Bifurcation Analysis of Planar Piecewise Smooth Systems with a Line of Discontinuity

2016 ◽  
Vol 26 (06) ◽  
pp. 1650104
Author(s):  
Dingheng Pi ◽  
Shihong Xu
Keyword(s):  
Hamiltonian System ◽  
Linear System ◽  
Qualitative Analysis ◽  
Bifurcation Analysis ◽  
Dynamical Behavior ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Differential Systems ◽  
Line Of Discontinuity ◽  
Smooth System

In this paper, we consider bifurcations of a class of planar piecewise smooth differential systems constituted by a general linear system and a quadratic Hamiltonian system. The linear system has four parameters. When the parameters vary in different regions, the left linear system can have a saddle, a node or a focus. For each case, we provide a completely qualitative analysis of the dynamical behavior for this piecewise smooth system. Our results generalize and improve the results in this direction.

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.

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.

Bifurcation analysis of a piecewise smooth system with non-linear characteristics

2005 ◽  
Vol 33 (4) ◽  
pp. 263-279 ◽  
Author(s):  
Takuji Kousaka ◽  
Tetsushi Ueta ◽  
Yue Ma ◽  
Hiroshi Kawakami
Keyword(s):  
Bifurcation Analysis ◽  
Piecewise Smooth ◽  
Piecewise Smooth System ◽  
Non Linear ◽  
Smooth System

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.

BORDER-COLLISION BIFURCATIONS AND CHAOTIC OSCILLATIONS IN A PIECEWISE-SMOOTH DYNAMICAL SYSTEM

2001 ◽  
Vol 11 (12) ◽  
pp. 2977-3001 ◽  
Author(s):  
ZHANYBAI T. ZHUSUBALIYEV ◽  
EVGENIY A. SOUKHOTERIN ◽  
ERIK MOSEKILDE
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Voltage Converter ◽  
Piecewise Smooth Systems ◽  
Border Collision Bifurcations ◽  
Solution Type ◽  
Nonautonomous Differential Equations ◽  
The Subject ◽  
Smooth System

Many problems of engineering and applied science result in the consideration of piecewise-smooth dynamical systems. Examples are relay and pulse-width control systems, impact oscillators, power converters, and various electronic circuits with piecewise-smooth characteristics. The subject of investigation in the present paper is the dynamical model of a constant voltage converter which represents a three-dimensional piecewise-smooth system of nonautonomous differential equations. A specific type of phenomena that arise in the dynamics of piecewise-smooth systems are the so-called border-collision bifurcations. The paper contains a detailed analysis of this type of bifurcational transition in the dynamics of the voltage converter, in particular, the merging and subsequent disappearance of cycles of different types, change of solution type, and period-doubling, -tripling, -quadrupling and -quintupling. We show that a denumerable set of unstable cycles can arise together with stable cycles at border-collision bifurcations. The characteristic peculiarities of border-collision bifurcational transitions in piecewise-smooth systems are described and we provide a comparison with some recent results.

MANDELBROT-LIKE BIFURCATION STRUCTURES IN CHAOS BAND SCENARIO OF SWITCHED CONVERTER WITH DELAYED-PWM CONTROL

2010 ◽  
Vol 20 (01) ◽  
pp. 99-119 ◽  
Author(s):  
J. A. TABORDA ◽  
F. ANGULO ◽  
G. OLIVAR
Keyword(s):  
Parameter Space ◽  
Dynamical Behavior ◽  
Power Converter ◽  
Piecewise Smooth ◽  
Pwm Control ◽  
Dimensional Parameter ◽  
Piecewise Smooth System ◽  
Pulse Width Modulator ◽  
Bifurcation Structures ◽  
Smooth System

In this paper, we report Mandelbrot-like bifurcation structures in a one-dimensional parameter space of real numbers corresponding to a dc-dc power converter modeled as a piecewise-smooth system with three zones. These fractal patterns have been studied in two-dimensional parameter space for smooth systems, but for nonsmooth systems has not been reported yet. The Mandelbrot-like sets we found are created in transition from the torus band to chaos band scenarios exhibited by a dc-dc buck power converter controlled by Delayed Pulse-Width Modulator (PWM) based on Zero Average Dynamics (or ZAD strategy), which corresponds to a piecewise-smooth system (PWS). The real parameter is provided by the PWM control strategy, namely ZAD strategy, and it can be varied in a large range, ideally (-∞, +∞). At -∞ and +∞ the dynamical behavior is the same, and thus we will describe the synamics in an ring-like parameter space. Mandelbrot-like borders are built by four chaotic bands, therefore these structures can be thought as instability islands where the state variables cannot be located. Using the Poincaré map approach we characterize the bifurcation structures and we describe recurrent patterns in different scales.

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.

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