Subharmonic Bifurcation for a Nonsmooth Oscillator

2017 ◽  
Vol 27 (10) ◽  
pp. 1750163 ◽  
Author(s):  
R. L. Tian ◽  
Z. J. Zhao ◽  
X. W. Yang ◽  
Y. F. Zhou
Keyword(s):  
Integral Method ◽  
Hamiltonian Function ◽  
Melnikov Function ◽  
Recursive Method ◽  
Subharmonic Bifurcation ◽  
Jump Discontinuities ◽  
Nonsmooth Systems ◽  
Pendulum Model ◽  
Existence Conditions ◽  
Smooth System

A nonsmooth pendulum model with multiple impulse effect is constructed to detect the bifurcation of a periodic orbit with multiple jump discontinuous points. Subharmonic Melnikov function of this kind of nonsmooth systems is studied. Differences of subharmonic Melnikov function between the nonsmooth system with multiple jump discontinuities and the smooth system are analyzed by using the Hamiltonian function and piecewise integral method. Applying the recursive method and perturbation principle, the effects of the jump discontinuous points on the subharmonic Melnikov function are converted to integral items which can be easily calculated. Hence, the subharmonic Melnikov function for the subharmonic orbit with multiple jump discontinuous points is obtained. Finally, the existence conditions for periodic motion of the subharmonic orbit are derived and the efficiency of the conclusions is verified via numerical simulations.

Heteroclinic Chaotic Threshold in a Nonsmooth System with Jump Discontinuities

2020 ◽  
Vol 30 (10) ◽  
pp. 2050141 ◽  
Author(s):  
R. L. Tian ◽  
T. Wang ◽  
Y. F. Zhou ◽  
J. Li ◽  
S. T. Zhu
Keyword(s):  
Melnikov Function ◽  
Heteroclinic Orbits ◽  
External Forcing ◽  
Jump Discontinuities ◽  
Nonsmooth Systems ◽  
Nonsmooth Dynamical System ◽  
Melnikov Theory ◽  
The One ◽  
Heteroclinic Solution ◽  
Unperturbed Part

In smooth systems, the form of the heteroclinic Melnikov chaotic threshold is similar to that of the homoclinic Melnikov chaotic threshold. However, this conclusion may not be valid in nonsmooth systems with jump discontinuities. In this paper, based on a newly constructed nonsmooth pendulum, a kind of impulsive differential system is introduced, whose unperturbed part possesses a nonsmooth heteroclinic solution with multiple jump discontinuities. Using the recursive method and the perturbation principle, the effects of the nonsmooth factors on the behaviors of the nonsmooth dynamical system are converted to the integral items which can be easily calculated. Furthermore, the extended Melnikov function is employed to obtain the nonsmooth heteroclinic Melnikov chaotic threshold, which implies that the existence of the nonsmooth heteroclinic orbits may be due to the breaking of the nonsmooth heteroclinic loops under the perturbation of damping, external forcing and nonsmooth factors. It is worth pointing out that the form of the nonsmooth heteroclinic Melnikov function is different from the one of the nonsmooth homoclinic Melnikov function, which is quite different from the classical Melnikov theory.

HOPF BIFURCATIONS FOR NEAR-HAMILTONIAN SYSTEMS

2009 ◽  
Vol 19 (12) ◽  
pp. 4117-4130 ◽  
Author(s):  
MAOAN HAN ◽  
JUNMIN YANG ◽  
PEI YU
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Efficient Algorithm ◽  
Hamiltonian Function ◽  
Melnikov Function ◽  
Quadratic System ◽  
New Method ◽  
Hopf Bifurcations ◽  
Computationally Efficient ◽  
Bifurcation Of Limit Cycles

In this paper, we consider bifurcation of limit cycles in near-Hamiltonian systems. A new method is developed to study the analytical property of the Melnikov function near the origin for such systems. Based on the new method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Moreover, we consider the case that the Hamiltonian function of the system depends on parameters, in addition to the coefficients involved in perturbations, which generates more limit cycles in the neighborhood of the origin. The results are applied to a quadratic system with cubic perturbations to show that the system can have five limit cycles in the vicinity of the origin.

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.

scholarly journals Algebraic Structure and Poisson Integral Method of Snake-Like Robot Systems

2021 ◽  
Vol 9 ◽  
Author(s):  
Fu Jing-Li ◽  
Xiang Chun ◽  
Meng Lei
Keyword(s):  
Algebraic Structure ◽  
Integral Method ◽  
Hamiltonian Function ◽  
First Integral ◽  
Poisson Integral ◽  
Integral Methods ◽  
Robot Systems ◽  
Generalized Momentum ◽  
Algebraic Forms

The algebraic structure and Poisson's integral of snake-like robot systems are studied. The generalized momentum, Hamiltonian function, generalized Hamilton canonical equations, and their contravariant algebraic forms are obtained for snake-like robot systems. The Lie-admissible algebra structures of the snake-like robot systems are proved and partial Poisson integral methods are applied to the snake-like robot systems. The first integral methods of the snake-like robot systems are given. An example is given to illustrate the results.

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.

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.

Melnikov-Type Method for a Class of Discontinuous Planar Systems and Applications

2014 ◽  
Vol 24 (02) ◽  
pp. 1450022 ◽  
Author(s):  
Shuangbao Li ◽  
Wei Zhang ◽  
Yuxin Hao
Keyword(s):  
Melnikov Method ◽  
Hamiltonian Function ◽  
Homoclinic Solution ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Planar System ◽  
Stable And Unstable Manifolds ◽  
Type Method ◽  
Planar Systems ◽  
Smooth System

In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed piecewise smooth planar system. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise smooth homoclinic solution transversally crossing the switching manifold. The Melnikov-type function is explicitly derived by using the Hamiltonian function to measure the distance of the perturbed stable and unstable manifolds. Finally, we apply the obtained results to study the chaotic dynamics of a concrete piecewise smooth system.

Chaos Generated by a Class of 3D Three-Zone Piecewise Affine Systems with Coexisting Singular Cycles

2020 ◽  
Vol 30 (14) ◽  
pp. 2050209
Author(s):  
Kai Lu ◽  
Wenjing Xu ◽  
Qigui Yang
Keyword(s):  
Mathematical Proof ◽  
Three Dimensional ◽  
Piecewise Affine Systems ◽  
Heteroclinic Cycles ◽  
Affine Systems ◽  
Numerical Examples ◽  
Piecewise Affine ◽  
Dynamical Behaviors ◽  
Nonsmooth Systems ◽  
Existence Conditions

It is a significant and challenging task to detect both the coexistence of singular cycles, mainly homoclinic and heteroclinic cycles, and chaos induced by the coexistence in nonsmooth systems. By analyzing the dynamical behaviors on manifolds, this paper proposes some criteria to accurately locate the coexistence of homoclinic cycles and of heteroclinic cycles in a class of three-dimensional (3D) piecewise affine systems (PASs), respectively. It further establishes the existence conditions of chaos arising from such coexistence, and presents a mathematical proof by analyzing the constructed Poincaré map. Finally, the simulations for two numerical examples are provided to validate the established results.

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