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.     

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.     

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.     

Implicit Difference Methods for First-Order Partial Differential Functional Equations

We present a new class of numerical methods for quasilinear first-order partial differential functional equations. The numerical methods are difference schemes implicit with respect to time variable. We give a complete convergence analysis for the methods and show by an example that the new methods are considerably better than explicit schemes. The proof of stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators with respect to the functional variable. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 2, pp. 201–215, April–June, 2005.     

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.     

Modulated Wave Trains in Lattice Differential Systems

The existence of weak sinks in mixed parabolic-lattice systems on the real line is established for systems that incorporate discrete coupling on an underlying lattice in addition to continuous diffusion. Sinks can be thought of as interfaces that separate two spatially periodic structures with different wave numbers: the corresponding modulated wave train is time periodic in the frame that moves with the speed of the interface. In this paper, the existence of weak sinks is proved that connect wave trains with almost identical wave number. The main difficulty is the global coupling between points on the underlying lattice, since its presence turns the equation solved by sinks into an ill-posed functional differential equation of mixed type.     

Systems of Linear Differential Equations of Rational Rank

We describe an application of the method of a perturbed characteristic equation to the asymptotic integration of systems of linear differential equations with fractional powers of a small parameter in the coefficients of the derivatives.     

C 1 Solutions for Semi-Implicit Systems of Differential Equations

The present note is a continuation of the author??s effort to study the existence of continuously differentiable solutions to the semi-implicit system of differential equations (1) $$f(x^{\prime}(t)) = g(t, x(t))$$ (2) $$\quad x(0) = x_0,$$ where

${\quad\Omega_g \subseteq \mathbb{R} \times\mathbb{R}^n}$ is an open set containing (0, x ₀) and ${g:\Omega_g \rightarrow\mathbb{R}^n}$ is a continuous function,

${\quad\Omega_f \subseteq \mathbb{R}^n}$ is an open set and ${f:\Omega_f\rightarrow\mathbb{R}^n}$ is a continuous function.

The transformation of (1)?C(2) into a solvable explicit system of differential equations is trivial if f is locally injective around an element ${\gamma\in \Omega_f\cap f^{-1}(g(0,x_0))}$ . In this paper, we study (1)?C(2) when such a translation is not possible because of the inherent multivalued nature of f ^?1.     

Unbounded Solutions to Systems of Differential Equations at Resonance

Journal of Dynamics and Differential Equations - We deal with a weakly coupled system of ODEs of the type $$begin{aligned} x_j'' + n_j^2 ,x_j + h_j(x_1,ldots ,x_d) = p_j(t), qquad...     

Stability for Nonautonomous Linear Differential Systems with Infinite Delay

Journal of Dynamics and Differential Equations - We study the stability of general n-dimensional nonautonomous linear differential equations with infinite delays. Delay independent criteria, as...     

Periodic Solutions for Some Systems of Delay Differential Equations

In this paper, we give sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay having 3, 4 or n equations. Moreover, we provide examples of delay systems satisfying the different sets of sufficient conditions.     

A New Approach to Asymptotic Diagonalization of Linear Differential Systems

We study the asymptotic diagonalization of a system consisting of an -matrix plus a finite number of -perturbations on an interval I ₀=[t ₀, ), where p, m _i[1, ). Using linear skew-product flows and spectral theory, we show that if the unperturbed system has full spectrum over its omega-limit set, then the entire system is asymptotically diagonalizable almost everywhere.     

Invariant Sets of Systems of Stochastic Differential Equations with Jumps

We introduce the notion of invariant surfaces for inhomogeneous stochastic differential equations with jumps. The results obtained enable one to determine invariant surfaces for stochastic differential equations of the type indicated. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 234–240, April–June, 2005.     

Differential Systems with Feedback: Time Discretizations and Lyapunov Functions

In this paper we examine a class of Eulerian time discretizations for a monotone cyclic feedback system with a time delay; see Mallet-Paret and Sell (1996a, 1996b) for background information. We construct an integer-valued function V for the discrete-time problem. The Main Theorem shows that V is a Lyapunov function, that is, V(x ⁿ⁺¹)≤V(x ⁿ ) along a solution {x ⁿ }^∞ _n=0, where the time steps can be relatively large.     

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.

     

On Asymptotics of Solutions of Semiexplicit Systems of Differential Equations

We study the problem of the existence of analytic solutions of a certain semiexplicit system of differential equations and obtain sufficient conditions for the existence of analytic solutions of the Cauchy problem in the neighborhood of a singular point.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 132–144, January–March, 2005.     

Projection-Iterative Method for Systems of Differential Equations with Delay and Restrictions

Compatibility conditions are established for systems of differential equations with constant delay and restrictions. We propose a new version of the projection-iterative method for such problems and present its justification.     

Incompressible Flows with Piecewise Constant Density

We investigate the incompressible Navier–Stokes equations with variable density. The aim is to prove existence and uniqueness results in the case of discontinuous initial density. In dimension n = 2,3, assuming only that the initial density is bounded and bounded away from zero, and that the initial velocity is smooth enough, we get the local-in-time existence of unique solutions. Uniqueness holds in any dimension and for a wider class of velocity fields. In particular, all those results are true for piecewise constant densities with arbitrarily large jumps. Global results are established in dimension two if the density is close enough to a positive constant, and in n dimensions if, in addition, the initial velocity is small. The Lagrangian formulation for describing the flow plays a key role in the analysis that is proposed in the present paper.