Influence of a small perturbation on Poincare-Andronov operators with not well defined topological degree

Let $P_e\in C^0(R^n,R^n)$ be the Poincare-Andronov operator over period $T>0$ of the $T$-periodically perturbed autonomous system $x'=f(x)+e g(t,x,e),$ where $e>0$ is small. Assuming that for $e=0$ this system has a $T$-periodic limit cycle $x_0$ we evaluate the topological degree $d(I-P_e,U)$ of $I-P_e$ on an open bounded set $U$ whose boundary contains $x_0([0,T])$ and does not contain other fixed points of $P_0.$ We give an explicit formula connecting $d(I-P_e,U)$ with topological indexes of zeros of the associated Malkin's bifurcation function. The goal of the paper is to prove Mawhin's conjecture which claims that $d(I-P_e,U)$ can be any integer in spite of the fact that the measure of the set of fixed points of $P_0$ on $\partial U$ is zero

Lipschitz perturbations of differentiable implicit functions

Let $y=f(x)$ be a continuously differentiable implicit function solving the equation $F(x,y)=0$ with continuously differentiable $F.$ In this paper we show that if F_\eps is a Lipschitz function such that the Lipschitz constant of F_\eps-F goes to 0 as \eps\to 0 then the equation F_\eps(x,y)=0 has a Lipschitz solution y=f_\eps(x) such that the Lipschitz constant of f_\eps-f goes to 0 as \eps\to 0 either. As an application we evaluate the length of time intervals where the right hand parts of some nonautonomous discontinuous systems of ODEs are continuously differentiable with respect to state variables. The latter is done as a preparatory step toward generalizing the second Bogolyubov's theorem for discontinuous systems.Comment: Submitte

Bifurcation of limit cycles from a fold-fold singularity in planar switched systems

An anti-lock braking system (ABS) is the primary motivation for this research. The ABS controller switches the actions of charging and discharging valves in the hydraulic actuator of the brake cylinder based on the wheels' angular speed and acceleration. The controller is, therefore, modeled by discontinuous differential equations where two smooth vector fields are separated by a switching manifold S. The goal of the controller is to maximize the tire-road friction force during braking (and, in particular, to prevent the wheel lock-up). Since the optimal slip L of the wheel is known rather approximately, the actual goal of the controller is to achieve such a switching strategy that makes the dynamics converging to a limit cycle surrounding the region of prospective values of L. In this paper we show that the required limit cycle can be obtained as a bifurcation from a point x0 of S when a suitable parameter D crosses 0. The point x0 turns out to be a fold-fold singularity (the vector fields on the two sides of S are tangent one another at x0) and the parameter D measures the deviation of the switching manifold from a hyperplane. The proposed result is based on an extension of the classical fold bifurcation theory available for smooth maps. Construction of a suitable smooth map is a crucial step of the proof.Comment: 23 page

Bifurcation of limit cycles from a switched equilibrium in planar switched systems and its application to power converters

We consider a switched system of two subsystems that are activated as the trajectory enters the regions $\{(x,y):x>\bar x\}$ and $\{(x,y):x<-\bar x\}$ respectively, where $\bar x$ is a positive parameter. We prove that a regular asymptotically stable equilibrium of the associated Filippov equation of sliding motion (corresponding to $\bar x=0$) yields an orbitally stable limit cycle for all $\bar x>0$ sufficiently small. The research is motivated by an application to a dc-dc power converter, where $\bar x>0$ is used in place of $\bar x=0$ to avoid sliding motions

Domain of attraction of asymptotically stable periodic solutions obtained via averaging principle

In this paper we propose an approach to evaluate the domain of attraction of asymptotically stable periodic solutions obtained via averaging principle (second Bogolubov's theorem or Mel'nikov's method). We discuss also how this result is extended in the case when the right hand part is nonsmooth.Comment: To appear in Proceedings of the 9th Conference on Dynamical Systems Theory and Applications (Lodz, 2007

A new test for stick-slip limit cycles in dry-friction oscillators with small nonlinear friction characteristics

