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.

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.

Melnikov Method for a Class of Planar Hybrid Piecewise-Smooth Systems

2016 ◽  
Vol 26 (02) ◽  
pp. 1650030 ◽  
Author(s):  
Shuangbao Li ◽  
Wensai Ma ◽  
Wei Zhang ◽  
Yuxin Hao
Keyword(s):  
Melnikov Method ◽  
Homoclinic Solution ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Stable And Unstable Manifolds ◽  
Straight Line ◽  
Unstable Manifolds ◽  
Smooth System

In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed planar hybrid piecewise-smooth systems. In this class, the switching manifold is a straight line which divides the plane into two zones, and the dynamics in each zone is governed by a smooth system. When a trajectory reaches the separation line, then a reset map is applied instantaneously before entering the trajectory in the other zone. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise-smooth homoclinic solution transversally crossing the switching manifold. Then, we study the persistence of the homoclinic orbit under a nonautonomous periodic perturbation and the reset map. To achieve this objective, we obtain the Melnikov function to measure the distance of the perturbed stable and unstable manifolds and present the theorem for homoclinic bifurcations for the class of planar hybrid piecewise-smooth systems. Furthermore, we employ the obtained Melnikov function to detect the chaotic boundaries for a concrete planar hybrid piecewise-smooth system.

Melnikov Method for a Three-Zonal Planar Hybrid Piecewise-Smooth System and Application

2016 ◽  
Vol 26 (01) ◽  
pp. 1650014
Author(s):  
Shuangbao Li ◽  
Wensai Ma ◽  
Wei Zhang ◽  
Yuxin Hao
Keyword(s):  
Periodic Orbits ◽  
Melnikov Method ◽  
Perturbation Parameter ◽  
Melnikov Function ◽  
Displacement Function ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
First Order ◽  
Piecewise Smooth System ◽  
Smooth System

In this paper, we extend the well-known Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth systems, defined in three domains separated by two switching manifolds [Formula: see text] and [Formula: see text]. The dynamics in each domain is governed by a smooth system. When an orbit reaches the separation lines, then a reset map describing an impacting rule applies instantaneously before the orbit enters into another domain. We assume that the unperturbed system has a continuum of periodic orbits transversally crossing the separation lines. Then, we wish to study the persistence of the periodic orbits under an autonomous perturbation and the reset map. To achieve this objective, we first choose four appropriate switching sections and build a Poincaré map, after that, we present a displacement function and carry on the Taylor expansion of the displacement function to the first-order in the perturbation parameter [Formula: see text] near [Formula: see text]. We denote the first coefficient in the expansion as the first-order Melnikov function whose zeros provide us the persistence of periodic orbits under perturbation. Finally, we study periodic orbits of a concrete planar hybrid piecewise-smooth system by the obtained Melnikov function.

BIFURCATION OF LIMIT CYCLES BY PERTURBING PIECEWISE HAMILTONIAN SYSTEMS

2010 ◽  
Vol 20 (05) ◽  
pp. 1379-1390 ◽  
Author(s):  
XIA LIU ◽  
MAOAN HAN
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Unperturbed System ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
General Perturbation ◽  
Smooth System

In this paper, the general perturbation of piecewise Hamiltonian systems on the plane is considered. When the unperturbed system has a family of periodic orbits, similar to the perturbations of smooth system, an expression of the first order Melnikov function is derived, which can be used to study the number of limit cycles bifurcated from the periodic orbits. As applications, the number of bifurcated limit cycles of several concrete piecewise systems are presented.

Stability and Limit Cycle Bifurcation for Two Kinds of Generalized Double Homoclinic Loops in Planar Piecewise Smooth Systems

2014 ◽  
Vol 24 (12) ◽  
pp. 1450153
Author(s):  
Feng Liang ◽  
Maoan Han
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Upper Bound ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Homoclinic Loop ◽  
Limit Cycle Bifurcation

In this paper, we present two kinds of generalized double homoclinic loops in planar piecewise smooth systems. For their stability a criterion is provided. Under nondegenerate conditions, we prove that for each case there are at most five limit cycles which can be bifurcated from the generalized double homoclinic loop. Especially, we construct two concrete systems to show that the upper bound can be achieved in both cases.

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 Nonhyperbolic Limit Cycles in Piecewise Smooth Planar Systems with Finitely Many Zones

2017 ◽  
Vol 27 (10) ◽  
pp. 1750162 ◽  
Author(s):  
Yurong Li ◽  
Liping Yuan ◽  
Zhengdong Du
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Bifurcation Of Limit Cycles ◽  
Planar System ◽  
Planar Systems ◽  
The Stability ◽  
Straight Lines ◽  
First Time

Like for smooth systems, it is very important to discuss the stability and bifurcation of limit cycles in a piecewise smooth planar system. Most of the previous works focus only on hyperbolic limit cycles. Few works have considered nonhyperbolic limit cycles. In fact, to date, no concrete examples of piecewise smooth planar system with nonhyperbolic limit cycles have been given in literature. In this paper, we consider for the first time the bifurcation of nonhyperbolic limit cycles in piecewise smooth planar systems with discontinuities on finitely many straight lines intersecting at the origin. We present a method of Melnikov type to derive two quantities which can be used to determine the stability and the number of limit cycles that can bifurcate from a nonhyperbolic limit cycle of a piecewise smooth planar system. As applications, we present two examples of piecewise smooth systems with two and three zones respectively whose unperturbed system has a nonhyperbolic limit cycle.

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.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

LIMIT CYCLE BIFURCATIONS IN NEAR-HAMILTONIAN SYSTEMS BY PERTURBING A NILPOTENT CENTER

2008 ◽  
Vol 18 (10) ◽  
pp. 3013-3027 ◽  
Author(s):  
MAOAN HAN ◽  
JIAO JIANG ◽  
HUAIPING ZHU
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Unperturbed System ◽  
Bifurcation Theorem ◽  
First Order ◽  
Cubic System ◽  
Limit Cycle Bifurcation ◽  
Almost All ◽  
Nilpotent Center ◽  
Computing Method

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented.

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