Hidden bifurcations and attractors in nonsmooth dynamical system

We investigate the role of hidden terms at switching surfaces in piecewise smooth vector fields. Hidden terms are zero everywhere except at the switching surfaces, but appear when blowing up the switching surface into a switching layer. When discontinuous systems do surprising things, we can often make sense of them by extending our intuition for smooth system to the switching layer. We illustrate the principle here with a few attractors that are hidden inside the switching layer, being evident in the flow, despite not being directly evident in the vector field outside the switching surface. These can occur either at a single switch (where we will introduce hidden terms somewhat artificially to demonstrate the principle), or at the intersection of multiple switches (where hidden terms arise inescapably). A more subtle role of hidden terms is in bifurcations, and we revisit some simple cases from previous literature here, showing that they exhibit degeneracies inside the switching layer, and that the degeneracies can be broken using hidden terms. We illustrate the principle in systems with one or two switches. </jats:p

Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems

Switches in real systems take many forms, such as impacts, electronic relays, mitosis, and the implementation of decisions or control strategies. To understand what is lost, and what can be retained, when we model a switch as an instantaneous event, requires a consideration of so-called hidden terms. These are asymptotically vanishing outside the switch, but can be encoded in the form of nonlinear switching terms. A general expression for the switch can be developed in the form of a series of sigmoid functions. We review the key steps in extending the Filippov's method of sliding modes to such systems. We show how even slight nonlinear effects can hugely alter the behaviour of an electronic control circuit, and lead to `hidden' attractors inside the switching surface.Comment: 12 page

Theory of differential inclusions and its application in mechanics

The following chapter deals with systems of differential equations with discontinuous right-hand sides. The key question is how to define the solutions of such systems. The most adequate approach is to treat discontinuous systems as systems with multivalued right-hand sides (differential inclusions). In this work three well-known definitions of solution of discontinuous system are considered. We will demonstrate the difference between these definitions and their application to different mechanical problems. Mathematical models of drilling systems with discontinuous friction torque characteristics are considered. Here, opposite to classical Coulomb symmetric friction law, the friction torque characteristic is asymmetrical. Problem of sudden load change is studied. Analytical methods of investigation of systems with such asymmetrical friction based on the use of Lyapunov functions are demonstrated. The Watt governor and Chua system are considered to show different aspects of computer modeling of discontinuous systems

Non-Smooth Spatio-Temporal Coordinates in Nonlinear Dynamics

This paper presents an overview of physical ideas and mathematical methods for implementing non-smooth and discontinuous substitutions in dynamical systems. General purpose of such substitutions is to bring the differential equations of motion to the form, which is convenient for further use of analytical and numerical methods of analyses. Three different types of nonsmooth transformations are discussed as follows: positional coordinate transformation, state variables transformation, and temporal transformations. Illustrating examples are provided.Comment: 15 figure

Bifurcation sequences in a discontinuous piecewise-smooth map combining constant-catch and threshold-based harvesting strategies

First Published in SIAM Journal of Applied Dynamical Systems in 2022, published by the Society for Industrial and Applied Mathematics (SIAM).We consider a harvesting strategy that allows constant catches if the population size is above a certain threshold value (to obtain predictable yield) and no catches if the population size is below the threshold (to protect the population). We refer to this strategy as threshold constant-catch (TCC) harvesting. We provide analytical and numerical results when applying TCC to monotone population growth models. TCC remedies the tendency to fishery collapse of pure constant-catch harvesting and provides a buffer for quotas larger than the maximum sustainable yield. From a dynamical systems point of view, TCC gives rise to a piecewise-smooth map with a discontinuity at the threshold population size. The dynamical behavior includes border-collision bifurcations, basin boundary metamorphoses, and boundary-collision bifurcation. We further find Farey trees, a slightly modified truncated skew tent map scenario, and the bandcount incrementing scenario. Our results underline, on the one hand, the protective function of thresholds in harvest control rules. On the other hand, they highlight the dynamical complexities due to discontinuities that can arise naturally in threshold-based harvesting strategies.The first author's work was partially supported by PhD scholarship FPU18/00719 (Ministerio de Ciencia, Innovación y Universidades, Spain) and research grants MTM2016-75140-P (AEI/FEDER, UE), ED431C2019/02 (Xunta de Galicia). Osnabrück University provided funding to the first author to visit the Institute of Environmental Systems Research for a period of six weeks, during which parts of this work were completed.S