Regularization of Planar Piecewise Smooth Systems with a Heteroclinic Loop

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Zhongjian Wang ◽  
Dingheng Pi
Keyword(s):  
Limit Cycles ◽  
Regular Point ◽  
Stable Limit Cycle ◽  
Piecewise Smooth ◽  
Stable Limit ◽  
Piecewise Smooth Systems ◽  
Heteroclinic Loop ◽  
Piecewise Smooth System ◽  
Stable Periodic ◽  
Smooth System

In this paper, we study bifurcations of the regularized systems of planar piecewise smooth systems, which have a visible fold-regular point and a sliding or grazing heteroclinic loop. Our results show that if the planar piecewise smooth system with a sliding heteroclinic loop undergoes sliding heteroclinic bifurcation, then the regularized system can bifurcate with a stable limit cycle passing through the regularized region and at most two limit cycles outside the regularized region. The regularized system can have at most three periodic orbits. When the upper subsystem is a Hamiltonian system, the regularized system can bifurcate with a semi-stable periodic orbit. Finally, we discuss two cases when the heteroclinic loop of a piecewise smooth system remains unbroken under a small perturbation. Our results show that the regularized system can bifurcate at most two limit cycles from an inner unstable grazing heteroclinic loop.

Bifurcation of Periodic Orbits Crossing Switching Manifolds Multiple Times in Planar Piecewise Smooth Systems

2019 ◽  
Vol 29 (12) ◽  
pp. 1950160
Author(s):  
Zhihui Fan ◽  
Zhengdong Du
Keyword(s):  
Limit Cycle ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Sufficient Conditions ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Smooth Curves ◽  
Switching Curve ◽  
Smooth System

In this paper, we discuss the bifurcation of periodic orbits in planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. We assume that the unperturbed system has either a limit cycle or a periodic annulus such that the limit cycle or each periodic orbit in the periodic annulus crosses every switching curve transversally multiple times. When the unperturbed system has a limit cycle, we give the conditions for its stability and persistence. When the unperturbed system has a periodic annulus, we obtain the expression of the first order Melnikov function and establish sufficient conditions under which limit cycles can bifurcate from the annulus. As an example, we construct a concrete nonlinear planar piecewise smooth system with three zones with 11 limit cycles bifurcated from the periodic annulus.

scholarly journals Bifurcations of Finite-Time Stable Limit Cycles from Focus Boundary Equilibria in Impacting Systems, Filippov Systems, and Sweeping Processes

2018 ◽  
Vol 28 (10) ◽  
pp. 1850126 ◽  
Author(s):  
Oleg Makarenkov ◽  
Lakmi Niwanthi Wadippuli Achchige
Keyword(s):  
Limit Cycles ◽  
Finite Time ◽  
Stable Limit Cycle ◽  
Piecewise Smooth ◽  
Stable Limit ◽  
Bifurcation Of Limit Cycles ◽  
Filippov Systems ◽  
Boundary Equilibrium ◽  
Sweeping Processes ◽  
Linearized System

We establish a theorem on bifurcation of limit cycles from a focus boundary equilibrium of an impacting system, which is universally applicable to prove the bifurcation of limit cycles from focus boundary equilibria in other types of piecewise-smooth systems, such as Filippov systems and sweeping processes. Specifically, we assume that one of the subsystems of the piecewise-smooth system under consideration admits a focus equilibrium that lie on the switching manifold at the bifurcation value of the parameter. In each of the three cases, we derive a linearized system which is capable of concluding the occurrence of a finite-time stable limit cycle from the above-mentioned focus equilibrium when the parameter crosses the bifurcation value. Examples illustrate how conditions of our theorems lead to closed-form formulas for the coefficients of the linearized system.

Sliding Bifurcation in Planar Piecewise Smooth Systems with Two Zones Separated by an Ellipse

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Fang Wu ◽  
Lihong Huang ◽  
Jiafu Wang
Keyword(s):  
Bifurcation Diagram ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Piecewise Smooth ◽  
Piecewise Smooth System ◽  
Sliding Bifurcation ◽  
Switching Curve ◽  
Necessary And Sufficient ◽  
Smooth System

The objective of this paper is to study the sliding bifurcation in a planar piecewise smooth system with an elliptic switching curve. Some new phenomena are observed, such as a crossing limit cycle containing four intersections with the switching curve, sliding cycles having four sliding segments, and sliding cycles consisting of the entire switching curve. Firstly, we investigate the bifurcation of sliding cycle from a sliding heteroclinic connection to two cusps and show the appearance of one sliding cycle with two folds. To plot the bifurcation diagram, a planar piecewise linear system with two zones separated by an ellipse are considered. Moreover, we study in more detail the unfolding of a sliding cycle connecting four cusps by exhibiting its complete bifurcation diagram. More precisely, we explore the necessary and sufficient conditions for the existence of limit cycles and derive the concrete bifurcation curves. Additionally, a simple piecewise smooth system with nonlinear subsystems is studied, which shows the possibility of the existence of two nested limit cycles. Finally, numerical simulations are given to confirm the theoretical analysis.

