The Compressible to Incompressible Limit of One Dimensional Euler Equations: The Non Smooth Case

We prove a rigorous convergence result for the compressible to incompressible limit of weak entropy solutions to the isothermal one dimensional Euler equations.     

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.     

Quasiperiodic Solutions of Differential-Difference Equations on a Torus

This paper is dedicated to Professor V. A. Pliss on his 70th birthday. In connection with problems considered here, one should note the famous result of V. A. Pliss concerning the structural stability of differential equations on a torus. The first part of the present paper is devoted to the extension to differential-difference equations on an m-dimensional torus of the V. I. Arnold's result [1] about the reducibility of analytic systems on an m-dimensional torus to pure rotation. This result of Arnold was first extended to smooth systems of differential equations on an m-dimensional torus by A. M. Samoilenko [2] and later, independently, by J. Moser [3]. The Nash smoothing method, which was used in [2] (as indicated in [3-5]), leads to a large loss of smoothness. In the second part of the present paper, we use the Moser method of approximation of smooth functions by analytic ones for the reduction of smooth differential-difference equations on an m-dimensional torus to the canonical form $$\dot \phi (t) = \omega + f(\phi (t),\;\phi (t - h)),{\text{}}f(\phi ,\;\phi - \omega h) = 0.$$ Note that the considered problem of investigation of differential-difference equations on an m-dimensional torus is important in theory [6, 7]. Moreover, the results obtained can be used for in the investigation of multifrequency oscillations of retarded systems.     

Multiplicity Results on Period Solutions to Higher Dimensional Differential Equations with Multiple Delays

This paper deals with the existence and multiplicity of periodic solutions to delay differential equations of the form

$\dot{z}(t)=-f(z(t-1))- f(z(t-2))-\cdots- f(z(t-2n+1)) $

where \({z\in {\bf R}^N, f\in C({\bf R}^{N}, {\bf R}^N)}\). By using the S ¹ pseudo geometrical index theory in the critical point theory, some known results for Kaplan–Yorke type differential delay equations are generalized to higher dimensional case. As a result, the Kaplan–Yorke’s conjecture is proved to be true in the case of higher dimensional systems.

     

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.     

Absence of Anomalous Dissipation of Energy in Forced Two Dimensional Fluid Equations

We prove the absence of anomalous dissipation of energy for long time averaged solutions of the forced critical surface quasi-geostrophic equation in two spatial dimensions.     

On the Multiplicity of Algebraic Limit Cycles

In the present paper we tackle the problem of determining the multiplicity of periodic orbit as limit cycle of a planar differential system. We consider the particular case of a circumference as periodic orbit. We show that the conditions of multiplicity can be almost algebraically solvable. There are parameters in which these conditions depend transcendentally, as happens in the degenerate center-focus problem. Even though this difficulty, these transcendental dependence can be, in some sense, controlled because only a basis of fundamental functions appear. The appearance of this fundamental basis opens the path to approach these types of problems. We present several examples of families for which these conditions can be computed.     

Holomorphic Flows on Simply Connected Regions Have No Limit Cycles

The dynamical system or flow = f(z), where f is holomorphic on C, is considered. The behavior of the flow at critical points coincides with the behavior of the linearization when the critical points are non-degenerate: there is no center-focus dichotomy. Periodic orbits about a center have the same period and form an open subset. The flow has no limit cycles in simply connected regions. The advance mapping is holomorphic where the flow is complete. The structure of the separatrices bounding the orbits surrounding a center is determined. Some examples are given including the following: if a quartic polynomial system has four distinct centers, then they are collinear.     

Symmetry Breaking for a System of Two Linear Second-Order Ordinary Differential Equations

A new canonical form for a system of two linear second-orderordinary differential equations (odes) is obtained. The latter isdecisive in unravelling symmetry structure of a system of two linearsecond-order odes. Namely we establish that the point symmetry Liealgebra of a system of two linear second-order odes can be5-, 6-, 7-, 8- or 15-dimensional. This result enhances both the richness andthe complexity of the symmetry structure of linear systems.     

Bifurcation of Limit Cycles from a Polynomial Non-global Center

Consider the planar ordinary differential equation , where the set consists of k non-zero points. In this paper we perturb this vector field with a general polynomial perturbation of degree n and study how many limit cycles bifurcate from the period annulus of the origin in terms of k and n. One of the key points of our approach is that the Abelian integral that controls the bifurcation can be explicitly obtained as an application of the integral representation formula of harmonic functions through the Poisson kernel. Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

A Smooth Center Manifold Theorem which Applies to Some Ill-Posed Partial Differential Equations with Unbounded Nonlinearities

We prove the existence of a smooth center manifold for several partial differential equations, including ill posed equations with unbounded nonlinearities. We also prove smooth dependence on parameters with respect to some perturbations, including unbounded ones. More concretely, we prove an abstract theorem and present applications to several concrete equations: ill posed Boussinesq, equation and system and nonlinear Laplace equations in cylindrical domains. We also consider the effect of some geometric structures.     

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.     

Robust Feedback Stabilization of Limit Cycles in PWM DC-DC Converters

Local feedback stabilization of limit cycles in PWM DC-DC converters isconsidered, using recently developed general sampled-data models. Thepaper focuses on converters for which the nominal operating conditionhas lost stability due to off-design operation. The results apply tostabilization of the nominal periodic operating condition. In addition,the same approach can be used to stabilize other limit cycles such asthose embedded in a possible chaotic trajectory. Two feedbackstabilization schemes are proposed and studied in detail. The firststabilization technique uses voltage reference compensation and thesecond uses dynamic ramp compensation. Both employ discrete-time washoutfilters to ensure preservation of the size and shape of the limit cycle.Washout filters ensure that the nominal operating branch is unaffectedby the control, without the need for accurate knowledge of the limitcycle.     

Limit Cycles Bifurcating from the Period Annulus of Quasi-Homogeneous Centers

We provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of any homogeneous and quasi-homogeneous center, which can be obtained using the Abelian integral method of first order. We show that these bounds are the best possible using the Abelian integral method of first order. We note that these centers are in general non-Hamiltonian. As a consequence of our study we provide the biggest known number of limit cycles surrounding a unique singular point in terms of the degree n of the system for arbitrary large n.      

Limit Cycles Appearing After Perturbation of Certain Multidimensional Vector Fields

Let an unperturbed multidimensional polynomial vector field have an invariant plane L and let the system restricted to this plane be Hamiltonian with a quadratic Hamilton function. Now take a polynomial perturbation of this system. The new system has an invariant surface close to L and the system restricted to it has a certain number of limit cycles. We strive to estimate this number. The linearization of this problem leads to estimation of the number of zeros of certain integral, which is a generalization of the abelian integral. We estimate this number of zeros by C ₁+C ₂ n, where n is the degree of the perturbation.     

A Reduced Order Cyclic Method for Computation of Limit Cycles

A reduced order cyclic method was developed to compute limit-cycle oscillations for large, nonlinear, multidisciplinary systems of equations. Method efficacy was demonstrated for two simplified models: a typical-section airfoil with nonlinear structural coupling and a nonlinear panel in high-speed flow. The cyclic method was verified to maintain second-order temporal accuracy, yield converged limit cycles in about 10 Newton iterates, and provide precise estimates of cycle frequency. This method was projected onto a low-order space using a set of variables governing the amplitudes of empirically derived modes, which were computed with the proper orthogonal decomposition. In this reduced order form, the cyclic Jacobian was greatly compressed, allowing accurate limit cycle solutions to be very efficiently computed.The U.S. Governments right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.