Complex mixed-mode vibration types triggered by the pitchfork bifurcation delay in a driven van der Pol-Duffing oscillator

2021 ◽  
Vol 411 ◽  
pp. 126522
Author(s):  
Xindong Ma ◽  
Yue Yu ◽  
Lifeng Wang
Keyword(s):  
Mixed Mode ◽  
Duffing Oscillator ◽  
Pitchfork Bifurcation ◽  
Van Der Pol ◽  
Mode Vibration ◽  
Bifurcation Delay

Canard-induced mixed mode oscillations as a mechanism for the Bonhoeffer-van der Pol circuit under parametric perturbation

Circuit World ◽  
2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Yue Yu ◽  
Cong Zhang ◽  
Zhenyu Chen ◽  
Zhengdi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbit ◽  
Large Amplitude ◽  
Mixed Mode ◽  
Periodic Perturbation ◽  
Van Der Pol ◽  
Content Type ◽  
Stable Focus ◽  
Mixed Mode Oscillations ◽  
Transition Mechanism

Purpose This paper aims to investigate the singular Hopf bifurcation and mixed mode oscillations (MMOs) in the perturbed Bonhoeffer-van der Pol (BVP) circuit. There is a singular periodic orbit constructed by the switching between the stable focus and large amplitude relaxation cycles. Using a generalized fast/slow analysis, the authors show the generation mechanism of two distinct kinds of MMOs. Design/methodology/approach The parametric modulation can be used to generate complicated dynamics. The BVP circuit is constructed as an example for second-order differential equation with periodic perturbation. Then the authors draw the bifurcation parameter diagram in terms of a containing two attractive regions, i.e. the stable relaxation cycle and the stable focus. The transition mechanism and characteristic features are investigated intensively by one-fast/two-slow analysis combined with bifurcation theory. Findings Periodic perturbation can suppress nonlinear circuit dynamic to a singular periodic orbit. The combination of these small oscillations with the large amplitude oscillations that occur due to canard cycles yields such MMOs. The results connect the theory of the singular Hopf bifurcation enabling easier calculations of where the oscillations occur. Originality/value By treating the perturbation as the second slow variable, the authors obtain that the MMOs are due to the canards in a supercritical case or in a subcritical case. This study can reveal the transition mechanism for multi-time scale characteristics in perturbed circuit. The information gained from such results can be extended to periodically perturbed circuits.

scholarly journals Hyperchaos and bifurcations in a driven Van der Pol–Duffing oscillator circuit

2014 ◽  
Vol 3 (4) ◽  
pp. 363-370 ◽  
Author(s):  
U. E. Vincent ◽  
B. R. Nana Nbendjo ◽  
A. A. Ajayi ◽  
A. N. Njah ◽  
P. V. E. McClintock
Keyword(s):  
Duffing Oscillator ◽  
Van Der Pol ◽  
Oscillator Circuit

Observe-Based Projective Synchronization of Chaotic Complex Modified Van Der Pol-Duffing Oscillator With Application to Secure Communication

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Ping Liu ◽  
Hongjun Song ◽  
Xiang Li
Keyword(s):  
Secure Communication ◽  
Chaotic Behavior ◽  
Design Method ◽  
Duffing Oscillator ◽  
Scaling Factor ◽  
Observer Design ◽  
Nonlinear Observer ◽  
Projective Synchronization ◽  
Chaotic Oscillator ◽  
Van Der Pol

This paper addresses the projective synchronization (PS) of the complex modified Van der Pol-Duffing (MVDPD for short) chaotic oscillator by using the nonlinear observer control and also discusses its applications to secure communication in theory. First, we construct the complex MVDPD oscillator and analysis its chaotic behavior. Moreover, an observer design method is applied to achieve PS of two identical MVDPD chaotic oscillators with complex offset terms, which are synchronized to the desired scale factor. The unpredictability of the scaling factor could further enhance the security of the communication. Finally, numerical simulations are given to demonstrate the effectiveness and feasibility of the proposed synchronization approach and also verify the success application to the proposed scheme’s in the secure communication.

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