scholarly journals Study of Periodic Orbits in Periodic Perturbations of Planar Reversible Filippov Systems Having a Twofold Cycle

2020 ◽  
Vol 19 (2) ◽  
pp. 1343-1371
Author(s):  
Douglas D. Novaes ◽  
Tere M. Seara ◽  
Marco A. Teixeira ◽  
Iris O. Zeli
Keyword(s):  
Periodic Orbits ◽  
Filippov Systems ◽  
Periodic Perturbations

COMPLEX DYNAMICS IN A PENDULUM EQUATION WITH A PHASE SHIFT

2012 ◽  
Vol 22 (12) ◽  
pp. 1250307 ◽  
Author(s):  
XIANWEI CHEN ◽  
XIANGLING FU ◽  
ZHUJUN JING
Keyword(s):  
Phase Shift ◽  
Periodic Orbits ◽  
Period Doubling ◽  
Periodic Perturbation ◽  
Chaotic Attractors ◽  
Melnikov's Method ◽  
Periodic Perturbations ◽  
Melnikov’S Method ◽  
Dynamical Behaviors ◽  
Period Doubling Bifurcations

Pendulum equation with a phase shift, parametric and external excitations is investigated in detail. By applying Melnikov's method, we prove the criteria of existence of chaos under periodic perturbation. Numerical simulations, including bifurcation diagrams of fixed points, bifurcation diagrams of the system in three- and two-dimensional spaces, homoclinic and heteroclinic bifurcation surfaces, Maximum Lyapunov exponents (ML), Fractal Dimension (FD), phase portraits, Poincaré maps are plotted to illustrate the theoretical analysis, and to expose the complex dynamical behaviors including the onset of chaos, sudden conversion of chaos to period orbits, interior crisis, periodic orbits, the symmetry-breaking of periodic orbits, jumping behaviors of periodic orbits, new chaotic attractors including two-three-four-five-six-eight-band chaotic attractors, nonchaotic attractors, period-doubling bifurcations from period-1, 2, 3 and 5 to chaos, reverse period-doubling bifurcations from period-3 and 5 to chaos, and so on.By applying the second-order averaging method and Melnikov's method, we obtain the criteria of existence of chaos in an averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 1, 2, 4, but cannot prove the criteria of existence of chaos in the averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 3, 5 – 15, by Melnikov's method, where ν is not rational to ω. By using numerical simulation, we have verified our theoretical analysis and studied the effect of parameters of the original system on the dynamical behaviors generated under quasi-periodic perturbations, such as the onset of chaos, jumping behaviors of quasi-periodic orbits, interleaving occurrence of chaotic behaviors and nonchaotic behaviors, interior crisis, quasi-periodic orbits to chaotic attractors, sudden conversion of chaos to quasi-periodic behaviors, nonchaotic attractors, and so on. However, we did not find period-doubling and reverse period-doubling bifurcations. We found that the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations, and the dynamics with a phase shift are different from the dynamics without phase shift.

Bifurcation of periodic orbits in a class of planar Filippov systems

2008 ◽  
Vol 69 (10) ◽  
pp. 3610-3628 ◽  
Author(s):  
Zhengdong Du ◽  
Yurong Li ◽  
Weinian Zhang
Keyword(s):  
Periodic Orbits ◽  
Filippov Systems

scholarly journals Sliding Bifurcations in the Memristive Murali–Lakshmanan–Chua Circuit and the Memristive Driven Chua Oscillator

2020 ◽  
Vol 30 (14) ◽  
pp. 2050214
Author(s):  
A. Ishaq Ahamed ◽  
M. Lakshmanan
Keyword(s):  
Numerical Simulations ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Filippov Systems ◽  
Chua Circuit ◽  
Linear Characteristic ◽  
Discontinuity Mapping ◽  
Chua Oscillator ◽  
Zero Time ◽  
Time Discontinuity

In this paper, we report the occurrence of sliding bifurcations admitted by the memristive Murali–Lakshmanan–Chua circuit [Ishaq & Lakshmanan, 2013] and the memristive driven Chua oscillator [Ishaq et al., 2011]. Both of these circuits have a flux-controlled active memristor designed by the authors in 2011, as their nonlinear element. The three-segment piecewise-linear characteristic of this memristor bestows on the circuits two discontinuity boundaries, dividing their phase spaces into three subregions. For proper choice of parameters, these circuits take on a degree of smoothness equal to one at each of their two discontinuities, thereby causing them to behave as Filippov systems. Sliding bifurcations, which are characteristic of Filippov systems, arise when the periodic orbits in each of the subregions, interact with the discontinuity boundaries, giving rise to many interesting dynamical phenomena. The numerical simulations are carried out after incorporating proper zero time discontinuity mapping (ZDM) corrections. These are found to agree well with the experimental observations which we report here appropriately.

Adaptive Control of Transient Dynamics to Periodic Orbits

2014 ◽  
Vol 2 ◽  
pp. 82-85
Author(s):  
Hiroyasu Ando ◽  
Kazuyuki Aihara
Keyword(s):  
Adaptive Control ◽  
Periodic Orbits ◽  
Transient Dynamics

scholarly journals Data-Driven Stabilization of Periodic Orbits

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Jason J. Bramburger ◽  
J. Nathan Kutz ◽  
Steven L. Brunton
Keyword(s):  
Periodic Orbits ◽  
Data Driven
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