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By introducing an ideal and active flux-controlled memristor and tangent function into an existing chaotic system, an interesting memristor-based self-replication chaotic system is proposed. The most striking feature is that this system has infinite line equilibria and exhibits the extreme multistability phenomenon of coexisting infinitely many attractors. In this paper, bifurcation diagrams and Lyapunov exponential spectrum are used to analyze in detail the influence of various parameter changes on the dynamic behavior of the system; it shows that the newly proposed chaotic system has the phenomenon of alternating chaos and limit cycle. Especially, transition behavior of the transient period with steady chaos can be also found for some initial conditions. Moreover, a hardware circuit is designed by PSpice and fabricated, and its experimental results effectively verify the truth of extreme multistability.

In 1963, the first chaotic system was discovered by Lorenz. Since then, many scientists have constructed many new chaotic systems, such as Chen system, Lu system, and Jerk system [

In addition to the sensitivity of the system to the parameters, it also depends on the initial value of the memristor. Due to the introduction of the ideal memristor [

Inspired by the abovementioned ideas, an interesting memristor-based chaotic system is constructed in this paper, which is achieved by introducing a tangent function

The rest of this paper is organized as follows. In Section

According to the chaotic system VB18 reported in [

A VB18 chaotic system is described as

The memductance function of the desired flux-controlled memristor is expressed as

Through introducing the newly proposed memristor featured by (

The voltage-current curves of the memristor with different values of the frequency

The memductance and pinched hysteresis loop. (a) The pinched hysteresis loop; (b) the memductance loop.

When _{0}, _{0}, _{0}, _{0}) are assigned as (1, 1, 0, 5). Matlab numerical simulations are performed and several useful results are obtained, as shown in Figure

Chaotic behavior characterized by an symmetric chaotic attractor of system (

When

According to the fourth dimension

The Jacobian matrix of system (

If the conditions in (

The dynamic properties of the system will be changed by the changing parameters, which will make the system show the phenomena of chaotic periodic divergence.

For system (

Lyapunov exponents and bifurcation diagram of system (

Lyapunov exponents and bifurcation diagram of system (

Phase portraits of attractors of system (

When the same parameter takes different initial values, two or more attractors are called coexistent attractors or multiple attractors, which is called the multistable state. The four-dimensional initial value of system (

In most memristor systems, the value of a memristor has some relations with the initial condition, so we have different oscillating dynamics depending on whether the initial data of the internal variable causes different oscillations. This phenomenon is widely studied and is known as extreme polystability. In system (_{0} varies in (−30, 30), system (_{0}, system (_{0} has a completely different dynamic behavior in the region around parameter 0. It is remarked that when the initial conditions of

Dynamics with respect to _{0} ]. (a) Bifurcation diagram of the state variable _{0}); (b) Lyapunov exponent spectra.

Bifurcation diagram of the state variable _{0}); (b) the initial condition (1 1 0 _{0}).

To better represent extreme multisteadiness, the typical phase portraits of attractors under different _{0} are shown in Figure _{0} is different, the state of system (

Coexisting attractor of system (_{0}]. (a) _{0} = 0.1 (green), _{0} = −12.5 (blue), and _{0} = −15 (red); (b) _{0} = 1.42 (red), _{0} = −1 (blue), and _{0} = −9 (green); (c) _{0} = −2 (green), _{0} = 3.69 (red), and _{0} = 12.35 (blue); (d) _{0} = 25 (red), _{0} = 3.69 (red), and _{0} = 22.5 (blue); (e) _{0} = −5 (red), _{0} = −10 (green), and _{0} = 4 (blue); and (f) _{0} = 3.5 (blue) and _{0} = −5.5 (red).

Attractors in system (_{0}].

