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We consider the simplest model of a passive biped walking down a slope given by the equations of switched coupled pendula (McGeer T. 1990 Passive dynamic walking. Int. J. Robot. Res. 9, 62-82. (doi:10.1177/027836499000900206)). Following the fundamental work by Garcia (Garcia et al. 1998 J. Biomech. Eng. 120, 281. (doi:10.1115/1.2798313)), we view the slope of the ground as a small parameter γ ≥ 0. When γ =0, the system can be solved in closed form and the existence of a family of cycles (i.e. potential walking cycles) can be computed in closed form. As observed in Garcia et al. (Garcia et al. 1998 J. Biomech. Eng. 120, 281. (doi:10.1115/1.2798313)), the family of cycles disappears when γ increases and only isolated asymptotically stable cycles (walking cycles) persist. However, no mathematically complete proofs of the existence and stability of walking cycles have been reported in the literature to date. The present paper proves the existence and stability of a walking cycle (long-period gait cycle, as termed by McGeer) by using the methods of perturbation theory for maps. In particular, we derive a perturbation theorem for the occurrence of stable fixed points from 1-parameter families in two-dimensional maps that can be of independent interest in applied sciences. [ABSTRACT FROM AUTHOR]

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]

This paper presents a two-phase control logic for anti-lock braking systems (ABS). ABS are by now a standard component in every modern car, preventing the wheels from going into a lock situation where the wheels are fixed by the brake and the stopping distances are greatly prolonged. There are different approaches to such control logics. An ABS design proposed in recent literature controls the wheel's slip by creating stable limit cycles in the corresponding phase space. This design is modified via an analytical approach that is derived from perturbation theory. Simulation results document shorter braking distance compared to available tests in the literature. [ABSTRACT FROM AUTHOR]

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: In this paper we consider a class of planar autonomous systems having an isolated limit cycle of smallest period such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem. [Copyright &y& Elsevier]