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.     

Regularizing Piecewise Smooth Differential Systems: Co-Dimension 2 Discontinuity Surface

In this paper, we are concerned with numerical solution of piecewise smooth initial value problems. Of specific interest is the case when the discontinuities occur on a smooth manifold of co-dimension 2, intersection of two co-dimension 1 singularity surfaces, and which is nodally attractive for nearby dynamics. In this case of a co-dimension 2 attracting sliding surface, we will give some results relative to two prototypical time and space regularizations. We will show that, unlike the case of co-dimension 1 discontinuity surface, in the case of co-dimension 2 discontinuity surface the behavior of the regularized problems is strikingly different. On the one hand, the time regularization approach will not select a unique sliding mode on the discontinuity surface, thus maintaining the general ambiguity of how to select a Filippov vector field in this case. On the other hand, the proposed space regularization approach is not ambiguous, and there will always be a unique solution associated to the regularized vector field, which will remain close to the original co-dimension 2 surface. We will further clarify the limiting behavior (as the regularization parameter goes to 0) of the proposed space regularization to the solution associated to the sliding vector field of Dieci and Lopez (Numer Math 117:779–811, 2011). Numerical examples will be given to illustrate the different cases and to provide some preliminary exploration in the case of co-dimension 3 discontinuity surface.     

Predicting Homoclinic Bifurcations in Planar Autonomous Systems

An analytical method to predict the homoclinic bifurcation in a planar autonomous self-excited weakly nonlinear oscillator is presented. The method is mainly based on the collision between the periodic orbit undergoing the homoclinic bifurcation and the saddle fixed point. To illustrate the analytical predictive criteria, two typical examples are investigated. The results obtained in this work are then compared to Melnikov's technique and to a previous criterion based on the vanishing of the frequency. Numerical simulations are also provided.     

Piecewise Implicit Differential Systems

In this article we deal with non-smooth dynamical systems expressed by a piecewise first order implicit differential equations of the form

$$\begin{aligned} \dot{x}=1,\quad \left( \dot{y}\right) ^2=\left\{ \begin{array}{lll} g_1(x,y) \quad \text{ if }\quad \varphi (x,y)\ge 0 \\ g_2(x,y) \quad \text{ if }\quad \varphi (x,y)\le 0 \end{array},\right. \end{aligned}$$

where \(g_1,g_2,\varphi :U\rightarrow \mathbb {R}\) are smooth functions and \(U\subseteq \mathbb {R}^2\) is an open set. The main concern is to study sliding modes of such systems around some typical singularities. The novelty of our approach is that some singular perturbation problems of the form

$$\begin{aligned} \dot{x}= f(x,y,\varepsilon ) ,\quad (\varepsilon \dot{ y})^2=g ( x,y,\varepsilon ) \end{aligned}$$

arise when the Sotomayor–Teixeira regularization is applied with \((x, y) \in U\) , \(\varepsilon \ge 0\), and f, g smooth in all variables.

     

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.     

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.     

Bifurcations in Nonlinear Discontinuous Systems

This paper treats bifurcations of periodic solutions indiscontinuous systems of the Filippov type. Furthermore, bifurcations offixed points in non-smooth continuous systems are addressed. Filippov'stheory for the definition of solutions of discontinuous systems issurveyed and jumps in fundamental solution matrices are discussed. It isshown how jumps in the fundamental solution matrix lead to jumps of theFloquet multipliers of periodic solutions. The Floquet multipliers canjump through the unit circle causing discontinuous bifurcations.Numerical examples are treated which show various discontinuousbifurcations. Also infinitely unstable periodic solutions are addressed.     

Centers and Uniform Isochronous Centers of Planar Polynomial Differential Systems

For planar polynomial vector fields of the form

$$\begin{aligned} (-y+X(x,y))\dfrac{\partial }{\partial x}+(x+Y(x,y))\dfrac{\partial }{\partial y}, \end{aligned}$$

where X and Y start at least with terms of second order in the variables x and y, we determine necessary and sufficient conditions under which the origin is a center or a uniform isochronous centers.

     

Bifurcations on a Five-Parameter Family of Planar Vector Field

In this paper we consider a five-parameter family of planar vector fields where μ = (μ ₁, μ ₂, μ ₃, μ ₄, μ ₅), which is a small parameter vector, and c(0) ≠ 0. The family X _μ represents the generic unfolding of a class of nilpotent cusp of codimension five. We discuss the local bifurcations of X _μ, which exhibits numerous kinds of bifurcation phenomena including Bogdanov-Takens bifurcations of codimension four in Li and Rousseau (J. Differ. Eq. 79, 132–167, 1989) and Dumortier and Fiddelaers (In: Global analysis of dynamical systems, 2001), and Bogdanov-Takens bifurcations of codimension three in Dumortier et al. (Ergodic Theory Dynam. Syst. 7, 375–413, 1987) and Dumortier et al. (Bifurcations of planar vector fields. Nilpotent singularities and Abelian integrals, 1991). After making some rescalings, we obtain the truncated systems of X _μ . For a truncated system, all possible bifurcation sets and related phase portraits are obtained. When the truncated system is a Hamiltonian system, the bifurcation diagram and the related phase portraits are given too. Hopf bifurcations are studied for another truncated system. And it shows that the system has the Hopf bifurcations of codimension at most three, and at most three limit cycles occur in the small neighborhood of the Hopf singularity. Dedicated to Professor Zhifen Zhang in the occasion of her 80th birthday     

