina.com; pjtyn@163.com


Introduction
In 1963, Lorenz discovered a chaotic system when studying the atmospheric convection [1].As the first chaotic model, Lorenz system has become a paradigm for chaos investigation [2][3][4][5][6][7].Many three-dimensional chaotic systems have been proposed by developing Lorenz chaotic system, such as Chen system [4], Liu system [5], Qi system [6], Lü system [7].Due to powerful applications in chemical reactions, cryptology, nonlinear circuits, secure communication and so on, researchers have paid great attention to generate new chaotic systems and analyse their dynamical behaviors and dynamical properties.Generally speaking, for these presented chaotic systems, the Lyapunov exponent spectrums vary gradually and rang from stable equilibrium points, periodic orbits to chaotic oscillations with the changing of system parameters.
As a basic and striking theory in chaotic dynamics, topological horseshoe with symbolic dynamics provides a powerful tool in rigorous studies of chaos in dynamical systems.Up till now, remarkable theoretical progress has been made in seeking sufficient conditions for the existence of horseshoes.Kennedy introduced an important chaos lemma which proposed a topological horseshoe theory in continuous map [8,9].Yang obtained another concernful criteria to find the topological horseshoe in non-continuous map [10,11], which have been applied successfully to some practical dynamical systems to present computer-assisted verification of chaos [12][13][14][15].Recently, Li introduced a new method for finding horseshoes in chaotic systems by using several simple results on topological horseshoes [16].However, it is still a challenge for researchers to seek a topological horseshoe in practical chaotic systems.
This paper proposes a new 3D chaotic system based on the construction pattern of Chen, Liu and Qi chaotic systems.Some basic dynamical characters of the presented chaotic system, such as phase portraits, equilibrium points, bifurcation diagram and Lyapunov exponents are investigated.It's found that with the variation of parameters a, b and c, the novel system can occur period doubling bifurcations.Of particular interest is that the Lyapunov exponent spectrum keeps invariable when the product term parameter e changes.When parameter e increases gradually, the amplitude of the signals x 1 , x 2 varies by the power function with a minus half index, but the third one keeps its amplitude in the same range.So we can control the signal amplitude artificially by adjusting e, but the dynamical system is always chaotic.Therefore, it can be concluded that this chaotic system has a more complicated dynamics.Finally, based on the topological horseshoe theory [16], we carefully pick a suitable cross-section with respect to the attractor, and find a topological horseshoe of the corresponding first-returned Poincaré map, thus giving a rigorous confirmation of the chaos existed in this dynamical system.

The proposed 3D chaotic system
In the light of construction pattern of Chen, Liu and Qi chaotic systems, a new 3D chaotic system is proposed from Lorenz chaotic system: where x 1 , x 2 , x 3 are the state variables, a, b, c, e are the positive parameters.

Equilibria and analysis of stability
In order to obtain the equilibria of system (1), let's suppose Then, if 2c − a 0, we get three equilibria of system (1): and if 2c − a < 0, there are five equilibria in system (1): Linearizing system (1) at any equilibrium (x 10 , x 20 , x 30 ), it yields the corresponding Jacobian matrix The characteristic equation is obtained as follows: where According to the Routh-Hurwitz criterion, only when C 2 > 0, C 1 > 0, C 0 > 0 and C 2 C 1 − C 0 > 0, the real parts of all the roots are negative.Thus, there are three unstable equilibria in system (1) when a = 40, b = 5, c = 30, e = 3.

Topological equivalence
denote the Lorenz (Chen, Liu or Qi) system.And the proposed system (1) is described by Definition 1. (See [17].)If systems (4) and ( 5) are said to be topological equivalent, then there would exist a diffeomorphism y = T (x), such that where J(x) = dT (x)/dx is the Jacobian matrix of T at the point x.
Let x 0 and y 0 = T (x 0 ) be the corresponding equilibria of g(x) and f (x), A(x 0 ) and B(y 0 ) denote the Jacobians of g(x) and f (x), respectively.If (4) and ( 5) are topological equivalent, then A(x 0 ) and B(y 0 ) are similar, i.e., their characteristic polynomials and eigenvalues should coincide.
Based on the concept and techniques of the equilibrium and resultant eigenvalue, we know that parameter e does not contribute to the eigenvalues of system (1) (see Section 3.3 for concrete analysis), and it is easy to actually verify that the system (5) with any parameter set is not smoothly equivalent to system (4).Therefore, systems (4) and ( 5) are not topological equivalent, although both have similar attractors.
C. Li et al.

Complex dynamics of novel 3D system by varying each parameter
This system has been found to exhibit complex dynamical behaviors by varying each parameter in a wide range, which is expatiated as below.Another interesting phenomenon is that, with the increasing of e, the signal amplitude adjusts in certain pattern.Speaking concretely, with the increasing of e, the amplitude of x 1 , x 2 decreases nonlinearly, but the amplitude of x 3 keeps in the same range.

Analysis of constant Lyapunov exponent spectrum
As argued above, the Lyapunov exponent spectrum keeps invariable with the varying of e.In fact, when putting the equilibrium in the coefficient expression (3), we can eliminate the influence of e.For example, when inserting equilibrium P 0 into (3), we obtain and when substituting equilibrium P 1 or P 2 into (3), we get where B 0 = (c − 2a) + (c 2 + 4ac) 0.5 .Therefore, the eigenvalues relating to characteristic equation ( 2) are irrespective parameter e, and when parameter e varies, the Lyapunov exponent spectrum remains constant.