Limit Cycles from Perturbing a Piecewise Smooth System with a Center and a Homoclinic Loop

2021 ◽  
Vol 31 (10) ◽  
pp. 2150159
Author(s):  
Ai Ke ◽  
Maoan Han
Keyword(s):  
Lower Bound ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Homoclinic Loop ◽  
First Order ◽  
Piecewise Smooth System ◽  
Period Annulus ◽  
Smooth System

We study bifurcations of limit cycles arising after perturbations of a special piecewise smooth system, which has a center and a homoclinic loop. By using the Picard–Fuchs equation, we give an upper bound of the maximum number of limit cycles bifurcated from the period annulus between the center and the homoclinic loop. Furthermore, by applying the method of first-order Melnikov function we obtain a lower bound of the maximum number of limit cycles bifurcated from the center.

scholarly journals Singularly Perturbed Oscillators with Exponential Nonlinearities

2021 ◽  
Author(s):  
S. Jelbart ◽  
K. U. Kristiansen ◽  
P. Szmolyan ◽  
M. Wechselberger
Keyword(s):  
Limit Cycles ◽  
Space Dimension ◽  
Blow Up ◽  
Exponential Convergence ◽  
Model System ◽  
Piecewise Smooth ◽  
Singularly Perturbed ◽  
Model Systems ◽  
Smooth System ◽  
Unique Limit

AbstractSingular exponential nonlinearities of the form $$e^{h(x)\epsilon ^{-1}}$$ e h ( x ) ϵ - 1 with $$\epsilon >0$$ ϵ > 0 small occur in many different applications. These terms have essential singularities for $$\epsilon =0$$ ϵ = 0 leading to very different behaviour depending on the sign of h. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the $$\epsilon \rightarrow 0$$ ϵ → 0 limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we use a blow-up approach to demonstrate that this exponential convergence can be harmless in some cases while in other scenarios it can lead to further degeneracies. For our second model system, we deal with such degeneracies due to exponentially small terms by extending the space dimension, following the approach in Kristiansen (Nonlinearity 30(5): 2138–2184, 2017), and prove—for both systems—existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.

Analysis of chemical kinetic systems over the entire parameter space - I. The Sal’nikov thermokinetic oscillator

1988 ◽  
Vol 416 (1851) ◽  
pp. 391-402 ◽  
Keyword(s):  
Steady State ◽  
Parameter Space ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Chemical Kinetic ◽  
Phase Portraits ◽  
Stable Steady State ◽  
Stable Limit ◽  
Unstable Steady State ◽  
Undamped Oscillations

In this series of papers we re-examine, using recently developed techniques, some chemical kinetic models that have appeared in the literature with a view to obtaining a complete description of all the qualitatively distinct behaviour that the system can exhibit. Each of the schemes is describable by two coupled ordinary differential equations and contain at most three independent parameters. We find that even with these relatively simple chemical schemes there are regions of parameter space in which the systems display behaviour not previously found. Quite often these regions are small and it seems unlikely that they would be found via classical methods. In part I of the series we consider one of the thermally coupled kinetic oscillator models studied by Sal’nikov. He showed that there is a region in parameter space in which the system would be in a state of undamped oscillations because the relevant phase portrait consists of an unstable steady state surrounded by a stable limit cycle. Our analysis has revealed two further regions in which the phase portraits contain, respectively, two limit cycles of opposite stability enclosing a stable steady state and three limit cycles of alternating stability surrounding an unstable steady state. This latter region is extremely small, so much so that it could be reasonably neglected in any predictions made from the model.

scholarly journals Global Dynamics of a Piecewise Smooth System for Brain Lactate Metabolism

2018 ◽  
Vol 18 (1) ◽  
pp. 315-332
Author(s):  
J.-P. Françoise ◽  
Hongjun Ji ◽  
Dongmei Xiao ◽  
Jiang Yu
Keyword(s):  
Piecewise Smooth ◽  
Global Dynamics ◽  
Lactate Metabolism ◽  
Piecewise Smooth System ◽  
Smooth System ◽  
Brain Lactate

Stochastically perturbed limit cycles

1978 ◽  
Vol 15 (02) ◽  
pp. 311-320
Author(s):  
Charles J. Holland
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Probability Measure ◽  
Limit Cycles ◽  
Noise Intensity ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Additive White Noise ◽  
Small Intensity ◽  
Deterministic Dynamical Systems

In this paper we examine the effects of perturbing certain deterministic dynamical systems possessing a stable limit cycle by an additive white noise term with small intensity. We place assumptions on the system guaranteeing that when noise is present the corresponding random process generates an ergodic probability measure. We then determine the behavior of the invariant measure when the noise intensity is small.

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