Homoclinic Saddle-Node Bifurcations in Singularly Perturbed Systems

In this paper we study the creation of homoclinic orbits by saddle-node bifurcations. Inspired on similar phenomena appearing in the analysis of so-called localized structures in modulation or amplitude equations, we consider a family of nearly integrable, singularly perturbed three dimensional vector fields with two bifurcation parameters a and b. The O() perturbation destroys a manifold consisting of a family of integrable homoclinic orbits: it breaks open into two manifolds, W ^s() and W ^u(), the stable and unstable manifolds of a slow manifold . Homoclinic orbits to correspond to intersections W ^s()W ^u(); W ^s()W ^u()= for a<a*, a pair of 1-pulse homoclinic orbits emerges as first intersection of W ^s() and W ^u() as a>a*. The bifurcation at a=a* is followed by a sequence of nearby, O( ²(log)²) close, homoclinic saddle-node bifurcations at which pairs of N-pulse homoclinic orbits are created (these orbits make N circuits through the fast field). The second parameter b distinguishes between two significantly different cases: in the cooperating (respectively counteracting) case the averaged effect of the fast field is in the same (respectively opposite) direction as the slow flow on . The structure of W ^s()W ^u() becomes highly complicated in the counteracting case: we show the existence of many new types of sometimes exponentially close homoclinic saddle-node bifurcations. The analysis in this paper is mainly of a geometrical nature.     

The Full Study of Planar Quadratic Differential Systems Possessing a Line of Singularities at Infinity

In this article we make a full study of the class of non-degenerate real planar quadratic differential systems having all points at infinity (in the Poincaré compactification) as singularities. We prove that all such systems have invariant affine lines of total multiplicity 3, give all their configurations of invariant lines and show that all these systems are integrable via the method of Darboux having cubic polynomials as inverse integrating factors. After constructing the topologically distinct phase portraits in this class we give invariant necessary and sufficient conditions in terms of the 12 coefficients of the systems for the realization of each one of them and give representatives of the orbits under the action of the affine group and time rescaling. We construct the moduli space of this class for this action and give the corresponding bifurcation diagram. Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

Algebraic Methods for Determining Hamiltonian Hopf Bifurcations in Three-Degree-of-Freedom Systems

A method is sketched to determine the presence of non-degenerate Hamiltonian Hopf bifurcations in three-degree-of-freedom systems by putting the bifurcation into standard form. Detailed computations are performed for the non-trivial example of the 3D Hénon–Heiles family. After a careful formulation of the local once reduced system in terms of properly chosen invariants the system can be compared to the standard form by the application of singularity theoretic results.     

Global Bifurcations in Parametrically Excited Systems with Zero-to-One Internal Resonance

In this work we study the existence of Silnikov homoclinicorbits in the averaged equations representing the modal interactionsbetween two modes with zero-to-one internal resonance. The fast mode isparametrically excited near its resonance frequency by a periodicforcing. The slow mode is coupled to the fast mode when the amplitude ofthe fast mode reaches a critical value so that the equilibrium of theslow mode loses stability. Using the analytical solutions of anunperturbed integrable Hamiltonian system, we evaluate a generalizedMelnikov function which measures the separation of the stable and theunstable manifolds of an annulus containing the resonance band of thefast mode. This Melnikov function is used together with the informationof the resonances of the fast mode to predict the region of physicalparameters for the existence of Silnikov homoclinic orbits.     

Parabolic Systems with Nowhere Smooth Solutions

We construct smooth 2×2 parabolic systems with smooth initial data and C^α right-hand side which admit solutions that are nowhere C¹. The elliptic part is in variational form and the corresponding energy ϕ is strongly quasiconvex and in particular satisfies a uniform Legendre-Hadamard (or strong ellipticity) condition.     

Attraction in Nonmonotone Planar Systems and Real-Life Models

Journal of Dynamics and Differential Equations - Let $$h:Vsubset {mathbb {R}}^{2}longrightarrow {mathbb {R}}^{2}$$ be an embedding. The aim of this paper is to analyze the dynamical behavior of...     

一类非线性系统在随机激励下的分叉

本文研究一类阻尼为线性，弹性恢复力为非线性的振动系统在随机外部激励作用下的随机分叉。文中采用广义稳态势和方法，求解系统响应的稳态联合概率密度函数。在此基础上根据由不变测度定义的随机分叉，讨论了具有权式分叉的确定性非线性系统在随机扰动下分叉行为。     

The Stability of the Equilibrium of Planar Hamiltonian and Reversible Systems

In this paper, we study the stability of the equilibrium of planar systems

L2 Well-Posedness of Planar Div-Curl Systems

Criteria for the existence and uniqueness of solutions of div-curl boundary value problems on bounded planar regions with nice boundaries are developed. The boundary conditions to be treated include prescribed normal component of the field, tangential component of the field and disjoint combinations of these conditions. Under natural assumptions on the data, when either tangential or normal components are given on the whole boundary, weak (finite-energy) solutions exist if and only if a compatibility condition holds. If the region is simply connected this solution is unique. When the region is multiply connected, there is a finite-dimensional family of solutions. The dimension of the solution space is the Betti number of the region. The problem is well-posed with a unique solution when certain line integrals are further prescribed. L ² estimates of the solutions are given. If mixed tangential, and normal, components of the field are specified on different parts of the boundary, no compatibility condition is required for solvability. In general, though, there is considerable non-uniqueness of solutions. Well-posedness is recovered by specifying certain line integrals. L ² estimates of the solutions are given. The dimensionality of the solution space depends on the topology of the boundary data. These results depend on certain weighted orthogonal decompositions of L ² vector fields on the region which are related to classical Hodge-Weyl decomposition results.