Analysis of amplitude control
It's known from the forementioned analysis, with the increasing of e, the signal amplitudes vary according to some criteria.
Theorem 1.The coefficient e of the square term x 2 2 is a local parameter of nonlinear amplitude adjuster, and with the increasing of e, the amplitude of x 1 , x 2 changes by the power function with an index of −1/2, but the third one keeps its amplitude in the same range.
therefore, the amplitude adjuster of signals x 1 , x 2 , x 3 correspond to the variety of parameter e, and when e increases successively, the amplitude of x 1 , x 2 changes by the power function with an index of −1/2, but the amplitude of x 3 keeps in the same range.As a numerical explanation, we consider the influence of parameter e on the amplitude of x 1 , x 2 , x 3 , as described in Fig. 4. Clearly, when e = 2, the amplitude of signals x 1 , x 2 is two times as large as the one with e = 4; and the amplitude of x 3 is same as the case with e = 4.

Review of topological horseshoe theorem
Before introducing the results on topological horseshoe, we should first recall the concept of m-shift map.
Let S m = {1, 2, . . ., m} be the set of non-negative integer from 1 to m.And let space m be the collection of all one-sided infinite sequences with the elements S m , i.e., each element s of m is of the following form: Now let's consider another sequence s(s ∈ S m ).Then the distance between s and s can be defined as follows: .
With the distance defined as above equation, m is a metric space.A m-shift map σ : m → m is defined as As a dynamical system defined on m , the shift map σ has: (a) a countable infinity of periodic orbits consisting of orbits of all periods; (b) an uncountable infinity of nonperiodic orbits; (c) a dense orbit.
A consequence of these three properties is that the dynamics generated by the shift map σ displays sensitive dependence on the initial conditions, therefore is chaotic.
Let S be a separable metric space and E ⊂ S is a compact subset, and there exist m mutually disjoint compact subsets E 1 , E 2 , . . ., E m of E. f : E → S is a map and f | E i , the restriction of f to each E i , is continuous.
i and γ ∩ E 2 i are nonempty and compact, we say a connected subset γ of E i connect E 1 i and E 2 i , and we denote this by (See [18].)Let γ be a connected subset of E i , if there exists a connected subset i and E 2 i .In this case, we denote it by f (γ) → E i .Definition 4. Let S and m be two topological spaces, f : S → S and g : m → m are continuous functions.If there exists a continuous surjection h : m → S such that foh = hog , we say that f is topologically semiconjugate to g .Lemma 1. (See [16].)Let m be a positive integer, if Theorem 2. (See [16].)If f p (E 1 ) → E 1 , f p (E 1 ) → E 2 and f q (E 2 ) → E 1 , then a compact invariant set Λ ⊂ E will be found, such that f 2p+p | Λ is semiconjugate to 2-shift dynamics, and we have ent(f ) (q + 1/2p) log 2 , here ent(f ) denotes the topological entropy of f .

Topological horseshoe in the novel chaotic system
It is no easy job to seek topological horseshoes in a dynamical system, especially to select a suitable polygon in the cross-section.In this section we will search the horseshoe existed in system (1) by using the topological horseshoe theorem in Section 4.
Precisely, let d1r and d1l denote the right and left sides of E 1 , respectively.Numerical simulations show that, under the map H, the image H(x)(x ∈ E 1 ) lies wholly across the quadrilateral E 1 , and H(d1r) lies on the left side of A 1 B 1 , H(d1l) lies on the right side of C 1 D 1 .For the details, see Fig. 6.
Similarly, we suppose d2r and d2l denote the right and left sides of E 2 , respectively.The first return Poincaré map H(E 2 ) is described as Fig. 7. Obviously, H(d2r) lies on the right side of C 1 D 1 , H(d2l) lies on the left side of A 1 B 1 , therefore, the image H(x)(x ∈ E 2 ) is wholly across the quadrangle E 1 .
Thus, according to Theorem 2, there exists a compact invariant set Λ ⊂ E, such that H 3 | Λ is semiconjugate to 2-shift dynamics, and ent(H) 1.5 log 2 > 0. Thereby it indicates that the attractor depicted in Fig. 5 is chaotic.

Conclusion
In this paper, a new chaotic system has been proposed from Lorenz system.Some basic properties of the presented chaotic system have been studied in terms of phase portraits, equilibrium points, bifurcation diagram and Lyapunov exponent spectrum.Careful investigation reveals that with the variation of parameters a, b and c, a period-doubling sequence of bifurcations leads to chaos.An interesting phenomenon is that the Lyapunov exponent keeps constant while parameter e varies.And with the increasing of parameter e, the amplitude of the signals x 1 , x 2 change by the power function with an index of −1/2, but the third one keeps in the same range.Finally, the horseshoe chaos in this system is investigated based on the topological horseshoe theory.

3. 1
Dynamics of novel system by varying parameters a, b, c Let a, b, c vary in some region, respectively, but the values of other parameters are set as in Section 2.2.The corresponding bifurcation diagrams are shown in Fig. 2. It's known from Fig. 2, that system (1) has a rich dynamical behavior, ranging from equilibrium points, periodic orbits to chaos, depending on the parameter values.

Fig. 4 .
Fig. 4. Phase portrait and time series of system (1) with different e: (a) phase portrait; (b) time series.