In this paper, we design a chaotic circuit with memristors, which consists of two flux-controlled memristors and a charge-controlled memristor, and the dimensionless mathematical model of the circuit was established. Using the conventional dynamic analysis methods, the equilibrium point set and stability of the chaotic system were analyzed, and the distribution of stable and unstable regions corresponding to the memristor initial states was determined. Then, we analyze the dynamical behaviors with the initial states of the memristors and the circuit parameter of the circuit system, respectively. By using spectral entropy (SE) and C_{0} complexity algorithms, the dynamic characteristics of the system were analyzed. In particular, the 2D and 3D complexity characteristics with multiple varying parameters were analyzed. Some peculiar physical phenomenon such as coexisting attractors was observed. Theoretical analysis and simulation results show that the chaotic circuit has rich dynamical behaviors. The complicated physical phenomenon in the new chaotic circuit enriches the related content of chaotic circuit with memristors.

According to the principle of completeness with variable combination, Professor Chua predicted the existence of memristor in 1971 [

Memristors are often divided into charge-controlled memristor and flux-controlled memristor. Both of them are typical nonlinear elements, and it is easy to generate a chaotic vibration signal by employing this element. So researchers pay more attention to the design and realization of memristive chaotic circuit [

In this paper, we focus on the mixed memristive chaotic circuit which has two smooth passive flux-controlled memristors and a charge-controlled memristor. It is organized as follows. The circuit model was designed and its dimensionless mathematical equation was deduced in Section

According to the mathematical model of flux-controlled memristor which was put forward in [

The volt-ampere characteristic curve of the smooth passive flux-controlled memristor driven by a 2.828 V and 1 Hz sinusoidal signal is shown in Figure

Volt-ampere characteristic curve of the smooth passive flux-controlled memristor.

According to the mathematical model of charge-controlled memristor which was proposed in [

When setting the parameters

Volt-ampere characteristic curve of the passive charge-controlled memristor.

From Figures

Based on three memristors and the Chua circuit, a nonlinear circuit is designed as shown in Figure

Circuit model based on memristors.

Let

Obviously, the new nonlinear memristive chaotic circuit is a six-dimensional circuit system. It can be described by (^{−4},0), and the time step is 0.01 s. We can get the chaotic attractors as shown in Figure

Chaotic attractor of the circuit with three memristors (a)

Let

It shows that the new memristive system has three zero eigenvalues and three nonzero eigenvalues. All of the cubic polynomial coefficients of (

If we chose parameter

If

If

If

As it is well known, Lyapunov exponents measure the exponential rates of divergence and convergence of nearby trajectories in a state space, and the Lyapunov exponent spectrum provides additional useful information for a nonlinear system. Keeping the circuit parameters the same as mentioned above, select the initial states

Lyapunov exponent spectrum with varying initial states

The Lyapunov exponent spectrum reflects the stability performance of the system. Meanwhile, the three zero eigenvalues of the system are also the important cause. For example, even though all the Lyapunov exponents of the system are less than or equal to zero in some case, the system is not stable point or steady state but a stable sink. All the chaos ranges with different initial values are summarized in Table

Chaos ranges with different initial states.

−0.099~−0.082 | −0.012~−0.019 | −0.167~−0.009 |

−0.084~−0.066 | −0.017~−0.013 | −0.007~0.119 |

−0.064~0.044 | −0.011~−0.109 | 0.121~0.143 |

0.046~0.059 | −0.107~−0.009 | 0.145~0.231 |

−0.005~0.087 | 0.241~0.279 | |

0.089~0.103 | 0.293~0.343 |

Let _{0} complexity curves are shown in Figures

Dynamical analysis with different circuit parameter _{0} complexity.

Dynamical behaviors of the system with different

Dynamical behaviors | |
---|---|

0~0.023 | Stable sink |

0.024~0.04 | Limit cycle |

0.041~0.0447 | Period-2 cycle |

0.0448~0.0449 | Period-2,-4 bifurcation |

0.045~0.059 | One-scroll chaotic attractor |

0.06~0.246 | Two-scroll chaotic attractor |

0.247~0.2717 | One-scroll chaotic attractor |

0.2718~0.2719 | Period-4,-2 bifurcation |

0.272~0.286 | Period-2 cycle |

0.287~0.431 | Limit cycle |

0.432~0.5 | Stable sink |

To further display its dynamic characteristics,

Phase portrait with different circuit parameter

Different initial values can produce different dynamic behaviors, and they can coexist in the same chaotic system. The phenomenon of coexisting attractors is a representation of coexisting oscillation. Select

Coexisting chaotic attractors with different initial state. (a) Coexisting chaotic attractors on the

Complexity characteristic is essentially the reflection of entropy with the chaotic sequence. Generally, the greater the degree of fluctuation in the sequence, the greater the entropy, which means the higher the value of complexity. Figures _{0} complexity has more complex characteristics because a complexity value is determined by three variable parameters. Figures _{0} complexity with three varying circuit parameters. Due to the integral tolerance problem of the system, the simulation results of 3D SE and C_{0} complexity have slight deviation when circuit parameter

Complexity characteristics of the system. (a) 2D SE complexity, (b) 2D C_{0} complexity, (c) 3D SE complexity, and (d) 3D C_{0} complexity.

In this paper, a new chaotic circuit system was derived from Chua’s chaotic oscillators by introducing a charge-controlled memristor and two flux-controlled memristors. It has an equilibrium point set located in the space constructed by the inner state variables of the three memristors, and the stable and unstable regions coexist in the space. By using spectral entropy (SE) and C_{0} complexity algorithms, the dynamic characteristics of the system are accurately analyzed. Using the numerical simulation tool, some complex dynamical phenomena such as coexisting attractors were analyzed. Therefore, the new chaotic circuit system is different from the general chaotic system. The dynamical characteristics of the new system accompany with the variation of the circuit parameter and depend on the initial states of the memristors. The new memristive circuit model and the new mathematical model can be widely used in the chaotic-related field. Next, we will try to find more chaotic characteristics in this new memristive chaotic circuit.

The data used to support the findings of this study are included within the article.

The authors declare that they have no competing interests.

Xiaolin Ye designed and performed experiments, analyzed data, and wrote this manuscript; Jun Mou made a theoretical guidance for this paper; Feifei Yang and Chunfeng Luo designed circuit experiments. All authors reviewed the manuscript; Yinghong Cao made a technical support for this paper.

This work supported by the Natural Science Foundation of Liaoning Province (Grant no. 20170540060), Basic Scientific Research Projects of Colleges and Universities of Liaoning Province (Grant no. 2017J045), Scientific Research Projects in General of Liaoning Province (Grant no. L2015043), and Doctoral Research Startup Fund Guidance Program of Liaoning Province (Grant no. 201601280).