Simple hyperchaotic memory system with large topological entropy

. The memory elements have been the research hot spot for a long time. However, there is a litter works on the linear memory element. This paper presents a study of a new memory system containing a linear memory element. The study shows that the system not only has two kinds of route to hyperchaos, but also exists many kinds of coexisting attractors in the phase space. Moreover, the system can generate more complex hyperchaotic behaviors. To prove it, we ﬁnd a new kind of topological horseshoes with two-directional expansions that consist of three disconnected compact sets. This new kind of horseshoes suggests that the topological entropy of the hyperchaotic attractor is larger than other hyperchaotic attractors reported before. For detailed study of the hyperchaotic invariant set, we also demonstrate a method to extract the orbits from the hyperchaotic horseshoes.


Introduction
The memory circuit element is a device with memory effect [6]. Because of the ability of storing readable information, the memory element is regarded to have potential phenomenon is meaningful for the memory system. We know that the memory patterns correspond to the attractors in mathematics. The coexistence of multi-attractors implies that the linear memory element can bring the system much larger capacity of memory than we thought before.
Third, we propose a general method based on mapping to extract the orbits in the topological horseshoes. To the knowledge of us, this is the first work that implements the mapping from symbolic sequences to orbits. This method provide a practical tool for observing and studying the construction of the hyperchaotic invariant set.
The rest of this paper is organized as follows. Section 2 introduces the new hyperchaotic system. Section 3 presents the bifurcation analysis. Section 4 carries out the topological entropy computation. Section 5 gives a method to extract the orbits in the horseshoes. Section 6 draws conclusion.

The proposed hyperchaotic memory system
In 2006, Qi et al. [19] proposed the Qi system, which was constructed by adding a crossproduct nonlinear term into the Lorenz system. It is defined aṡ Typically, in parameter setting a = 35, b = 7 and c = 25, the system is chaotic.
In this paper, for generality, we do not select the most used memristor, but adopt the more general model, namely, the linear memory element. According to [3], this kind of memory element can be defined by the following equations: h(t) = 1 + w(t) y(t).
Here y(t) and h(t) are input and output of a memory element, respectively; w(t) is the internal state variable. Obviously, if y(t) and h(t) correspond to voltage and current of a two-terminal electronic element, respectively, the relations of (2) define a flux-controlled memristor with linear memductance 1 + w(t).
Based on the chaotic system and the linear memory element above, a system containing the linear memory element can be constructed. Its dynamical equation is presented bẏ where α = 47, β = 28, γ = 3, h = (1 + w)y and k is a positive system parameter that indicates the strength of the linear memory element. When taking k = 0.5, system (3) is in the state of hyperchaos. The corresponding attractor, Poincaré maps and power spectra are shown in Figs. 1, 2 and 3, respectively.   Obviously, system (3) keeps the symmetry of the Qi system, namely, it is invariant under the transformation (x, y, z, w) ↔ (−x, −y, z, −w). Since the divergence of flow ∇V = ∂ẋ/∂x + ∂ẏ/∂y + ∂ż/∂z + ∂ẇ/∂w = −53 < 0, system (3) is dissipative and exists an attractor in its phase space.
In the following, the stability of the equilibrium points is presented. Firstly, let the right-side of Eq. (3) to be 0 and obtain By solving the algebraic equations above, it is found that the new system has infinite number of equilibrium points. We denote the equilibrium points as O = (0, 0, 0, s), in which s is an arbitrary real number.
And then, by linearizing system (3), the Jacobian matrix on O is obtained: Since the eigenvalues of J O are λ 1 = −3, λ 2 = 0, λ 3 = −66.9535, λ 4 = 18.9535, the equilibrium points of system (3) are unstable. Moreover, their stability is not related to the strength of the feedback k and the value of s.

Bifurcation analysis 3.1 Complex dynamical behaviors
In most works, the Lyapunov exponents are used as the indicator of systems state.
5) Meanwhile, it is can be seen that the LEs are discontinuous for 0.053 < k 0.159, 0.181 < k 0.198 and 0.426 < k 0.467, which implies that there exist multi-attractor. Using the method of attraction basin, we find out that there are many kinds of coexisting   attractors including a) coexisting hyperchaos and torus, b) coexisting limit cycle and hyperchaos, c) coexisting limit cycle and torus, d) coexisting limit cycle and limit cycle, e) coexisting torus and torus. These findings are listed in Table 1

The routes to hyperchaos
In order to observe the routes to hyperchaos, we draw the bifurcation diagram of system (3) when 0 k 0.64 as shown in Fig. 11. In the figure, to eliminate the transient behaviors, the first 30000 points are canceled. From the figure it can be seen that the bifurcation coincides with the LEs spectrum shown in Fig. 4. Moreover, there exist two different routes to hyperchaos. The one is the typical quasiperiod route to hyperchaos. With the increase of parameter k, system (3) goes into the three hyperchaotic regions according to same route: limit cycle, quasi periodic torus, chaos and hyperchaos in succession.
The another one is the intermittency route [8] to hyperchos. With the decrease of parameter k, system (3) enters hyperchaos state from limit cycle directly. In order to determinate the kind of route to hyperchaos, we investigate the dynamical behavior of the system when k takes a critical value. As shown in Fig. 12, when k = 0.3034, the state of limit cycle and the state of hyperchaos appear alternately. The situation indicates that the system (3) goes through a transition from limit cycle to hyperchaos via an intermittency process.

