PHYSICAL REVIEW E 88, 042924 (2013) Stability and dynamics of nonautonomous systems with pulsed nonlinearity Yannis Kominis 1,2 and Tassos Bountis 2 1 School of Electrical and Computer Engineering, National Technical University of Athens, Zographou GR-15773, Greece 2 Department of Mathematics, University of Patras, Patras GR-26500, Greece (Received 11 July 2013; revised manuscript received 25 September 2013; published 30 October 2013) We study the dynamics of a class of nonautonomous systems with pulsed nonlinearity that consist of a periodic sequence of linear and nonlinear autonomous systems, each one acting alone in a different time or space interval. We focus on the investigation of control capabilities of such systems in terms of altering their fundamental dynamical properties by appropriate parameter selections. For the case of single oscillators, the stability of the zero solution as well as the phase space topology is shown to drastically depend on parameters such as the frequency of the linear oscillations and the durations of the linear and nonlinear intervals. In cases of chain of coupled oscillators with pulsed onsite nonlinearity, it is shown that appropriate parameter selections can stabilize an otherwise unstable zero background allowing for the existence of dynamically robust localized excitations, whose evolution properties can now be explicitly determined and controlled. DOI: 10.1103/PhysRevE.88.042924 PACS number(s): 05.45.Gg, 05.45.Yv, 45.05.+x I. INTRODUCTION Nonautonomous dynamical systems are commonly used as models for a large variety of natural or man-made systems. In such systems the independent variable measuring the evolution appears explicitly in the model equations. The independent variable can either have the dimensions of time or space. In the first case, it describes the explicit time variation of one of the system parameters, which may include a direct external driving force acting on the system. This time dependence usually comes from interaction with a different dynamical system, which is considered as external and practically unaffected from the system under investigation. A large number of systems fall into this category, including forced oscillators and oscillators with time-varying parameters, which are the basic paradigm in the theory of complex dynamical systems [1,2]. In the second case, when the independent variable corresponds to a spa- tial dimension, the nonautonomous systems describe spatial dynamics in spatially inhomogeneous media. This category includes the large class of models describing periodic or solitary wave formation in transversely [3,4] or longitudinally [5] inhomogeneous structures. In all cases, the introduction of the explicit dependence on the independent variable results in an additional degree of freedom in comparison to the respective autonomous system. As a result, a symmetry property is lost, the correspond- ing invariant quantity of the autonomous system becomes time-varying, and the phase space of the system undergoes a qualitative modification. In the case of an autonomous integrable Hamiltonian system, it is well known that even small nonautonomous perturbations lead, in general, to nonintegra- bility and irregular dynamics. The utilization of perturbation methods allows for the investigation of the relation between the dynamics of the original autonomous integrable system and the perturbed nonautonomous systems. The Poincare-Birkhoff theorem as well as Melnikov’s theory for periodic orbits [6,7] predict that when certain resonance conditions between the unperturbed system and the time-dependent perturbation are satisfied, a finite discrete family of periodic solutions persist under perturbation. Additionally, the Melnikov’s theory for homoclinic orbits [6,7] relates the existence of an unperturbed orbit homoclinic to a fixed point with the persistence of a discrete number of orbits homoclinic to a periodic orbit. We study the dynamics of a class of nonautonomous systems for which the explicit dependence on the independent variable is due to a periodically pulsed nonlinearity. Alterna- tively, these systems can be considered as nonlinear systems whose action is interrupted by a periodically interposed linear system. Such systems consist of a periodic sequence of au- tonomous linear and nonlinear systems, each one acting alone in a different time or space interval. We consider a periodic sequence of an either integrable or nonintegrable nonlinear and a linear Hamiltonian system. The special conditions under which the interposition of the linear system does not drastically modify the dynamics of the respective autonomous nonlinear system has been previously studied [2,8]. It has been shown that under such conditions the solutions of the total nonautonomous systems consists of the solutions of its nonlinear part interrupted by complete oscillations due to the linear part. In this work, we consider the general case where these conditions are not fullfiled, so that the dynamics of the nonautonomous system can be drastically different from those of its autonomous nonlinear part, depending crucially on the parameters. This dependence suggests a control mech- anism capable of determining some of the most characteristic features of nonlinear systems, that is their behavior under the presence of noise and the extent of regions with chaotic dynamics in the phase space as well as the propagation of localized excitations in spatially inhomogeneous systems and chains of oscillators. In Sec. II we consider single oscillators with pulsed nonlinearity. The stability type of the zero solution determines their behavior under the presence of noise and the extent of the chaotic region of the phase space, which can be therefore controlled by appropriate choices of the durations of the intervals at which the nonlinear and linear parts of the system act. Among several applications of such systems we can refer to the modeling of particle beam dynamics in storage rings of high-energy accelerators, where the nonlinear part of the system corresponds to focusing elements and the linear part to the beam propagation between them, with the crucial issue 042924-1 1539-3755/2013/88(4)/042924(7) ©2013 American Physical Society