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We establish a theorem on bifurcation of limit cycles from a focus boundary equilibrium of an impacting system, which is universally applicable to prove the bifurcation of limit cycles from focus boundary equilibria in other types of piecewise-smooth systems, such as Filippov systems and sweeping processes. Specifically, we assume that one of the subsystems of the piecewise-smooth system under consideration admits a focus equilibrium that lie on the switching manifold at the bifurcation value of the parameter. In each of the three cases, we derive a linearized system which is capable of concluding the occurrence of a finite-time stable limit cycle from the above-mentioned focus equilibrium when the parameter crosses the bifurcation value. Examples illustrate how conditions of our theorems lead to closed-form formulas for the coefficients of the linearized system. [ABSTRACT FROM AUTHOR]

We use a bifurcation approach to investigate the dynamics of a planar switched system that alternates between two smooth systems of ODEs denoted as (L) and (R), respectively. For an x ∊ R that plays a role of bifurcation parameter, a switch to (R) occurs when the trajectory hits the switching line {x}xR and a switch to (L) occurs when the trajectory hits the switching line {-x}xR. This type of switching is known as relay, hysteresis, or hybrid switching in control. The main result of the paper gives suffcient conditions for bifurcation of an attracting or repelling limit cycle from a point O ∊ {0} xRR when x crosses 0. The result is achieved by identifying a region where the dynamics of the system is described by the map Px(y) = y + y2 +fβx/=y +o(y2): The map Px is obtained as a local return map induced by the switching line {x}x R. The fixed points of Px correspond to small amplitude limit cycles surrounding O. Motivated by applications to antilock braking systems, we focus on a particular class of switched systems where, for x=0, the point O is a so-called fold-fold singularity, i.e., the vector fields of both systems (L) and (R) are parallel to {0}xR at O. The result of the paper can be used for the design of switched control strategies that ensure limit cycling around a given point of the phase space. In particular, we illustrate the main theorem by establishing limit cycling behavior in relay affine systems (that model, e.g., power converters). Furthermore, we design a two-rule switched control to guarantee the existence of such an attracting limit cycle in antilock braking systems, whose magnitude can be made as small as necessary. This diminishes the need of additional switching rules used in other papers to not exceed the actuator technical capabilities. [ABSTRACT FROM AUTHOR]

Abstract: In this survey we discuss current directions of research in the dynamics of nonsmooth systems, with emphasis on bifurcation theory. An introduction to the state-of-the-art (also for non-specialists) is complemented by a presentation of the main open problems. We illustrate the theory by means of elementary examples. The main focus is on piecewise smooth systems, which have recently attracted a lot of attention, but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities. This extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context. A dedicated section is devoted to concrete applications that stimulate the development of the field. This survey is concluded by an extensive bibliography. [Copyright &y& Elsevier]

The goal of this paper is to present a new method to prove bifurcation of a branch of asymptotically stable periodic solutions of a T-periodically perturbed autonomous system from a T-periodic limit cycle of the autonomous unperturbed system. The method is based on a linear scaling of the state variables to convert, under suitable conditions, the singular Poincaré map (with two singularity conditions) associated to the perturbed autonomous system into an equivalent non-singular equation to which the classical implicit function theorem applies directly. As a result we obtain the existence of a unique branch of T-periodic solutions (usually found for bifurcations of co-dimension 2) as well as a relevant property of the spectrum of their derivatives. Finally, by a suitable representation formula of the classical Malkin bifurcation function, we show that our conditions are equivalent to the existence of a non-degenerate simple zero of the Malkin function. The novelty of the method is that it permits to solve the problem without explicit reduction of the dimension of the state space as it is usually done in the literature by the Lyapunov-Schmidt method. [ABSTRACT FROM AUTHOR]

Abstract: Classical conditions for asymptotic stability of periodic solutions bifurcating from a limit cycle rely on the sign of the derivative of the associated bifurcation function at a zero. In this paper we show that, for analytic systems, this result is of topological nature. This means that it is enough to impose a change of sign at the zero, without any assumption on the succesive derivatives. [Copyright &y& Elsevier]

Abstract: The paper addresses the problem of bifurcation of periodic solutions from a normally nondegenerate family of periodic solutions of ordinary differential equations under perturbations. The approach to solve this problem can be described as transforming (by a Lyapunov–Schmidt reduction) the initial system into one which is in the standard form of averaging, and subsequently applying the averaging principle. This approach encounters a fundamental problem when the perturbation is only Lipschitz (nonsmooth) as we do not longer have smooth Lyapunov–Schmidt projectors. The situation of Lipschitz perturbations has been addressed in the literature lately and the results obtained conclude the existence of the bifurcated branch of periodic solutions. Motivated by recent challenges in control theory, we are interested in the uniqueness problem. We achieve this in the case when the Lipschitz constant of the perturbation obeys a suitable estimate. [Copyright &y& Elsevier]