Piecewise Hamiltonian System Research Articles

Melnikov functions and limit cycle bifurcations for a class of piecewise Hamiltonian systems

<abstract><p>This study evaluated the number of limit cycles for a class of piecewise Hamiltonian systems with two zones separated by two semi-straight lines. First, we obtained explicit expressions of higher Melnikov functions. Then we applied these expressions to find the upper bounds of the number of limit cycles bifurcated from a period annulus of a piecewise polynomial Hamiltonian system.</p></abstract>

Crossing Limit Cycles of Planar Piecewise Hamiltonian Systems with Linear Centers Separated by Two Parallel Straight Lines

Crossing Limit Cycles of Planar Piecewise Hamiltonian Systems with Linear Centers Separated by Two Parallel Straight Lines

Melnikov-type method for a class of planar hybrid piecewise-smooth systems with impulsive effect and noise excitation: Heteroclinic orbits.

The classical Melnikov method for heteroclinic orbits is extended theoretically to a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation. We assume that the unperturbed system is a piecewise Hamiltonian system with a pair of heteroclinic orbits. The heteroclinic orbit transversally jumps across the first switching manifold by an impulsive effect and crosses the second switching manifold continuously. In effect, the trajectory of the corresponding perturbed system crosses the second switching manifold by applying the reset map describing the impact rule instantaneously. The random Melnikov process of such systems is then derived by measuring the distance of perturbed stable and unstable manifolds, and the criteria for the onset of chaos with or without noise excitation is established. In this derivation process, we overcome the difficulty that the derivation method of the corresponding homoclinic case cannot be directly used due to the difference between the symmetry of the homoclinic orbit and the asymmetry of the heteroclinic orbit. Finally, we investigate the complicated dynamics of a particular piecewise-smooth system with and without noise excitation under the reset maps, impulsive effect, and non-autonomous periodic and damping perturbations by this new extended method and numerical simulations. The numerical results verify the correctness of the theoretical results and demonstrate that this extended method is simple and effective for studying the dynamics of such systems.

Melnikov-type method for a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation: Homoclinic orbits.

The Melnikov method is extended to a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation when an unperturbed system is a piecewise Hamiltonian system with a homoclinic orbit. The homoclinic orbit continuously crosses the first switching manifold and transversally jumps across the second switching manifold by the impulsive effect. The trajectory of the corresponding perturbed system crosses the first switching manifold by applying the reset map describing the impact rule instantaneously. Then, the random Melnikov process of such systems is derived and the criteria for the onset of chaos with or without noise excitation are established. In addition, the complicated dynamics of concrete piecewise-smooth systems with or without noise excitation under the reset maps, impulsive effect, and non-autonomous periodic and damping perturbations are investigated by this extended method and numerical simulations.

Limit Cycles Bifurcating from a Periodic Annulus in Discontinuous Planar Piecewise Linear Hamiltonian Differential System with Three Zones

In this paper, we study the number of limit cycles that can bifurcate from a periodic annulus in a discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines. We prove that if the central subsystem, i.e. the system defined between the two parallel lines, has a real center and the other subsystems have centers or saddles, then we have at least three limit cycles that appear after perturbations of the periodic annulus. For this, we study the number of zeros of a Melnikov function for piecewise Hamiltonian system and present a normal form for this system in order to simplify the computations.

Melnikov functions of arbitrary order for piecewise smooth differential systems in [formula omitted] and applications

In this paper we develop an arbitrary order Melnikov function to study limit cycles bifurcating from a periodic submanifold for autonomous piecewise smooth differential systems in Rn with two zones separated by a hyperplane. This result not only extends some of the known results on the Melnikov theory in dimension and order but also compensates for some defects of the averaging theory in studying the limit cycle bifurcation of autonomous systems from a periodic submanifold. To demonstrate the application of our theoretical result and its superiority for some systems to the existing averaging theory, we study the maximum number of limit cycles bifurcating from an n-dimensional periodic submanifold caused by non-smooth centers of the fold-fold type, providing an upper bound for any order piecewise polynomial perturbations of degree m. Concerning the planar case of the unperturbed system, a piecewise Hamiltonian system, we obtain a better upper bound for piecewise polynomial Hamiltonian perturbations up to order two. The realizability of these upper bounds is also discussed.

