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In this paper, a four-dimensional (4-D) memristor-based Colpitts system is reaped by employing an ideal memristor to substitute the exponential nonlinear term of original three-dimensional (3-D) Colpitts oscillator model, from which the initials-dependent extreme multistability is exhibited by phase portraits and local basins of attraction. To explore dynamical mechanism, an equivalent 3-D dimensionality reduction model is built using the state variable mapping (SVM) method, which allows the implicit initials of the 4-D memristor-based Colpitts system to be changed into the corresponding explicitly initials-related system parameters of the 3-D dimensionality reduction model. The initials-related equilibria of the 3-D dimensionality reduction model are derived and their initials-related stabilities are discussed, upon which the dynamical mechanism is quantitatively explored. Furthermore, the initials-dependent extreme multistability is depicted by two-parameter plots and the coexistence of infinitely many attractors is demonstrated by phase portraits, which is confirmed by PSIM circuit simulations based on a physical circuit.

Chua’s circuit [

The careful dynamical analyses of these constructed memristive systems show that the memristor initials do play a crucial role in dynamical characteristics of these systems [

The initials-dependent multistability [

Latterly, to solve the abovementioned problem, flux-charge analysis method [

The aforementioned analytic strategies have been preliminarily verified in several memristor-based Chua’s circuits [

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

The constructing scheme is adopted through imitating the method narrated in [

A classic 3-D Colpitts oscillator model with an exponential nonlinear term was reported in [

Based on the 3-D Colpitts oscillator model presented in (

The ideal memristor (

Attractor types with different initials of system (

Initials | Attractor types | Phase portraits |
---|---|---|

(10^{−9}, 0, 0, ±3.6) | Asymmetric chaotic double-scroll attractors | Figure |

(10^{−9}, 0, 0, ±3.3) | Period-2 limit cycles | Figure |

(−1, 2, 0, 2.55) | Spiral chaotic attractor | Figure |

(10^{−9}, 0, 0, 0) | Symmetric chaotic double-scroll attractor | Figure |

(10^{−9}, −1.5, 0, −2) | Period-2 limit cycle | Figure |

(10^{−9}, 0, 0, −0.8) | Period-3 limit cycle | Figure |

(−1, 0, 0, −3.4) | Period-1 limit cycle | Figure |

(−1, 0, 0, −3.2) | Point attractor | Figure |

(−1, 2, 0, 2.2) | Unbounded orbit | Figure |

Phase portraits of coexisting infinitely many attractors in the_{3} −_{4} plane for different initials (_{1}(0),_{2}(0),_{3}(0),_{4}(0)). (a) Asymmetric chaotic double-scroll attractors for (10^{−9}, 0, 0, ±3.6). (b) Period-2 limit cycles for (10^{−9}, 0, 0, ±3.3). (c) Chaotic attractors with different topologies for (10^{−9}, 0, 0, 0) and (−1, 2, 0, 2.55). (d) Period-2 and period-3 limit cycles for (10^{−9}, −1.5, 0, −2) and (10^{−9}, 0, 0, −0.8). (e) Period-1 limit cycle and point attractor for (−1, 0, 0, −3.2) and (−1, 0, 0, −3.4). (f) Unbounded orbit for (−1, 2, 0, 2.2).

The phase portraits of coexisting infinitely many attractors in Figure _{1} are considered and the topologies and locations of the attractors are ignored here. The red regions marked by CH represent chaotic behaviors. The black and blue regions labeled by DE and P0 denote unbounded divergent and stable point behaviors respectively. Whereas the other color regions labeled by P1 ~ P4 stand for periodic behaviors with different periodicities. Therefore, the emergence of extreme multistability is disclosed, indicating the coexistence of infinitely many attractors in the 4-D memristor-based Colpitts system.

