Averaging Theory of Arbitrary Order for Piecewise Smooth Differential Systems and Its Application

The averaging theory for studying periodic orbits of smooth differential systems has a long history. Whereas the averaging theory for piecewise smooth differential systems appeared only in recent years, where the unperturbed systems are smooth. When the unperturbed systems are only piecewise smooth, there is not an existing averaging theory to study existence of periodic orbits of their perturbed systems. Here we establish such a theory for one dimensional perturbed piecewise smooth periodic differential equations. Then we show how to transform planar perturbed piecewise smooth differential systems to one dimensional piecewise smooth periodic differential equations when the unperturbed planar piecewise smooth differential systems have a family of periodic orbits. Finally as application of our theory we study limit cycle bifurcation of planar piecewise differential systems which are perturbation of a \(\Sigma \)-center.     

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.     

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.     

Dynamics of a nearly symmetrical piecewise linear oscillator close to grazing incidence: Modelling and experimental verification

Experimental studies and mathematical modelling have been carried out for a nearly symmetrical piecewise linear oscillator to examine the bifurcation scenarios close to grazing. Higher period responses are found after grazing, although the period adding windows predicted as a generic feature of one-sided impacting systems are not observed. It appears that the presence of the second high stiffness spring stabilises additional periodic orbits. The global solution for a piecewise smooth model is developed by stitching locally valid maps. For the symmetrical case the highest period of response is three, if asymmetry in the gap and/or stiffness is introduced then higher periodic orbits are observed. Only small asymmetries are required to achieve a good correspondence with experiments. Further examination shows that many attractors are not stable to even small changes in the symmetry of the system.     

Border collision bifurcations in 3D piecewise smooth chaotic circuit

A variety of border collision bifurcations in a three-dimensional (3D) piecewise smooth chaotic electrical circuit are investigated. The existence and stability of the equilibrium points are analyzed. It is found that there are two kinds of non-smooth fold bifurcations. The existence of periodic orbits is also proved to show the occurrence of non-smooth Hopf bifurcations. As a composite of non-smooth fold and Hopf bifurcations, the multiple crossing bifurcation is studied by the generalized Jacobian matrix. Some interesting phenomena which cannot occur in smooth bifurcations are also considered.     

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.     

一类分段光滑非线性平面运动系统响应特性研究

针对由一个线性子系统和一个非线性子系统构成的两自由度非自治分段光滑非线性平面运动系统的响应特性开展了研究. 该分段光滑非线性模型可用来确定对称转子/定子系统的主要碰摩响应, 且在反映非光滑系统典型特性上具有明显的特征:(1) 切换分界面是由两自由度坐标共同决定的一个幅值曲面;(2) 子系统周期解与分界面的擦碰, 不是发生在一个点上, 而是同时发生在解的所有点上;(3) 完整系统未发现由两子系统共同作用而产生的周期解. 因此, 对于该非光滑系统响应特性的研究, 很难直接利用目前有关非光滑系统平衡点和周期解分岔分析的方法. 为此, 尝试了根据子系统的响应特性, 划分出完整系统响应对分界面处切换的敏感区和非敏感区, 并针对非敏感区可由子系统解的特性求得完整系统的响应, 而针对敏感区通过子系统动力学特征的分析有助于解释完整系统响应的生成机制.     

The Sliding Bifurcations in Planar Piecewise Smooth Differential Systems

In this paper we are mainly interested in the bifurcation phenomena for a class of planar piecewise smooth differential systems, where a new phenomenon, i.e. sliding heteroclinic bifurcation, is found. Furthermore we will show that the involved systems can present many interesting bifurcation phenomena, such as the (sliding) heteroclinic bifurcation, sliding (homoclinic) cycle bifurcation and semistable limit cycle bifurcation and so on. The system can have two hyperbolic limit cycles, which are bifurcated in one way from a semistable limit cycle, and in another way from a heteroclinic cycle and a sliding cycle. In the proof of our main results, we will use the geometric singular perturbation theory to analyze the dynamics near the sliding region.     

