Limit Cycles of Piecewise Smooth Differential Equations on Two Dimensional Torus

In this paper we study the limit cycles of some classes of piecewise smooth vector fields defined in the two dimensional torus. The piecewise smooth vector fields that we consider are composed by linear, Ricatti with constant coefficients and perturbations of these one, which are given in (3). Considering these piecewise smooth vector fields we characterize the global dynamics, studying the upper bound of number of limit cycles, the existence of non-trivial recurrence and a continuum of periodic orbits. We also present a family of piecewise smooth vector fields that posses a finite number of fold points and, for this family we prove that for any 2k number of limit cycles there exists a piecewise smooth vector fields in this family that presents k number of limit cycles and prove that some classes of piecewise smooth vector fields presents a non-trivial recurrence or a continuum of periodic orbits.     

Periodic Orbits for Planar Piecewise Smooth Systems with a Line of Discontinuity

In this work we examine the existence of periodic orbits for planar piecewise smooth dynamical systems with a line of discontinuity. Unlike existing works, we consider the case where the line does not contain the equilibrium point. Most of the analysis is for a family of piecewise linear systems, and we discover new phenomena which produce the birth of periodic orbits, as well as new bifurcation phenomena of the periodic orbits themselves. A model nonlinear piecewise smooth systems is examined as well.     

Elucidating the escape dynamics of the four hill potential

This paper treads discontinuous bifurcation in piecewise smooth systems of Filippov type. These bifurcations occur when a fixed point or a periodic orbit crosses with the border between two regions of smooth behavior. A detailed analysis of generalization Poincaré map and monodromy matrix which are related shows that subfamily of system with invariant cone-like objects is foliated by periodic orbits and determines its stability. In addition, we introduce a theoretical framework for analyzing 3D perturbed nonlinear piecewise smooth systems and give necessary conditions so that different types of bifurcations occur. The analysis identifies criteria for the existence of a novel bifurcation based on sensitively the location of the return map. Moreover, the piecewise smooth Melnikov function and sufficient conditions of the existence of the periodic orbits for nonlinear perturbed system are explicitly obtained.     

Homoclinic Trajectories in Discontinuous Systems

We study bifurcations of bounded solutions from homoclinic orbits for time-perturbed discontinuous systems. Functional analytic method is used. An illustrative example of a periodically perturbed piecewise linear differential equation in is presented.      

Analysis of the periodic solutions of a smooth and discontinuous oscillator

In this paper, the periodic solutions of the smooth and discontinuous (SD) oscillator, which is a strongly irrational nonlinear system are discussed for the system having a viscous damping and an external harmonic excitation. A four dimensional averaging method is employed by using the complete Jacobian elliptic integrals directly to obtain the perturbed primary responses which bifurcate from both the hyperbolic saddle and the non-hyperbolic centres of the unperturbed system. The stability of these periodic solutions is analysed by examining the four dimensional averaged equation using Lyapunov method. The results presented herein this paper are valid for both smooth ( α > 0) and discontinuous ( α = 0) stages providing the answer to the question why the averaging theorem spectacularly fails for the case of medium strength of external forcing in the Duffing system analysed by Holmes. Numerical calculations show a good agreement with the theoretical predictions and an excellent efficiency of the analysis for this particular system, which also suggests the analysis is applicable to strongly nonlinear systems.     

A note on the periodic orbits of a self excited rigid body

The aim of the present paper is to study the periodic orbits of a perturbed self excited rigid body with a fixed point. For studying these periodic orbits we shall use averaging theory of first order.     

New 1:1:1 periodic solutions in 3-dimensional galactic-type Hamiltonian systems

Applying the averaging theory, we prove the existence of new families of periodic orbits for \(3\) -dimensional type-galactic Hamiltonian systems.     

A Modified Poincaré Method for the Persistence of Periodic Orbits and Applications

This paper is devoted to the persistence of periodic orbits under perturbations in dynamical systems generated by evolutionary equations, which are not smoothing in finite time, but only asymptotically smoothing. When the periodic orbit of the unperturbed system is non-degenerate, we show the existence and uniqueness of a periodic orbit (with a minimal period near the minimal period of the unperturbed problem) by using “modified” Poincaré methods. Examples of applications, including the perturbed hyperbolic Navier–Stokes equations, systems of damped wave equations and the system of second grade fluids, are given.     

