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This paper investigates extreme multistability and its controllability for an ideal voltage-controlled memristor emulator-based canonical Chua’s circuit. With the voltage-current model, the initial condition-dependent extreme multistability is explored through analyzing the stability distribution of line equilibrium point and then the coexisting infinitely many attractors are numerically uncovered in such a memristive circuit by the attraction basin and phase portraits. Furthermore, based on the accurate constitutive relation of the memristor emulator, a set of incremental flux-charge describing equations for the memristor-based canonical Chua’s circuit are formulated and a dimensionality reduction model is thus established. As a result, the initial condition-dependent dynamics in the voltage-current domain is converted into the system parameter-associated dynamics in the flux-charge domain, which is confirmed by numerical simulations and circuit simulations. Therefore, a controllable strategy for extreme multistability can be expediently implemented, which is greatly significant for seeking chaos-based engineering applications of multistable memristive circuits.

Initial condition-dependent extreme multistability, first encountered in several coupled nonlinear dynamical systems [

Extreme multistability is a fantastic kind of multistability, which makes a nonlinear dynamical circuit or system supply great flexibility for its potential uses in chaos-based engineering applications [

Besides, for a memristor-based circuit or system with line equilibrium point or plane equilibrium point, its stability at the equilibrium point is very difficult to be determined due to the existence of one or two zero eigenvalues [

Flux-charge analysis method was first postulated as a tool of dimensionality reduction [

The rest of the paper is structured as follows. In Section

Based on a canonical Chua’s circuit and an ideal voltage-controlled memristor emulator, a new memristor-based canonical Chua’s circuit is constructed, as shown in Figure

Circuit parameters of memristor-based canonical Chua’s circuit.

Parameters | Significations | Values |
---|---|---|

| Capacitance | 1 nF |

| Capacitance | 4.7 nF |

| Capacitance | 33 nF |

| Inductor | 30 mH |

| Total gain | 0.2 V^{−2} |

| Resistance | 4 kΩ |

| Resistance | 1.5 kΩ |

| Resistance | 2 kΩ |

| Resistance | 1.5 kΩ |

| Resistance | 2 kΩ |

Memristor-based canonical Chua’s circuit. (a) Circuit schematic with simple topology. (b) Ideal voltage-controlled memristor emulator implemented with discrete components.

For the ideal voltage-controlled memristor emulator in Figure

Introduce four new state variables and scale the circuit parameters as

With the circuit parameters in Table

Similar to the memristive Chua’s circuit containing an ideal voltage-controlled memristor emulator [

At the line equilibrium point

Stability distributions classified by the real parts of three nonzero eigenvalues of the line equilibrium point

Due to the existence of the zero eigenvalue, the stability of the memristor-based canonical Chua’s circuit can not be simply determined by the three nonzero eigenvalues of the line equilibrium point. The following numerical simulations demonstrate that the zero eigenvalue also has influence on the dynamics of the circuit under some circuit parameters [

The initial conditions for numerical simulations of the coexisting attractors’ behaviors are taken as [^{−9}, 0, 0]; that is, only the memristor initial condition

Nonzero eigenvalues and attractor types for different memristor initial conditions.

| Nonzero eigenvalues | Stability regions | Attractor types |
---|---|---|---|

0 | | Region I: 3P0N | Double-scroll chaotic |

(Unstable node-foci) | |||

2 | | Region I: 3P0N | Infinite |

(Unstable node-foci) | |||

2.4 | | Region II: 1P2N | Chaotic spiral attractor |

(Unstable saddle-foci) | |||

3.2 | | Region III: 0P3N | Stable point attractor |

(Stable node-foci) | |||

4.3 | | Region VI: 2P1N | Stable point attractor |

(Unstable saddle-foci) | |||

4.5 | | Region VI: 2P1N | Limit cycle with period 1 |

(Unstable saddle-foci) | |||

4.8 | | Region VI: 2P1N | Asymmetric double- |

(Unstable saddle-foci) |

With reference to the stability distributions in Figure

For the normalized parameters in (

Attraction basin in the

Corresponding to different color areas, different types of coexisting attractors are listed in Table

Different color regions and the corresponding attractor types.

Colors | Coexisting attractor types | Examples in Figure |
---|---|---|

Blue and forest green | Right- and left-point attractors | Figure |

Cyan and lime green | Right- and left-period-1 limit cycles with small size | Figure |

Cadet blue and lawn green | Left- and right-period-1 limit cycles with large size | Figure |

Tan and yellow | Left- and right-multi-period limit cycles | Figure |

Orchid and coral | Left- and right-chaotic spiral attractors | Figure |

Medium slate blue and fuchsia | Left- and right-half-baked double-scroll chaotic attractors | Figure |

Red | Standard double-scroll chaotic attractor | Figure |

Black | Unbounded orbit | Figure |

Phase portraits of coexisting infinitely many attractors in the

It should be mentioned that just like the ideal flux/voltage-controlled memristor-based chaotic circuits [

Due to the existence of the line equilibrium point, the memristor-based canonical Chua’s circuit can exhibit the special phenomenon of extreme multistability under different initial conditions. For seeking the potential uses of the multistable memristive circuit in chaos-based engineering applications [

The accurate constitutive relation of the ideal voltage-controlled memristor emulator in Figure

Suppose that

Integrating the second, third, and fourth equations of (

Analogously, introduce three new state variables and scale the circuit parameters as

It should be emphasized that the initial conditions of (

For the circuit parameters given in Table

For the normalized model (

Define

The Jacobian matrix at equilibrium point

Based on (

Take

For the zero equilibrium point

With model (

The normalized system parameters

Phase portraits of various types of attractors distributed in different locations of the parameter space of

With model (

Equivalent circuit of model (

According to the fundamental theory of circuit, the circuit state equations of Figure

The op-amps OP07CP and multipliers AD633JNZ with ±15 V power supplies are utilized. The integrating time constant is selected as

To better present the control effect of the multistable states generated from the equivalent circuit in Figure ^{−9 }V can be achieved by a slightly induced voltage in the equivalent circuit, so the value of

Multisim intercepted phase portraits of various types of attractors for different values of

By replacing Chua’s diode in the canonical Chua’s circuit with an ideal voltage-controlled memristor emulator, a memristor-based canonical Chua’s circuit is presented in this paper. Because of the existence of a line equilibrium point, the initial condition-dependent extreme multistability easily emerged in such a memristive circuit, resulting in the coexistence of infinitely many attractors. To implement the controllability of the extreme multistability, an incremental flux-charge model for the memristive circuit is formulated through deriving the accurate constitutive relation of the memristor emulator. Thus, the initial condition-dependent dynamics in the voltage-current domain is converted into the system parameter-associated dynamics in the flux-charge domain, that is, the implicit expression of the initial conditions in the voltage-current model can be transformed into the explicit representation of the system parameters in the flux-charge model, leading to the fact that the multiple steady states emerging in the memristive circuit can be consequently controlled by changing the initial condition-related system parameters. The feasibility of the controllable strategy for extreme multistability is confirmed by numerical simulations and circuit simulations, which is greatly significant for seeking the potential uses of the multistable memristive circuits in chaos-based engineering applications.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundations of China under Grants nos. 51777016, 61601062, 51607013, and 11602035 and the Natural Science Foundations of Jiangsu Province, China, under Grant no. BK20160282.