Two-dimensional local basins of attraction of system (_{1}(0),_{2}(0),_{3}(0),_{4}(0)). (a) The_{3}(0) –_{4}(0) initial plane with_{1}(0) =_{2}(0) = 0. (b) The_{1}(0) –_{2}(0) initial plane with_{3}(0) =_{4}(0) = 0. (c) The_{2}(0) –_{4}(0) initial plane with_{2}(0) =_{3}(0) = 0. (d) The_{1}(0) –_{4}(0) initial plane with_{2}(0) =_{3}(0) = 0.

In addition, lots of unbounded divergent behaviors can be observed in Figure

To explore dynamical mechanism of the initials-dependent extreme multistability emerged in system (_{4}, i.e., the forth equation of (

Similar to [

Noteworthily, the implicit initials_{i}(0) of the 4-D memristor-based Colpitts system are mapped as explicitly initials-related system parameters_{i} appearing in the 3-D dimensionality reduction model. What needs illustration is that, under the situation_{1}(0) =_{2}(0) =_{3}(0) = 0, system (_{1},_{2},_{3},_{4} as the extrinsic initials-related system parameters. It follows that the aforementioned 3-D dimensionality reduction model can be utilized for quantitatively investigating the initials-dependent dynamics of the 4-D memristor-based Colpitts system by changing the initials-related system parameters_{i}.

System (_{1} = 10^{−9} and_{2} =_{3} =_{4}_{1} −_{3} plane, as shown in Figure _{1}(0) −_{2}(0) initial plane with_{3}(0) = 0 is depicted, as shown in Figure

Illustrations for the bounded chaotic behavior (red) and unbounded divergent behavior (yellow). (a) Phase portraits under the initials (0, 5, 0) and (−9, 0, 0). (b) Local basin of attraction in the_{1}(0) −_{2}(0) initial plane with_{3}(0) = 0.

By setting ^{2} + (^{3} [

The detailed breakdowns of the equilibrium_{i} (

The equilibrium of the dimensionality reduction model (

Δ | | | Equilibrium |
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Δ > 0 | | | |

Δ = 0 | | | |

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Δ < 0 | | | |

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| |

Take_{1} = 10^{−9} and_{2} =_{3} = 0 as an example. When the initials-related system parameter_{4} varies within _{3,1},_{3,2}, and_{3,3}; the _{1} is presented in Figure _{1}(0),_{2}(0),_{3}(0)] = _{4} varies from −4 to 0, and the lower is bifurcation diagram such that_{4} varies from 0 to 4. It can be seen that the representing dynamics in Figure

Dynamics with the variation of_{4} at_{1} =10^{−9},_{2} =_{3} = 0. (a) The _{1}, the upper:_{4} _{4}

Since the trajectory of system (_{4} in the region I, but are asymmetric in the regions II and III. More narrowly, in the region I, when_{4} varies within _{3,1},_{3,2}, and_{3,3} are all unstable, such that the system orbit may be randomly pushed toward one of these three unstable equilibria. And system (_{3,2} becomes a stable equilibrium, _{4} lead to the emergence of complex dynamical behaviors in system (

Observed from Figure _{1},_{2},_{3}, and_{4}. For intuitively manifesting the coexistence of infinitely many attractors, two-parameter bifurcation plots in different initials-related parameter planes are plotted, as shown in Figure _{1}, which are different from the parameter-space plots given in [

Two-parameter bifurcation plots depicted by the periodicities of the state variable_{1} in different initials-related parameter planes for four sets of the initials-related system parameters (_{1},_{2},_{3},_{4}). (a) Bifurcation plot in the_{3} –_{4} plane with_{1} =_{2} = 0. (b) Bifurcation plot in the_{1} –_{2} plane with_{3} =_{4} = 0. (c) Bifurcation plot in the_{2} –_{4} plane with_{1} =_{3} = 0. (d) Bifurcation plot in the_{1} –_{4} plane with_{2} =_{3} = 0.