Up to second order Melnikov functions for general piecewise Hamiltonian systems with nonregular separation line

In this paper, we consider a perturbation of planar general piecewise Hamiltonian systems with nonregular separation line. A general explicit expression of the first and second order Melnikov functions is presented for such piecewise Hamiltonian systems. In addition, the maximum number of limit cycles for piecewise linear differential system is an important topic that many researchers are concerned with. Different from considering a piecewise linear perturbation of a linear center in the previous research, we apply our explicit expression to estimate the number of limit cycles bifurcated from a class of piecewise linear Hamiltonian systems. More specifically, the unperturbed system we study is piecewise smooth and constructed by a linear center and a constant differential system, which is difficult to be considered by a polar coordinate transformation. And we obtain the upper bounds to be 4 and 5 for the limit cycles bifurcated from the periodic orbits by using the first and second order Melnikov functions, respectively. Moreover the upper bounds are sharp.

Limit Cycles Bifurcating from an Invisible Fold–Fold in Planar Piecewise Hamiltonian Systems

The aim of this article is twofold. Firstly, we study the existence of limit cycles in a family of piecewise smooth vector fields corresponding to an unfolding of an invisible fold–fold singularity. More precisely, given a positive integer k, we prove that this family has exactly k hyperbolic crossing limit cycles in a suitable neighborhood of this singularity. Secondly, we provide a complete study relating the existence and stability of these crossing limit cycles with the limit cycles of the family of smooth vector fields obtained by the regularization method. This relationship is obtained by studying the equivalence between the signs of the Lyapunov coefficients of the family of piecewise smooth vector fields and the ones of its regularization.

Second Order Melnikov Functions of Piecewise Hamiltonian Systems

In this paper, we consider the general perturbations of piecewise Hamiltonian systems. A formula for the second order Melnikov functions is derived when the first order Melnikov functions vanish. As an application, we can improve an upper bound of the number of bifurcated limit cycles of a piecewise Hamiltonian system with quadratic polynomial perturbations.

Lyapunov coefficients for an invisible fold-fold singularity in planar piecewise Hamiltonian systems

We study the stability of an invisible fold-fold singularity of planar piecewise smooth Hamiltonian vector fields by computing some kind of Lyapunov coefficients. We obtain the general expressions for the first five Lyapunov coefficients. As a consequence, the bifurcation diagrams, illustrating the number, types and positions of the bifurcating small amplitude crossing limit cycles for these vector fields, are determined.

Random Disordered Periodical Input Induced Chaos in Discontinuous Systems

In this paper, we extend the random Melnikov method from stochastic systems with a continuous vector field to discontinuous systems driven by a random disordered periodic input under the assumption that the unperturbed system is a piecewise Hamiltonian system. By measuring the distance of the perturbed stable and unstable manifolds, the nonsmooth random Melnikov process can be derived in detail, and then the mean square criterion for the onset of chaos is established in the statistical sense. It is shown that the threshold for the onset of chaos depends on the stochastic force and a scalar function of hypersurface. Finally, an example is given to analyze the chaotic dynamics using this extended approach, and discuss the effects of noise intensity on the dynamical behaviors of the system. The results indicate that the increase of the noise intensity will result in a chaotic motion of the discontinuous stochastic system and the changes of possible chaotic degree in the phase space. At the same time, the effects of noise intensity on chaos are further investigated through the system response including time history and phase portraits, Poincaré maps and [Formula: see text]-[Formula: see text] test.

