† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 51377124) and the Science Fund for New Star of Youth Science and Technology of Shaanxi Province, China (Grant No. 2016KJXX-40).
The self-excited attractors and hidden attractors in a memcapacitive system which has three elements are studied in this paper. The critical parameter of stable and unstable states is calculated by identifying the eigenvalues of Jacobian matrix. Besides, complex dynamical behaviors are investigated in the system, such as coexisting attractors, hidden attractors, coexisting bifurcation modes, intermittent chaos, and multistability. From the theoretical analyses and numerical simulations, it is found that there are four different kinds of transient transition behaviors in the memcapacitive system. Finally, field programmable gate array (FPGA) is used to implement the proposed chaotic system.
Over the past few decades, chaos has been studied by many researchers. The first chaotic system was founded by Lorenz in 1963.[1] Since then, various chaotic systems have been proposed, such as Rössler,[2] Chen,[3] Lü,[4] Chua,[5] and Liu systems.[6] All these chaotic systems are self-excited systems, which is a concept defined by Kuznetsov and Leonov.[7] Also, in this reference, they divided these chaotic attractors into two types, self-excited attractors and hidden attractors. For the hidden attractor, its definition is: the basin of attraction of hidden attractor will not connect with unstable equilibrium point.[8,9] But for the self-excited attractor, it results from an unstable equilibrium point.[10,11] Since then, many achievements have been obtained by researchers interested in hidden attractor chaotic system, such as the chaotic systems with no equilibrium point, with stable equilibrium points, with infinite number of equilibrium points. For example, Zhou presented a three-dimensional (3D) chaotic system without equilibrium point but can generate hidden attractors.[12] Pham introduced a nonlinear system with only one stable equilibrium point.[13] Sprott investigated the coexistence of point, period, and chaos in a chaotic system with stable equilibrium point.[14] Barati proposed chaotic system with curves of equilibrium points.[15]
In addition, under fixed parameters, different chaotic attractors in different initial conditions can be found in coexisting attractor systems, which is defined as multistability in chaotic systems. For example, Lai presented a four-dimensional (4D) chaotic system with different kinds of coexisting attractors, such as attractors with four limit cycles, single-scroll attractors, and double-scroll attractors.[16] Pham studied a chaotic system with coexisting attractors and without equilibrium point.[17] Different kinds of coexisting attractors can be found in Ref. [18], such as point attractors, hidden attractors, limit cycles, and hyperchaos. In addition, the most important characteristic of Ref. [19] is that it possesses an infinite number of equilibrium points in a six-element-memcapacitor-based circuit with coexisting attractors and extreme multistability. Yuan investigated nonlinear dynamics of coexisting attractors in a memcapacitive system with different starting conditions.[20] Hyperchaotic system with infinite equilibrium points and coexisting attractors in a memcapacitor oscillator is accomplished in Ref. [21]. The hyperchaotic system was implemented by field programmable gate array (FPGA).
Furthermore, in recent years, there has been an interest about the concept of transient transition behavior in academic circles. In a nonlinear system, transient transition behaviors are temporal evolutions before steady states. Transient transition behaviors can be more relevant than the steady states of the system in terms of the observation, modeling, prediction, and control of the system. As a result, transient transition behaviors are important to nonlinear systems such as physics, chemistry, biology, engineering, and economics. What is more, it is useful to study this subject because transient behaviors can arise in many fields: shimmy of the front wheels of motorcycles and airplanes,[22] a compass forced by a magnetic field,[23] lasers,[24] and electronic oscillators.[25] The phenomenon of transient transition behavior in the chaotic system could be divided into two types, transient chaos and transient period. On the one hand, transient chaos,[26,27] is a common phenomenon, which can be observed in different nonlinear systems, such as Lorenz system, Rabinovich–Fabrikant system,[28] and memristor system.[29] The traditional transient chaos theory considers that the orbits of a nonlinear system come from the state of chaos firstly and then the transition of orbits would stop in a periodic orbit or a stable point. In other words, the transition of time-domain waveform is from chaos to period or from chaos to a stable point. However, the novel phenomenon of transient chaos is from chaos to period to a stable point, which is founded in this paper. On the other hand, transient period, a new phenomenon different from transient chaos, refers to a system exhibiting periodic orbit firstly and then it converts into chaos at some point. For example, it was firstly reported in Ref. [30] in a memristor-based nonlinear dynamical system with coexisting infinitely many attractors by Bao, who regarded it as an unusual transition behavior. A new 3D chaotic system without equilibrium point but with hidden attractors was proposed in Ref. [12] in which transient period was also found.
Traditional chaotic system was realized by an analog circuit, but there are several problems in the analog circuit, such as component aging, temperature drifting, and voltage changing,[31,32] which has limited the development of the chaotic system. In addition, chaotic the system can be realized by digital signal processing (DSP),[12] advanced RISC machine (ARM),[33] Arduino,[34] and FPGA.[35,36] However, FPGA has advantages compared with DSP, ARM, and Arduino, such as high logic density and versatility.[37]
The contribution of this paper is the finding of a memcapacitive system with stable equilibrium points, multistability, and different kinds of transient transition behaviors. The remaining sections are organized as follows. Section
The charge-controlled memcapacitive system is described as[38]
Since C–1 and
There are three elements in the simple memcapacitive system: a linear resistor, a linear inductor, and a memcapacitor. The circuit schematic diagram is shown in Fig.