Lyapunov and LMI analysis and feedback control of border collision bifurcations

Feedback control of piecewise smooth discrete-time systems that undergo border collision bifurcations is considered. These bifurcations occur when a fixed point or a periodic orbit of a piecewise smooth system crosses or collides with the border between two regions of smooth operation as a system parameter is quasistatically varied. The class of systems studied is piecewise smooth maps that depend on a parameter, where the system dimension n can take any value. The goal of the control effort in this work is to replace the bifurcation so that in the closed-loop system, the steady state remains locally attracting and locally unique (“nonbifurcation with persistent stability”). To achieve this, Lyapunov and linear matrix inequality (LMI) techniques are used to derive a sufficient condition for nonbifurcation with persistent stability. The derived condition is stated in terms of LMIs. This condition is then used as a basis for the design of feedback controls to eliminate border collision bifurcations in piecewise smooth maps and to produce the desirable behavior noted earlier. Numerical examples that demonstrate the effectiveness of the proposed control techniques are given.     

受周期和白噪声激励的分段线性系统的吸引域与离出问题研究

离出行为是随机非线性系统的重要现象之一,而离出问题是除随机动力系统理论以外考察随机非线性系统随机稳定性的另一种重要的方法.分段线性系统是一个经典的非线性动力学模型,受随机激励后成为随机系统,但并不是严格的随机动力系统,因而此时随机动力系统理论也不适用.为了研究同时受周期和白噪声激励的分段线性系统,首先使用Poincaré截面模拟其在无噪声时确定性的动力学行为,然后使用Monte Carlo模拟对其在白噪声激励下的离出行为进行了数值仿真分析.其次,为了考察离出问题中的重要参数,系统的平均首次通过时间（mean first-passage time,MFPT）,使用van der Pol变换,随机平均法,奇异摄动法和射线方法进行了量化计算.通过对理论结果与模拟结果的对比分析,得到结论:当系统吸引子对应的吸引域边界出现碎片化时,理论结果与模拟结果的误差极大;而当吸引域边界足够光滑的以后,理论结果与模拟结果才会相当吻合.      

RESEARCH FOR ATTRACTING REGION AND EXIT PROBLEM OF A PIECEWISE LINEAR SYSTEM UNDER PERIODIC AND WHITE NOISE EXCITATIONS

离出行为是随机非线性系统的重要现象之一,而离出问题是除随机动力系统理论以外考察随机非线性系统随机稳定性的另一种重要的方法.分段线性系统是一个经典的非线性动力学模型,受随机激励后成为随机系统,但并不是严格的随机动力系统,因而此时随机动力系统理论也不适用.为了研究同时受周期和白噪声激励的分段线性系统,首先使用Poincaré截面模拟其在无噪声时确定性的动力学行为,然后使用Monte Carlo模拟对其在白噪声激励下的离出行为进行了数值仿真分析.其次,为了考察离出问题中的重要参数,系统的平均首次通过时间（mean first-passage time,MFPT）,使用van der Pol变换,随机平均法,奇异摄动法和射线方法进行了量化计算.通过对理论结果与模拟结果的对比分析,得到结论：当系统吸引子对应的吸引域边界出现碎片化时,理论结果与模拟结果的误差极大;而当吸引域边界足够光滑的以后,理论结果与模拟结果才会相当吻合.     

On the existence of low-period orbits in <Emphasis Type="Italic">n</Emphasis>-dimensional piecewise linear discontinuous maps

Discontinuous maps occur in many practical systems, and yet bifurcation phenomena in such maps is quite poorly understood. In this paper, we report some important results that help in analyzing the border collision bifurcations that occur in n-dimensional discontinuous maps. For this purpose, we use the piecewise linear approximation in the neighborhood of the plane of discontinuity. Earlier, Feigin had made a similar analysis for general n-dimensional piecewise smooth continuous maps. In this paper, we extend that line of work for maps with discontinuity to obtain the general conditions of existence of period-1 and period-2 fixed points before and after a border collision bifurcation. The application of the method is then illustrated using a specific example of a two-dimensional discontinuous map. This work was supported in part by the BRNS, Department of Atomic Energy (DAE), Government of India under project no. 2003/37/11/BRNS.     

Poincaré maps modeling and local orbital stability analysis of discontinuous piecewise affine periodically driven systems

This paper presents a methodology to study the local stability of periodic orbits (orbital stability) in switched discontinuous piecewise affine (DPWA) periodically driven multiple-input multiple-output (MIMO) systems. The switched system of interest has a bilinear state space representation where the controller inputs are binary signals taking values in the set {0,1}. These systems are characterized by a set of affine differential equations together with switching rules to commute between them. These switching rules are described by switching functions that are periodic in time and linear in state. The methodology is based on obtaining a discrete time model (Poincaré map), its steady state operation points, and its Jacobian matrix. This provides a powerful tool for studying their stability and to predict some kind of instability phenomena that the system can undergo like subharmonic oscillations. The proposed approach is applied to a power electronic circuit which toggles among six different system equations with five switching boundaries within a switching cycle. This work was supported by the Spanish Ministerio de Educación y Ciencia under Grant TEC-2004-05608-C02-02.     