Periodic and Homoclinic Motions in Forced,Coupled Oscillators

We study periodic and homoclinic motions in periodically forced, weakly coupled oscillators with a form of perturbations of two independent planar Hamiltonian systems. First, we extend the subharmonic Melnikov method, and give existence, stability and bifurcation theorems for periodic orbits. Second, we directly apply or modify a version of the homoclinic Melnikov method for orbits homoclinic to two types of periodic orbits. The first type of periodic orbit results from persistence of the unperturbed hyperbolic periodic orbit, and the second type is born out of resonances in the unperturbed invariant manifolds. So we see that some different types of homoclinic motions occur. The relationship between the subharmonic and homoclinic Melnikov theories is also discussed. We apply these theories to the weakly coupled Duffing oscillators.     

On the dynamics of the rigid body with a fixed point: periodic orbits and integrability

The aim of the present paper is to study the periodic orbits of a rigid body with a fixed point and quasi-spherical shape under the effect of a Newtonian force field given by different small potentials. For studying these periodic orbits, we shall use averaging theory. Moreover, we provide information on the $\mathcal{C}^{1}$ -integrability of these motions.     

Subharmonic Melnikov method of six-dimensional nonlinear systems and application to a laminated composite piezoelectric rectangular plate

In this paper, the bifurcations of subharmonic orbits are investigated for six-dimensional non-autonomous nonlinear systems using the improved subharmonic Melnikov method. The unperturbed system is composed of three independent planar Hamiltonian systems such that the unperturbed system has a family of periodic orbits. The key problem at hand is the determination of the sufficient conditions on some of the periodic orbits for the unperturbed system to generate the subharmonic orbits after the periodic perturbations. Using the periodic transformations and the Poincaré map, an improved subharmonic Melnikov method is presented. Two theorems are obtained and can be used to analyze the subharmonic dynamic responses of six-dimensional non-autonomous nonlinear systems. The subharmonic Melnikov method is directly utilized to investigate the subharmonic orbits of the six-dimensional non-autonomous nonlinear system for a laminated composite piezoelectric rectangular plate. Using the subharmonic Melnikov method, the bifurcation function of the subharmonic orbit is obtained. Numerical simulations are used to verify the analytical predictions. The results of the numerical simulation also indicate the existence of the subharmonic orbits for the laminated composite piezoelectric rectangular plate.     

Analytic Tools to Bound the Criticality at the Outer Boundary of the Period Annulus

In this paper we consider planar potential differential systems and we study the bifurcation of critical periodic orbits from the outer boundary of the period annulus of a center. In the literature the usual approach to tackle this problem is to obtain a uniform asymptotic expansion of the period function near the outer boundary. The novelty in the present paper is that we directly embed the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary. We obtain in this way explicit sufficient conditions in order that at most \(n\geqslant 0\) critical periodic orbits bifurcate from the outer boundary. These theoretical results are then applied to study the bifurcation diagram of the period function of the family \(\ddot{x}=x^p-x^q,\) \(p,q\in {\mathbb {R}}\) with \(p>q\).     

Limit Cycles for a Mechanical System Coming from the Perturbation of a Four–Dimensional Linear Center

We study the bifurcation of limit cycles from the periodic orbits of a four-dimensional center in a class of differential systems coming from the Mechanics. The tool for proving these results is the averaging theory.     

Continuation of the Periodic Orbits for the Differential Equation with Discontinuous Right Hand Side

In this paper the discontinuous system with one parameter perturbation is considered. Here the unperturbed system is supposed to have at least either one periodic orbit or a limit cycle. The aim is to prove the continuation of the periodic orbits under perturbation by means of the bifurcation map and the zeroes of this map imply the periodic orbits for the perturbed system. The tools for this problem are jumps of fundamental matrix solutions and the Poincare map for discontinuous systems. Therefore, we develop the Diliberto theorem and variation lemma for the system with discontinuous right hand side. At the end, as application of our method, the effect of discontinuous damping on Van der pol equation, and the effect of small force on the discontinuous linear oscillator with add a ·sgn(x) are considered.     

