ON THE NUMBER OF LIMIT CYCLES BY PERTURBING A PIECEWISE SMOOTH HAMILTON SYSTEM WITH TWO STRAIGHT LINES OF SEPARATION

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.

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Singularly Perturbed Oscillators with Exponential Nonlinearities

Journal of Dynamics and Differential Equations ◽

10.1007/s10884-021-10041-1 ◽

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.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416502047 ◽

2016 ◽

Vol 26 (12) ◽

pp. 1650204 ◽

Cited By ~ 6

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.

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Crossing Limit Cycles of Planar Piecewise Linear Hamiltonian Systems without Equilibrium Points

Mathematics ◽

10.3390/math8050755 ◽

2020 ◽

Vol 8 (5) ◽

pp. 755

Author(s):

Rebiha Benterki ◽

Jaume LLibre

Keyword(s):

Hamiltonian Systems ◽

Limit Cycles ◽

Piecewise Linear ◽

Equilibrium Points ◽

Upper Bounds ◽

Linear Hamiltonian Systems ◽

Straight Lines

In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they are separated by a hyperbola or an ellipse, they can have at most three crossing limit cycles. Additionally, we prove that these upper bounds are reached. Secondly, we show that there is an example of two crossing limit cycles when these systems have four zones separated by three straight lines.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501608 ◽

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.

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The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian Systems with Polycycles S(2) or S(3) Under Perturbations of Piecewise Smooth Polynomials

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420502302 ◽

2020 ◽

Vol 30 (15) ◽

pp. 2050230

Author(s):

Jiaxin Wang ◽

Liqin Zhao

Keyword(s):

Hamiltonian Systems ◽

Limit Cycles ◽

Upper Bounds ◽

Melnikov Function ◽

Piecewise Smooth ◽

Bifurcation Of Limit Cycles ◽

First Order ◽

Period Annulus ◽

Quadratic Hamiltonian

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles [Formula: see text] or [Formula: see text] under the perturbations of piecewise smooth polynomials with degree [Formula: see text]. Roughly speaking, for [Formula: see text], a polycycle [Formula: see text] is cyclically ordered collection of [Formula: see text] saddles together with orbits connecting them in specified order. The discontinuity is on the line [Formula: see text]. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary [Formula: see text] and [Formula: see text] are respectively [Formula: see text] and [Formula: see text] (taking into account the multiplicity).

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Uniqueness of Limit Cycles for a Class of Cubic Systems with Two Invariant Straight Lines

Discrete Dynamics in Nature and Society ◽

10.1155/2010/737068 ◽

2010 ◽

Vol 2010 ◽

pp. 1-17

Author(s):

Xiangdong Xie ◽

Fengde Chen ◽

Qingyi Zhan

Keyword(s):

Limit Cycle ◽

Limit Cycles ◽

Uniqueness Of Limit Cycles ◽

Cubic System ◽

Straight Lines ◽

Cubic Systems

A class of cubic systems with two invariant straight linesdx/dt=y(1-x2), dy/dt=-x+δy+nx2+mxy+ly2+bxy2.is studied. It is obtained that the focal quantities ofO(0,0)are,W0=δ; ifW0=0, thenW1=m(n+l); ifW0=W1=0, thenW2=−nm(b+1); ifW0=W1=W2=0, thenOis a center, and it has been proved that the above mentioned cubic system has at most one limit cycle surrounding weak focalO(0,0). This paper also aims to solve the remaining issues in the work of Zheng and Xie (2009).

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Topological Classification of Limit Cycles of Piecewise Smooth Dynamical Systems and Its Associated Non-Standard Bifurcations

Entropy ◽

10.3390/e16041949 ◽

2014 ◽

Vol 16 (4) ◽

pp. 1949-1968 ◽

Cited By ~ 1

Author(s):

John Taborda ◽

Ivan Arango

Keyword(s):

Dynamical Systems ◽

Limit Cycles ◽

Piecewise Smooth ◽

Topological Classification

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On the existence and cusp singularity of solutions to semilinear generalized Tricomi equations with discontinuous initial data

Communications in Contemporary Mathematics ◽

10.1142/s021919971450028x ◽

2015 ◽

Vol 17 (03) ◽

pp. 1450028 ◽

Cited By ~ 8

Author(s):

Zhuoping Ruan ◽

Ingo Witt ◽

Huicheng Yin

Keyword(s):

Initial Data ◽

Local Existence ◽

Piecewise Smooth ◽

Degree Zero ◽

Essential Difference ◽

Existence Of A Solution ◽

Discontinuous Initial Data ◽

Tricomi Equation ◽

Low Regularity Solution ◽

Straight Lines

In this paper, we are concerned with the local existence and singularity structures of low regularity solution to the semilinear generalized Tricomi equation [Formula: see text] with typical discontinuous initial data (u(0, x), ∂tu(0, x)) = (0, φ(x)), where m ∈ ℕ, x = (x1,…,xn), n ≥ 2, and f(t, x, u) is C∞ smooth on its arguments. When the initial data φ(x) is homogeneous of degree zero or piecewise smooth along the hyperplane {t = x1 = 0}, it is shown that the local solution u(t, x) ∈ L∞([0, T] × ℝn) exists and is C∞ away from the forward cuspidal conic surface [Formula: see text] or the cuspidal wedge-shaped surfaces [Formula: see text] respectively. On the other hand, for n = 2 and piecewise smooth initial data φ(x) along the two straight lines {t = x1 = 0} and {t = x2 = 0}, we establish the local existence of a solution [Formula: see text] and further show that [Formula: see text] in general due to the degenerate character of the equation under study, where [Formula: see text]. This is an essential difference to the well-known result for solution [Formula: see text] to the two-dimensional semilinear wave equation [Formula: see text] with (v(0, x), ∂tv(0, x)) = (0, φ(x)), where Σ0 = {t = |x|}, [Formula: see text] and [Formula: see text].

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