We consider a dry friction oscillator on a moving belt with both the Coulomb friction and a small nonlinear addition which can model e.g. the Stribeck effect. By using the perturbation theory, we establish a new sufficient condition for the nonlinearity to ensure the occurrence of a stick-slip limit cycle. The test obtained is more accurate compared to what one gets by building upon the divergence test.Comment: 7 figure

Existence and stability of a limit cycle in the model of a planar passive biped walking down a slope

We consider the simplest model of a passive biped walking down a slope given by the equations of switched coupled pendula (McGeer, 1990). Following the fundamental work by Garcia et al (1998), we view the slope of the ground as a small parameter $\gamma\ge 0$. When $\gamma=0$ the system can be solved in closed form and the existence of a family of limit cycles (i.e. potential walking cycles) can be established explicitly. As observed in Garcia et al (1998), the family of limit cycles disappears when $\gamma$ increases and only isolated asymptotically stable cycles (walking cycles) persist. However, no rigorous proofs of such a bifurcation (often referred to as Melnikov bifurcation) have ever been reported. The present paper fills in this gap in the field and offers the required proof.Comment: 1

Dwell time for switched systems with multiple equilibria on a finite time-interval

We describe the behavior of solutions of switched systems with multiple globally exponentially stable equilibria. We introduce an ideal attractor and show that the solutions of the switched system stay in any given $\varepsilon$-inflation of the ideal attractor if the frequency of switchings is slower than a suitable dwell time $T$. In addition, we give conditions to ensure that the $\varepsilon$-inflation is a global attractor. Finally, we investigate the effect of the increase of the number of switchings on the total time that the solutions need to go from one region to another.Comment: 5 figure

Stabilization of the response of cyclically loaded lattice spring models with plasticity

This paper develops an analytic framework to design both stress-controlled and displacement-controlled T-periodic loadings which make the quasistatic evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function t-> (e(t),p(t)), where e_i(t) and p_i(t) are the elastic and plastic deformations of spring i, defined on [t0,\infty) by the initial condition (e(t0),p(t0)). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron C(t) in a vector space E of dimension d, it becomes natural to expect (based on a result by Krejci) that the solution t->(e(t),p(t)) always converges to a T-periodic function. The achievement of this paper is in spotting a class of loadings where the Krejci's limit doesn't depend on the initial condition (e(t0),p(t0)) and so all the trajectories approach the same T-periodic regime. The proposed class of sweeping processes is the one for which the normal vectors of any d different facets of the moving polyhedron C(t) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. We discover that the normal vectors of any d different facets of the moving polyhedron C(t) are linearly independent, if the number of displacement-controlled loadings is two less the number of nodes of the given network of springs and when the magnitude of the stress-controlled loading is sufficiently large (but admissible). The result can be viewed as an analogue of the high-gain control method for elastoplastic systems. In continuum theory of plasticity, the respective result is known as Frederick-Armstrong theorem.Comment: 41 pages, 10 figure

VecHGrad for Solving Accurately Complex Tensor Decomposition

Tensor decomposition, a collection of factorization techniques for multidimensional arrays, are among the most general and powerful tools for scientific analysis. However, because of their increasing size, today's data sets require more complex tensor decomposition involving factorization with multiple matrices and diagonal tensors such as DEDICOM or PARATUCK2. Traditional tensor resolution algorithms such as Stochastic Gradient Descent (SGD), Non-linear Conjugate Gradient descent (NCG) or Alternating Least Square (ALS), cannot be easily applied to complex tensor decomposition or often lead to poor accuracy at convergence. We propose a new resolution algorithm, called VecHGrad, for accurate and efficient stochastic resolution over all existing tensor decomposition, specifically designed for complex decomposition. VecHGrad relies on gradient, Hessian-vector product and adaptive line search to ensure the convergence during optimization. Our experiments on five real-world data sets with the state-of-the-art deep learning gradient optimization models show that VecHGrad is capable of converging considerably faster because of its superior theoretical convergence rate per step. Therefore, VecHGrad targets as well deep learning optimizer algorithms. The experiments are performed for various tensor decomposition including CP, DEDICOM and PARATUCK2. Although it involves a slightly more complex update rule, VecHGrad's runtime is similar in practice to that of gradient methods such as SGD, Adam or RMSProp