Sensitive Dependence on Initial Conditions of Strongly Nonlinear Periodic Orbits of the Forced Pendulum

We employ nonsmooth transformations of the independent coordinate to analytically construct families of strongly nonlinear periodic solutions of the harmonically forced nonlinear pendulum. Each family is parametrized by the period of oscillation, and the solutions are based on piecewise constant generating solutions. By examining the behavior of the constructed solutions for large periods, we find that the periodic orbits develop sensitive dependence on initial conditions. As a result, for small perturbations of the initial conditions the response of the system can jump from one periodic orbit to another and the dynamics become unpredictable. An analytical procedure is described which permits the study of the generation of periodic orbits as the period increases. The periodic solutions constructed in this work provide insight into the sensitive dependence on initial conditions of chaotic trajectories close to transverse intersections of invariant manifolds of saddle orbits of forced nonlinear oscillators.     

Identification of momentum life cycle stage of stock price

In this article, we study the continuous and discontinuous planar piecewise differential systems formed only by linear centers separated by one or two parallel straight lines. When these piecewise differential systems are continuous, they have no limit cycles. Also if they are discontinuous separated by a unique straight line, they do not have limit cycles. But when the piecewise differential systems are discontinuous separated two parallel straight lines, we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle.     

The Conley Index for Fast-Slow Systems I. One-Dimensional Slow Variable

We develop a qualitative theory for fast-slow systems with a one-dimensional slow variable. Using Conley index theory for singularity perturbed systems, conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0, these orbits persist when the derivative of the slow variable is small and nonzero.     

一类非自治动力系统超次谐周期解的不存在性

在非线性动力系统的研究中， Melnikov函数被广泛地用来作为微扰哈密顿系统是否发生次谐或超次谐分岔乃至混沌的判 据. 但是在大多数情况下，经典的Melnikov方法往往只给出存在次谐周期解的结论. 产生 该结果的原因被归之为在经典的Melnikov方法中只采取了一阶近似，因而高阶Melnikov方 法被发展用来判断超次谐周期解的存在性. 本文对一类非自治微分动力系统进行了研究，证 明了在这样一类系统中如果存在周期解则只可能是次谐周期解，超次谐周期解不可能存在， 并进一步证明了在一类平面问题中所定义的旋转(R)型超次谐周期解同样不可能存在.作为 该结论的一个应用，文中考察了几个典型的算例，结果表明现有的二阶Melnikov方法判断 平面扰动系统是否存在超次谐周期解的结论是不恰当的，并提供了一个简单的几何上的解释.     

Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

In this paper we investigate the bifurcations and the chaos of a piecewise linear discontinuous (PWLD) system based upon a rig-coupled SD oscillator, which can be smooth or discontinuous (SD) depending on the value of a system parameter, proposed in [18], showing the equilibrium bifurcations and the transitions between single, double and triple well dynamics for smooth regions. All solutions of the perturbed PWLD system, including equilibria, periodic orbits and homoclinic-like and heteroclinic-like orbits, are obtained and also the chaotic solutions are given analytically for this system. This allows us to employ the Melnikov method to detect the chaotic criterion analytically from the breaking of the homoclinic-like and heteroclinic-like orbits in the presence of viscous damping and an external harmonic driving force. The results presented here in this paper show the complicated dynamics for PWLD system of the subharmonic solutions, chaotic solutions and the coexistence of multiple solutions for the single well system, double well system and the triple well dynamics.     

Bifurcation of periodic solutions and invariant tori for a four-dimensional system

In this paper, a four-dimensional system of autonomous ordinary differential equations depending on a small parameter is considered. Suppose that the unperturbed system is composed of two planar systems: one is a Hamiltonian system and another system has a focus. By using the Poincaré map and the integral manifold theory, sufficient conditions for the existence of periodic solutions and invariant tori of the four-dimensional system are obtained. An application of our results to a nonlinearly coupled Van der Pol–Duffing oscillator system is given.