Topological entropy estimation
According to the topological horseshoe theory, the topological entropy closely depends on the construction of topological horseshoe. Therefore, we arrange this section as follows: Firstly, the symbolic dynamics is introduced briefly. And then we recall the topological horseshoes theory. Finally, we detect the topological horseshoe with two-directional expanding and compute the topological entropy.
Let From [31] we know that Σ m is compact, totally disconnected and perfect. Therefore, Σ m is a Cantor set, which frequently appears in the characterization of complex structures of chaotic invariant sets in a chaotic dynamical system.
Let σ : Σ m → Σ m to be m-shift map, it satisfies σ(s i ) = s i+m .
Proposition 1. (See [31].) The shift map σ satisfies σ(Σ m ) = Σ m and is continuous. As a dynamical system defined on Σ m , σ has three properties: 1) σ has a countable infinity of periodic orbits consisting of orbits of all periods; 2) σ has an uncountable infinity of aperiodic orbits; 3) σ has a dense orbit.
From the three conditions above it can be deduced that dynamics generated by the shift map σ is sensitive to initial conditions. Therefore, σ is chaotic.
Let X be a metric space, D be a compact subset of X and f : D → X be a map.  [13].) Let Γ ⊂ D i be a subset, we say that f (Γ ) separates D j with respect to D 1 j and D 2 j if Γ contains a compact subsetΓ such that f (Γ ) (D 1 j , D 2 j ). In this case, we denote it by f (Γ ) ‡ D j . In case that f (Γ ) ‡ D j holds true for every subset [13].) If the codimension-one crossing relation f (D i ) ‡ D j holds for 1 i, j m, then there exists a compact invariant set K ⊂ D such that f |K is semiconjugate to the m-shift map, which is denoted by σ|Σ m .
Since f is topologically semiconjugate to σ, which means there exists a continuous surjection g : Σ m → X such that f • g = g • σ, f must be also sensitive to initial conditions, therefore, f is chaotic. By means of Theorem 1, we can not only rigorously proof the existence of hyperchaos, but also assess the intensity of hyperchaos by topological entropy, which can be computed by following proposition. [21].) Let f : X → X be a continuous map. If there exists an invariant set K ⊂ X such that f |K is semi-conjugate to the m-shift σ|Σ m , then ent(f ) ent(σ) = log m. In addition, for every positive integer k, ent(f k ) = k · ent(f ) log m. Therefore, in the case, ent(f ) (log m)/k.

Proposition 2. (See
By choosing the hyperplane Υ = {(x, y, z, w): x − 10 = 0,ẋ < 0} as the crosssection, we obtain the Poincaré map of system (3). It is defined as: Let R : Υ → Υ to be a map, for each x = (y, z, w) ∈ Υ , R n (x) is taken to be the nth return point in Υ under the flow of system (3) with the initial condition x.
According to the method proposed in [11], we detect the three-dimensional horseshoes with two-directional expansions by three steps as follows.
1) Since that the attractor is very close to a curved surface, denoted by y = g(z, w), the map R can be transformed to a two-dimensional map P that satisfies P(z, w) = R(y = g(z, w), z, w). 2) By a serial of attempts, we find a horseshoe with two directional expansions of the map P in the phase space z × w. The vertices of quadrilaterals C 1 , C 2 and C 3 in terms of (z, w) are listed in Table 2.  Proof. From Proposition 1, to get the topological entropy of R, we must proof the conclusion: for the Poincaré map R, there exists a compact invariant set Λ ⊂ (A 1 ∪A 2 ∪A 3 ), on which R 3 |Λ is semi-conjugate to the 3-shift map. Since that the uniqueness and continuity of solution are ensured owing to the smoothness of system (3). Therefore, we only need to show the existence of the following three relations:  Fig. 13, under the map R 3 , A 1 is compressed to R 3 (A 1 ), which transversely intersects {A i , i = 1, 2, 3}. The details show that R 3 (A 1 ) across {A i , i = 1, 2, 3} at their middle part (see Fig.13(b)) and the four side surfaces are mapped outside {A i , i = 1, 2, 3} (see Fig. 13(c)). Let (A t i , A b i ) be the top and bottom surface of A i . We can have the conclusion: for each separation S of (A Refer to Definitions 1 and 2, we have Analogously, from Figs. 14, 15 we can proof relations 2) and 3) mentioned above. Therefore, it is true that the codimension-one crossing relation R(D i ) ‡ D j holds for 1 i, j 3.
From Theorem 1 we know that on the compact invariant set Λ ⊂ (A 1 ∪ A 2 ∪ A 3 ), the map R 3 is semi-conjugate to the 3-shift map. And referring to Proposition 1, we proof that the topological entropy ent(R) (log 3)/3.
Additionally, since that R 3 (A i ), i = 1, 2, 3, expands in two directions, the expansions along each trajectory in Λ are also in two directions. So, there must exist two positive Lyapunov exponents. Therefore, the system is hyperchaotic. Here, in order to verify that there indeed exist the two-directional expanding in the R 3 (A i ), the eigenvalues of the Jacobian matrices J Ai on A i , i = 1, 2, 3, are computed. As listed in Table 3 minimums of the first two absolute eigenvalues are larger than one.
Here we list the topological entropy of some typical hyperchaotic systems in Table 4. Among them, system (a) is the Rössler [22], which is the first hyperchaotic system; system (b) is the hyperchaotic Hénon map [1] that is a typical discrete-time hperchaotic system; system (c) is the famous Saito [24] circuit, which is a hysteresis chaos generator; system (d) is the a Lü-like hyperchaotic system [37] with an infinite number of equilibrium points; system (e) is a modified Lorenz system [18] with large LEs; system (f) is a memristive system [14], in which the introduced memristor is nonlinear. Compared with the previous systems, system (3) has the maximal topological entropy.