Initial condition (_{0}) | Colour | Attractor | Figure number |
---|---|---|---|

_{0} = 0.1, −9, −2, −10 | g | Chaotic | ( |

_{0} = −12.5 | b | Cycle-4 | ( |

_{0} = −15 | r | Cycle-2 | ( |

_{0} = 1.42, −3.68 | r | Quasiperiodicity | ( |

_{0} = −1 | b | Chaotic | ( |

_{0} = 2.25 | b | Cycle-1 | ( |

_{0} = 22.5 | g | Cycle-2 | ( |

_{0} = 12.35 | b | Quasiperiodicity | ( |

_{0} = 25 | r | Cycle-1 | ( |

_{0} = −5 | r | Chaotic | ( |

_{0} = 4 | b | Chaotic | ( |

_{0} = 35, −5.5 | b, r | Cycle-1 | ( |

The control parameters of system (_{0} 0 5), and the initial condition _{0} is taken as the bifurcation parameter. When _{0} is varied in the region [−10, 10], the bifurcation diagram of the state variable _{0} and its Lyapunov exponent spectra are plotted in Figures _{0} is increased from 0, system (

Lyapunov exponents and bifurcation diagram of system (_{0}, 0, 5]. (a) Bifurcation diagram with

Near _{0} = 1.85, system (_{0} = 2.09, system (_{0} is further increased, system (_{0} = 2.5. In the parameter region [1.85, 2.09] of _{0}, system (_{0}, system (_{0} = 2.24, 3.22. Therefore, system (_{0} on the system. It can be seen that the initial value _{0} is positive and negative, and system (_{0} changes in the interval [−10, 10], the phase diagram of the system is symmetrically distributed in the _{0} is 1.85, the system is in the cycle-3 stats, when the value of _{0} is 2.215, the system is in the cycle-4 stats, and when the value of _{0} is 3.214, the system is in the cycle-6 stats. When the value of _{0} is 1, the system is in a chaotic state. Thus, the results of Figure

Coexisting and symmetry attractor of system (_{0}, 0,5 ]. (a) Cycle-1 symmetry; (b) cycle-2 symmetry; (c) cycle-4 symmetry; and (d) chaotic symmetry.

Furthermore, for the periodic tangent function, all the infinite countless attractors can be self-reproduced in the dimension of _{0}. System (_{0} is representatively 0, ±pi, and

The self-replicating attractor of system (

The bifurcation diagram and Lyapunov exponent spectrum under the change of the initial value _{0} are shown in Figure _{0} takes different values, system (_{0} is in the interval [−0.55, 0.8], the system is in a state of chaos and periodic state. Typical phase portraits of attractors of system (_{0} is −0.55, the system is in the cycle-1 state, when the value of _{0} is −0.49, the system is in the cycle-2 state, when the value of _{0} is 0.12, the system is in the cycle-3 state, when the value of _{0} is −0.32, the system is in the cycle-4 state, and when the value of _{0} is 0.24, the system is in the quasiperiodic attractor state. When the value of _{0} is 0.8, the system is in a chaotic state. Thus, the results of Figure

Lyapunov exponents and bifurcation diagram of system (_{0}, 5 ]. (a) Bifurcation diagram; (b) Lyapunov exponents.

Coexisting attractor of system (_{0}, 5]. (a) _{0} = −0.55; (b) _{0} = −0.49; (c) _{0} = 0.12; (d) _{0} = −0.32; (e) _{0} = 0.24; and (f) _{0} = 0.8.

The analog circuit of system (

Hardware circuit implementation of memristive system (

The circuit consists of four channels to realize the integration, addition, subtraction, and nonlinear operations including absolute value function and quadratic nonlinearity, as shown in Figure _{1} = _{2} = 1 V and V_{2} = −1 V, _{3} = 0 V, and _{4} = 5 V, and the circuit simulation is shown in Figure _{1} = _{2} = 1 V, _{3} = _{3} = −_{4} = 5 V, and the circuit simulation is shown in Figure

Pinched hysteresis loop of the memristor described by equation (

When

For different initial conditions, phase portraits in the different plane by PSpice: (a)

A new memristive self-replicating attractor system was obtained by introducing a tangent function and memristor into the classical system. In this case, an infinite line of equilibrium is produced, causing the phenomenon of alternating chaos and period. We analyzed the change of the initial value of each variable in the four-dimensional system, a dynamic behavior in which the transient period and steady state chaos alternately appear depending on the initial value change is newly discovered, and the super multistable state was discussed. The phenomenon of attractor self-replication appears due to the introduction of the tangent function. The results of the new system imply that there is coexisting infinitely many attractors’ behavior. The initial-condition-dependent dynamical behaviors of coexisting infinitely many attractors and transient period are finally validated by hardware experiments and PSpice circuit simulations, which could enhance security for possible secure communication.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported in part by the Natural Science Foundation of Shandong Province under Grant ZR2020KA007.