Limit Cycle Bifurcations Near a Piecewise Smooth Generalized Homoclinic Loop with a Saddle-Fold Point

In this paper, we suppose that a planar piecewise Hamiltonian system, with a straight line of separation, has a piecewise generalized homoclinic loop passing through a Saddle-Fold point, and assume that there exists a family of piecewise smooth periodic orbits near the loop. By studying the asymptotic expansion of the first order Melnikov function corresponding to the period annulus, we obtain the formulas of the first six coefficients in the expansion, based on which, we provide a lower bound for the maximal number of limit cycles bifurcated from the period annulus. As applications, two concrete systems are considered. Especially, the first one reveals that a quadratic piecewise Hamiltonian system can have five limit cycles near a generalized homoclinic loop under a quadratic piecewise smooth perturbation. Compared with the smooth case [Horozov & Iliev, 1994; Han et al., 1999], three more limit cycles are found.

Limit Cycle Bifurcations by Perturbing a Piecewise Hamiltonian System with a Double Homoclinic Loop

This paper is concerned with the bifurcation problem of limit cycles by perturbing a piecewise Hamiltonian system with a double homoclinic loop. First, the derivative of the first Melnikov function is provided. Then, we use it, together with the analytic method, to derive the asymptotic expansion of the first Melnikov function near the loop. Meanwhile, we present the first coefficients in the expansion, which can be applied to study the limit cycle bifurcation near the loop. We give sufficient conditions for this system to have [Formula: see text] limit cycles in the neighborhood of the loop. As an application, a piecewise polynomial Liénard system is investigated, finding six limit cycles with the help of the obtained method.

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

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.

Limit cycle bifurcations near a generalized homoclinic loop in piecewise smooth systems with a hyperbolic saddle on a switch line

In this paper, we suppose a planar piecewise Hamiltonian system has a generalized homoclinic loop with a hyperbolic saddle on a switch line, and assume that there are two families of periodic orbits on both sides of the loop. Under perturbations we first give the expansion of the first Melnikov function near the loop. Then by using the first coefficients in the expansion, we study the number of limit cycles bifurcating from the homoclinic loop.

Limit cycles for perturbing a piecewise linear Hamiltonian system with one or two saddles

We study bifurcations of limit cycles arising after perturbations of linear piecewise Hamiltonian systems. In this paper we find bounds for the numbers of limit cycles for several families of which phase portraits were classified in Xiong and Han (2013) [14].

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

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.

Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems

In this paper, we study the bifurcation of limit cycles in piecewise smooth systems by perturbing a piecewise Hamiltonian system with a generalized homoclinic or generalized double homoclinic loop. We first obtain the form of the expansion of the first Melnikov function. Then by using the first coefficients in the expansion, we give some new results on the number of limit cycles bifurcated from a periodic annulus near the generalized (double) homoclinic loop. As applications, we study the number of limit cycles of a piecewise near-Hamiltonian systems with a generalized homoclinic loop and a central symmetric piecewise smooth system with a generalized double homoclinic loop.

BIFURCATION OF LIMIT CYCLES BY PERTURBING PIECEWISE HAMILTONIAN SYSTEMS

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.

Three dimensional mirror

In this paper we develop an arbitrary order Melnikov function to study limit cycles bifurcating from a periodic submanifold for autonomous piecewise smooth differential systems in Rn with two zones separated by a hyperplane. This result not only extends some of the known results on the Melnikov theory in dimension and order but also compensates for some defects of the averaging theory in studying the limit cycle bifurcation of autonomous systems from a periodic submanifold. To demonstrate the application of our theoretical result and its superiority for some systems to the existing averaging theory, we study the maximum number of limit cycles bifurcating from an n-dimensional periodic submanifold caused by non-smooth centers of the fold-fold type, providing an upper bound for any order piecewise polynomial perturbations of degree m. Concerning the planar case of the unperturbed system, a piecewise Hamiltonian system, we obtain a better upper bound for piecewise polynomial Hamiltonian perturbations up to order two. The realizability of these upper bounds is also discussed.