The orbits in the hyperchaotic horseshoes
It is well known that the orbits are meaningful for the study of dynamic system. From the theory of topological horseshoes we know that the horseshoes contain infinite periodic and chaotic orbits. Those orbits not only can help us to intuitively understand the behavior of the system, but also can be applied in many applications such as the control of robot gait and the information security. In this section, we propose a simple method to extract the orbits in the new hyperchaotic horseshoes, which is obtained in Section 4.
First, the initial point is detected. Let F be the three-times mapping R 3 . Let x 0 be the initial point. We mark the sets A 1 , A 2 and A 3 with three symbols a, b and c, respectively. The given symbolic sequence with length n is defined by If a sequence of point satisfies the relationship as following: the start point of s is the initial point x 0 . In this paper, this sequence s is called a realization of the symbolic sequence. Obviously, it is very hard to directly detect those points in the phase space. However, by the mapping relationship of topological horseshoes, an initial compact space, which contain the initial point x 0 can be detected. In order to obtain the initial space, we construct the following sequences of sets according to the order represented by s : Then the initial space can be found by the following iteration process: where Ω is the initial space, F −1 indicates one-time reverse mapping of F. After finding the initial space, we take an arbitrary point x ∈ Ω to be x 0 . Under the initial condition, the orbit correspond to symbolic sequence s is obtained.  In this paper, for simplicity, we only give the basic sequences, i.e. the length of sequence n = 3. Suppose that there is a given symbolic sequence {a, b, c}, with the method mentioned above, we get a realization, which is denoted as {x 0 , x 1 , x 2 }, where x 0 = [−15.0077, 30.6462, −19.0963] T as shown Fig. 17(a). Correspondingly, the continuous orbit is represented in Fig. 17(b). Similarly, Fig. 16 shows the orbits corresponding to some typical symbolic sequences, and the initial points are listed in Table 5.

Conclusion
In this paper, we present a new four-dimensional hyperchaotic memory system, which is modified from the Qi chaotic system by introducing a simple linear memory element. For the complex dynamical behavior, we take a study in detail. By theoretical analysis, it is shown that the new system presented in this paper is symmetry, dissipative and exists an infinite number of unstable equilibrium points. The bifurcation analysis shows that the system enters the hyperchaos through two different routes and many kinds of coexisting attractors exist. In addition, in the phase space of the new system, a new kind of hyperchaotic horseshoes consisting of three compact-sets is found, by which the topological entropy is estimated.
In the end, a method is proposed to extract the orbits in the hyperchaotic horseshoes. This work provide a new view for many application. For example, in the field of chaosbased encryption, we can use the orbits in the horseshoes to modulate the information represented by the symbolic sequences. Moreover, since that the method is based on the mapping relationship, it is easy to implement by numerical method and applied directly in other kinds of horseshoes. Meanwhile, this method has some limitations. When the length of the symbolic sequences are large, the initial region of the corresponding orbits become very small owing to the fractal structure of the horseshoes. Limited to the numerical precision of the computer, it is hard to accurately express the initial region and its images. Therefore, this method is not suitable to extract the orbits for the long symbolic sequences. In future, we will pay the attentions on this problem.