When the initials-related system parameters_{1} =_{2} = 0, the coexistence of infinitely many attractors in the_{3} −_{4} parameter plane can be observed in Figure _{4}, the system can generate asymmetric chaotic double-scroll attractors. In contrast, the intuition of the region _{4} is that the system can generate symmetric chaotic double-scroll attractors. Furthermore, when_{3} =_{4} = 0,_{1} =_{3} = 0, and_{2} =_{3} = 0, Figures

Corresponding to the part of different color areas in Figure _{1},_{2},_{3},_{4}) in different color areas of Figure _{1} −_{2} plane are numerically simulated, as displayed in Figure

Different color regions and the coexisting attractor types.

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

Red | Upper- and lower-asymmetric chaotic double-scroll attractors | Figure |

Cyan | Upper- and lower-period-2 limit cycles | Figure |

Red | Symmetric chaotic double-scroll attractor and chaotic spiral attractor | Figure |

Cyan and yellow | Period-2 and period-3 limit cycles | Figure |

Cadet blue and blue | Period-1 limit cycle and point attractor | Figure |

Black | Unbounded orbit | Figure |

MATLAB simulated phase portraits of coexisting infinitely many attractors in the_{1} −_{2} plane for different initials-related system parameters (_{1},_{2},_{3},_{4}). (a) Upper- and lower-asymmetric chaotic double-scroll attractors. (b) Upper- and lower-period-2 limit cycles. (c) Chaotic attractor with different topologies. (d) Period-2 and period-3 limit cycles. (e) Period-1 limit cycle and point attractor. (f) Unbounded orbit.

The 3-D dimensionality reduction model described by (_{1} and_{2} are set as 1. According to basic circuit theory, the circuit state equations are formulated in a general form as_{1},_{2}, and_{3} represent the state variables and_{1},_{2},_{3,} and_{4} are implemented by additional DC voltage sources or directly linking to the ground. Note that the_{6} in Figure

Physical circuit implementing the 3-D dimensionality reduction model (

To better confirm the extreme multistability generated from the equivalent circuit in Figure _{1} =_{4}^{2}_{2} =_{3} =_{4},_{4} = 3_{5} = 2_{6} =_{1}(0),_{2}(0),_{3}(0)] are assigned as (0 V, 0 V, 0 V) and the initials-related system parameters (_{1},_{2},_{3},_{4}) are assigned as the same values by referring to those in Figure _{1} −_{2} plane are shown in Figure

PSIM simulated phase portraits of coexisting infinitely many attractors in the_{1} −_{2} plane for different initials-related system parameters (_{1},_{2},_{3},_{4}). (a) Upper- and lower-asymmetric chaotic double-scroll attractors. (b) Upper- and lower-period-2 limit cycles. (c) Chaotic attractor with different topologies. (d) Period-2 and period-3 limit cycles. (e) Period-1 limit cycle and point attractor. (f) Unbounded orbit.

In this paper, a dimensionality reduction reconstitution scheme for extreme multistability in memristor-based Colpitts system was introduced. By employing an ideal memristor to substitute the exponential nonlinear term of original 3-D Colpitts oscillator model, a novel 4-D memristor-based Colpitts system was obtained. The initials-dependent extreme multistability of the proposed system was exhibited via phase portraits and local basins of attraction. To explore dynamical mechanism, an equivalent 3-D dimensionality reduction model was constructed using SVM method. As a consequence, the implicit initials of the 4-D memristor-based Colpitts system were transformed into the explicitly initials-related system parameters of the 3-D dimensionality reduction model. Meanwhile, the dynamical mechanism was quantitatively explored by deriving the initials-related equilibria and discussing the equilibrium stabilities in the 3-D dimensionality reduction model. Furthermore, the initials-dependent extreme multistability was verified by two-parameter bifurcation plots and the coexistence of infinitely many attractors was demonstrated by phase portraits and confirmed by PSIM circuit simulations based on a physical circuit. To sum up, this work has multiple advantages:

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

The authors declare that they have no conflicts of interest.

This research issue was supported by the grants from the National Natural Science Foundations of China under Grant Nos. 61671245, 51777016, 51607013, and 61601062.