On the purposeful coarsening of smooth vector fields

This paper argues for the possibility of purposely approximating smooth vector fields with highly localized variability in terms of piecewise smooth vector fields for the purpose of analyzing the bifurcation characteristics of the corresponding dynamical systems. Here, emphasis is placed on the changes in system response that result as a periodic trajectory begins to incorporate a brief flow segment in the region of high variability under variations in some system parameter. In particular, it is shown that tools from the theory of grazing bifurcations in piecewise-smooth systems may be employed to qualitatively predict the bifurcation scenario associated with such a transition both in terms of the shape of the branch of periodic trajectories and in terms of the persistence of a local attractor in the vicinity of the original periodic trajectory.     

Global Dynamics of a Rapidly Forced Cart and Pendulum

In this paper, we study emergent behaviors elicited by applying open-loop, high-frequency oscillatory forcing to nonlinear control systems. First, we study hovering motions, which are periodic orbits associated with stable fixed points of the averaged system which are not fixed points of the forced system. We use the method of successive approximations to establish the existence of hovering motions, as well as compute analytical approximations of their locations, for the cart and pendulum on an inclined plane. Moreover, when small-amplitude dissipation is added, we show that the hovering motions are asymptotically stable. We compare the results for all of the local analysis with results of simulating Poincaré maps. Second, we perform a complete global analysis on this cart and pendulum system. Toward this end, the same iteration scheme we use to establish the existence of the hovering periodic orbits also yields the existence of periodic orbits near saddle equilibria of the averaged system. These latter periodic orbits are shown to be saddle periodic orbits, and in turn they have stable and unstable manifolds that form homoclinic tangles. A quantitative global analysis of these tangles is carried out. Three distinguished limiting cases are analyzed. Melnikov theory is applied in one case, and an extension of a recent result about exponentially small splitting of separatrices is developed and applied in another case. Finally, the influence of small damping is studied. This global analysis is useful in the design of open-loop control laws.     

Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems

This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integrable Hamiltonian systems. It presents a constructive method for establishing whether small conservative perturbations of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria will yield transverse homoclinic connections between periodic orbits in the resonance band resulting from the perturbation. In any given example, this method may be used to prove the existence of such transverse homoclinic orbits, as well as to determine their precise shape, their asymptotic behavior, and their possible bifurcations. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.     

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.     

Periodic orbits of a conservative 2-DOF vibro-impact system by piecewise continuation: bifurcations and fractals

Nonlinear Dynamics - The exact periodic orbits of a conservative 2-degree-of-freedom vibro-impact system with stereo-mechanical impact model is studied using a piecewise continuation method....     

Global analysis of the dynamics of a mathematical model to intermittent HIV treatment

The aim of this paper is to study the qualitative dynamics of a piecewise smooth system modeling the intermittent treatment of the human immunodeficiency virus. Typical singularities and closed orbits are observable, and we quantitatively explore the dynamics around those singularities and closed orbits. Moreover, we conclude that this protocol always will be successful since the trajectory passing through any initial condition converges to one of these distinguished orbits. Our formal mathematical results corroborate the real-world observation, where the virus is not eliminated, but the number of infected cells is controlled around a specific value.

     

Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control

This paper presents a detailed analysis on the dynamics of a delayed oscillator with negative damping and delayed feedback control. Firstly, a linear stability analysis for the trivial equilibrium is given. Then, the direction of Hopf bifurcation and stability of periodic solutions bifurcating from trivial equilibrium are determined by using the normal form theory and center manifold theorem. It shows that with properly chosen delay and gain in the delayed feedback path, this controlled delayed system may have stable equilibrium, or periodic solutions, or quasi-periodic solutions, or coexisting stable solutions. In addition, the controlled system may exhibit period-doubling bifurcation which eventually leads to chaos. Finally, some new interesting phenomena, such as the coexistence of periodic orbits and chaotic attractors, have been observed. The results indicate that delayed feedback control can make systems with state delay produce more complicated dynamics.