Extended Abstracts Spring 2016

The main objective of this paper is to study bifurcations of a vector field in a neighborhood of a cycle having a homoclinic-like connection at a saddle-regular point. In order to perform such a study it is necessary to analyze how the cycle can be broken, in this way the approach is to look separately at local bifurcations and at the structure of the first return map defined near the cycle.

In this paper, the properties of codimension–2 discontinuity sufaces of vector fields are presented which can arise from e.g., spatial Coulomb friction. Concepts of sliding region and sliding dynamics are defined for these systems.

Asynchronous networks form a natural framework for many classes of dynamical networks encountered in technology, engineering and biology. Typically, nodes can evolve independently, be constrained, stop, and later restart, and interactions between components of the network may depend on time, state, and stochastic effects. We outline some of the main ideas, motivations and a basic result.

Regularization was a big topic at the 2016 CRM Intensive Research Program on Advances in Nonsmooth Dynamics. There are many open questions concerning well known kinds of regularization (e.g., by smoothing or hysteresis). Here, we propose a framework for an alternative and important kind of regularization, by external variables that shadow either the state or the switch of the original system. The shadow systems are derived from and inspired by various applications in electronic control, predator-prey preference, time delay, and genetic regulation.

During the workshop on Climate Modeling in Nonsmooth Systems, one of the major discussions involved investigating including more realistic elements, such as fluctuations and time variation, in nonsmooth models that undergo a sudden transition, with an emphasis on conceptual climate models. A number of models were discussed, including the Stommel 1961 model, the Paillard 1997 model, the Eisenman–Wettlaufer 2009 model, and the Hogg 2008 model.

In this work, we show some global bifurcations for a class of three-dimensional discontinuous piecewise linear (DPWL) systems having a unique two-fold point of visible-invisible type. We consider the simplest case of DPWL systems, with two vector fields separated by a switching plane with a unique equilibrium point in each half-space.

We describe a geometric approach to understand the mechanism of creation and annihilation of single-impact periodic orbits close to grazing for a general second-order one degree of freedom impact oscillator. Here, non-degenerate grazing (nonzero acceleration) is assumed, with approaches to degenerate grazing also outlined: this is work in progress. The method in principle extends to more degrees of freedom (coupled oscillators, for example) and to a variety of restitution rules such as soft impacts, delays or sticking.

Piecewise smooth systems are frequently used as an alternative representation of mena with multiple time scales. One would of course expect that the qualitative behaviour of a model be independent of the choice of a smooth or piecewise smooth representation. We address this issue by building on some classical results from piecewise smooth systems theory.

This paper establishes a link between closed-loop controls for heterogeneous systems and sliding mode controls. We demonstrate that sliding mode analysis matches with experimental results from dielectric charge controllers. This approach provides a new way to analyze the behaviour of different heterogeneous systems.

We consider a specific family of three-dimensional differential systems whose vector field is continuous and piecewise linear, with two regions separated by a plane. After detecting a center configuration at infinity, we look for possible limit cycle bifurcation from such a center, by allowing parameter variations that do not destroy the center configuration.

We study the dynamics of a one dimensional discontinuous linear-power map. It has a vertical asymptote giving rise to new kinds of border collision bifurcations. We explain the peculiar periods of attracting cycles, appearing due to cascades of alternating smooth and nonsmooth bifurcations. Robust unbounded chaotic attractors are also described.

Motivated by a problem from pharmacology, we consider a general two parameter slow–fast system in which the critical set consists of a one dimensional manifold and a two dimensional manifold, intersecting transversally at the origin. Using geometric desingularisation, we show that for a subset of the parameter set there is an exchange of stabilities between the attracting components of the critical set and the direction of the continuation can be expressed in terms of the parameters.

The analysis of piecewise smooth bifurcations reveals an alarming proliferation of cases as the dimension of phase space increases. This suggests that a different approach needs to be taken when trying to describe bifurcations. In particular, it may not be helpful to analyze particular bifurcations at the level of detail that is standard for smooth systems.

