A numerical approach for the bifurcation analysis of nonsmooth delay equations

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record .Mathematical models based on nonsmooth dynamical systems with delay are widely used to understand complex phenomena, specially in biology, mechanics and control. Due to the infinite-dimensional nature of dynamical systems with delay, analytical studies of such models are difficult and can provide in general only limited results, in particular when some kind of nonsmooth phenomenon is involved, such as impacts, switches, impulses, etc. Consequently, numerical approximations are fundamental to gain both a quantitative and qualitative insight into the model dynamics, for instance via numerical continuation techniques. Due to the complex analytical framework and numerical challenges related to delayed nonsmooth systems, there exists so far no dedicated software package to carry out numerical continuation for such type of models. In the present work, we propose an approximation scheme for nonsmooth dynamical systems with delay that allows a numerical bifurcation analysis via continuation (path-following) methods, using existing numerical packages, such as COCO (Dankowicz and Schilder). The approximation scheme is based on the well-known fact that delay differential equations can be approximated via large systems of ODEs. The effectiveness of the proposed numerical scheme is tested on a case study given by a periodically forced impact oscillator driven by a time-delayed feedback controller.Engineering and Physical Sciences Research Council (EPSRC

Dynamics of Simple Balancing Models with State Dependent Switching Control

Time-delayed control in a balancing problem may be a nonsmooth function for a variety of reasons. In this paper we study a simple model of the control of an inverted pendulum by either a connected movable cart or an applied torque for which the control is turned off when the pendulum is located within certain regions of phase space. Without applying a small angle approximation for deviations about the vertical position, we see structurally stable periodic orbits which may be attracting or repelling. Due to the nonsmooth nature of the control, these periodic orbits are born in various discontinuity-induced bifurcations. Also we show that a coincidence of switching events can produce complicated periodic and aperiodic solutions.Comment: 36 pages, 12 figure

Calculating the Lyapunov exponents of a piecewise smooth soft impacting system with a time-delayed feedback controller

This is the final version. Available on open access from Elsevier via the DOI in this record.Lyapunov exponent is a widely used tool for studying dynamical systems. When calculating Lyapunov exponents for piecewise smooth systems with time delayed arguments one faces two difficulties: a high dimension of the discretized state space and a lack of continuity of the variational problem. This paper shows how to build a variational equation for the efficient construction of Jacobians along trajectories of the delayed nonsmooth system. Trajectories of a piecewise smooth system may encounter the so-called grazing events, where the trajectory approaches discontinuity surfaces in the state space in a non-transversal manner. For these events we develop a grazing point estimation algorithm to ensure the accuracy of trajectories for the nonlinear and the variational equations. We show that the eigenvalues of the Jacobian matrices computed by the algorithm converge with an order consistent with the order of the numerical integration method. Finally, we demonstrate the proposed method for a periodically forced impacting oscillator under a time-delayed feedback control, which exhibits grazing and crossing of the impact surface.EPSRCEuropean Union

Tutorial of numerical continuation and bifurcation theory for systems and synthetic biology

Mathematical modelling allows us to concisely describe fundamental principles in biology. Analysis of models can help to both explain known phenomena, and predict the existence of new, unseen behaviours. Model analysis is often a complex task, such that we have little choice but to approach the problem with computational methods. Numerical continuation is a computational method for analysing the dynamics of nonlinear models by algorithmically detecting bifurcations. Here we aim to promote the use of numerical continuation tools by providing an introduction to nonlinear dynamics and numerical bifurcation analysis. Many numerical continuation packages are available, covering a wide range of system classes; a review of these packages is provided, to help both new and experienced practitioners in choosing the appropriate software tools for their needs.Comment: 14 pages, 2 figures, 2 table