Based on Kirchhoff’s current laws, the following equations can be derived
![](cpb_29_5_058502/cpb_29_5_058502_ieqn3.gif)
![](cpb_29_5_058502/cpb_29_5_058502_ieqn4.gif)
![](cpb_29_5_058502/cpb_29_5_058502_ieqn5.gif)
![](cpb_29_5_058502/cpb_29_5_058502_ieqn6.gif)
The Jacobian matrix of the system (
Substituting S+ and S− into Eq. (
Thus, we will get the following conclusion. The two nonzero equilibrium points are unstable when c > 0.25b(b + d), stable when c < 0.25b(b + d).
Substituting a = 1, b = 1.3, d = 0.1, and e = 0 in system (
When the initial condition is chosen as (0.1, 0.1, 0.1), the bifurcation diagram and Lyapunov exponents are plotted in Fig.
![]() | Fig. 3. Bifurcation diagram and Lyapunov exponents with respect to c. (a) Bifurcation diagram of y, (b) Lyapunov exponents. |
When 0 < c < 0.44, three Lyapunov exponents are negative, and the memcapacitive system shows a point attractor as shown in Fig.
![]() | Fig. 4. Phase diagrams in x–y plane with respect to c: (a) c = 0.4, (b) c = 0.5, (c) c = 1, (d) c = 1.5, (e) c = 2.2, (f) c = 3. |
![]() | Table 1. Dynamic behaviors of memcapacitive chaotic system for parameter c. . |
When a = 1, b = 1.3, d = 0.1, and e = 0, and c varies from 0.6 to 3, coexisting self-excited attractors are shown in Fig.
When c = 0.6, the coexistence of quasiperiodic attractors is shown in Fig.
The phenomenon of coexisting bifurcation shown in the small blank area is shown in Fig.
![]() | Fig. 6. Coexisting bifurcation mode under initial conditions of (0.1, 0.5, 0.1) (blue) and (0.1, −0.5, 0.1) (red). |
In system (
In Fig.
Transient chaos theory considers that the orbits of a nonlinear system come from the state of chaos firstly and then the transition of orbits would stop in a periodic orbit or a stable point. Intermittency is the irregular alternation of periodic and chaotic dynamics of chaotic system. Intermittent chaos and transient chaos could be determined in periodic window.[22] In a periodic window, transient chaos is caused by a nonattracting chaotic set, while intermittent chaos caused by an interior crisis. The crisis refers to sudden changes in the chaotic attractor when the system parameter is varied. Crises in the chaotic system could be divided into two types, boundary crises, and interior crises. When the chaotic attractor moves to the unstable orbit of the boundary, an attracting chaotic set would change into a nonattracting chaotic set. Transient chaos lies in the nonattracting chaotic set. This phenomenon is called boundary crisis. Interior crisis is the phenomenon that a chaotic attractor moves to a nonattracting chaotic set. After that, a smaller chaotic attractor becomes a larger one.
In Fig.
![]() | Fig. 9. Intermittent chaos (a) time-domain waveform, (b) chaos-1 and p-1, (c) chaos-2 and p-2, (d) chaos-3 and p-3. |
Two different kinds of transient chaos can be found in the self-excited system. The first transient chaos is from chaos to period. When setting parameters a = 1, b = 1.3, c = 2.76, d = 0.1, and e = 0 with the initial condition (0.1, 0.1, 0.1), the system displays chaos firstly and then switches over to periodic state in Fig.
![]() | Fig. 10. Transient chaos from chaos to period (a) time-domain waveform, (b) chaotic attractor when 0 < t < 1000, (c) periodic attractor when 1500 < t < 3000. |
The second transient chaos is a novel phenomenon. When parameters a = 1, b = 1.3, c = 1.5, d = 0.1, and e = –0.4, the initial condition is set as (−0.2, 3, −2). In Fig.
Substitute a = 1, b = 1.3, c = 0.437, d = 0.1, and e = 0 in Eq. (
As indicated in Ref. [43], hidden chaotic attractors are presented in a generalized Lorenz system of fractional order with an unstable point and two saddle points. In order to verify that the memcapacitive system has hidden attractors, the existence of small neighborhood of the unstable equilibrium S0, from which orbits are all attracted by the stable equilibriums S+ or S−, should be verified. As shown in Fig.
The multistability of the hidden attractors refers to different attractors in different initial conditions with fixed parameters. In Fig.
In hidden chaotic system, when setting parameters as a = 1, b = 1.3, c = 0.435, d = 0.1, and e = 0, the initial condition as (0, −1.5, 0), the time-domain waveform of state variable x is shown in Fig.
When setting parameters as a = 1, b = 1.3, c = 0.44, d = 0.1, and e = 0, the initial condition as (0.1, −0.1, 0.7), the time-domain waveform of state variable x is shown in Fig.
In recent years, many chaotic systems are realized by field programmable gate array (FPGA) in academic circles.[32,34,44–46] So, it is a growing tendency that a chaotic system will be implemented by FPGA technology. As a result, in this paper, the FPGA technology is used to realize the chaotic system.
Our FPGA development board is AC620, which includes main chip EP4CE10F17C8 N. In addition, digital to analog converter chip is ACM9767, which is connected with the oscilloscope. In FPGA, the second-order Runge–Kutta algorithm is used to realize the chaotic system.
Setting a = 1, b = 1.3, d = 0.1, e = 0, and h = 1/128, the memcapacitive system is discretized as
In this paper, hidden and self-excited attractors are studied in a simple memcapacitive system. First, the critical parameter between self-excited attractor and hidden attractor is obtained by the calculation of equilibrium points and stability. In addition, there are only three elements in the system, but the system has complex chaotic dynamical behaviors, such as coexisting attractors, hidden attractors, coexisting bifurcation modes, intermittent chaos, and multistability. What is more, four different kinds of transient transition behaviors are found in this system: chaos to period, chaos to a stable point, chaos to period to a stable point, and period to chaos.
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