The analysis of piecewise smooth bifurcations reveals an alarming proliferation of cases as the dimension of phase space increases. Rather than attempt the derivation of exhaustive lists of possibilities, we describe ways of giving less detailed, but possibly more useful, results.

We introduce the notion of semi-local structural stability which detects if a nonsmooth system is structurally stable around the switching manifold. More specifically, we characterize the semi-local structurally stable systems in a class of Filippov systems on a compact 3-manifold which has a simply connected switching manifold.

We use the piecewise linear McKean model to present a proof-of-concept to address the estimation of synaptic conductances when a neuron is spiking. Using standard techniques of non-smooth dynamical systems, we obtain an approximation of the period in terms of the parameters of the system which allows to estimate the steady synaptic conductance of the spiking neuron. The method gives also fairly good estimations when the synaptic conductances vary slowly in time.

Nonsmooth systems are typically studied with smooth or piecewise-smooth boundaries between smooth vector fields, especially with linear or hyper-planar boundaries. What happens when there is a boundary that is not as simple, for example a fractal? Can a solution to such a system slide or “chatter” along this boundary? It turns out that the dynamics is rather fascinating, and yet contained within A.F. Filippov’s theory (as promised in Utkin, Comments for the continuation method by A.F. Filippov for discontinuous systems, parts I and II, [2] from this volume).

Perhaps we should wrap up this volume by asking why nonsmooth dynamics is the subject of a three month Intensive Research Program at the CRM (February to April 2016), why it was the subject of more than 2000 papers published in 2015 (and only 700 in the year 2000; data from Thomson Reuters Web of Science), and why it is a growing presence at international conferences involving mathematics and its applications. We briefly survey here why discontinuity is not only important in modeling real-world systems, but is also a fundamental property of many nonlinear systems.

It took nearly 30 years from the translation of Filippov’s seminal book to be able to say that the two-fold singularity is understood. We now know that its structural stability requires nonlinear switching or hidden terms, and that it comes in three main flavours, with numerous subclasses between which bifurcations can occur. We know that it is neither an attractor nor a repellor, but a bridge between attracting and repelling sliding and, in certain cases, is a source of determinacy-breaking.

Sliding mode controllers are ideally modeled as responding to the state of a system when, in practice, only a measurement of the state is available, provided by non-ideal sensors. We provide an equivalent control model for a buck converter system that includes the dynamics of the sensors. The results demonstrate some limitations of the basic equivalent control method in determining the stability of systems with sensors.

The discrete time variational principle is applied to the Lagrangian formulation of multidomain nonsmooth dynamics to produce a stable time stepping scheme. Examples from electronics are used to demonstrate how to construct pseudo-potentials of nonsmooth devices such as transistors.

In this work we discuss the appearance of minimal trajectories for the flow of piecewise smooth dynamical systems defined in the two dimensional torus and sphere in such a way that the switching manifold breaks the manifold into two connected components. We show that the number of pseudo-singularities of the sliding vector field is an invariant for the structural stability and study global bifurcations. Using a generic normal form, we prove that these systems can present chaotic behavior.

A new discontinuity-induced bifurcation, referred to as nonsmooth Hopf-type bifurcation, observed in a nonautonomous impacting hybrid systems in $$\mathbb {R}^4$$ is presented. The system studied models the bouncing motion, repeated instantaneous impacts with friction, in rotating machines with magnetic bearing support. At the nonsmooth Hopf-type bifurcation point a stable regular equilibrium and two unstable small amplitude 1-impact periodic orbits arise. The existence of this bifurcation scenario depends on a complex relationship between damping, the restitution, and the friction coefficient.

Recently, some upper bounds were found for the maximum number of limit cycles for some non-generic classes of planar piecewise linear differential systems with two zones separated by a straight line. However, many distinct cases were considered. Here the main properties of those classes are identified, this allows us to unify the approach and to extend the results. We also study a new class of differential systems.

We use Bell polynomials to provide an alternative formula for the averaged functions. This new formula can make the computational implementation of the averaged functions easier.

In this work we discuss a piecewise-smooth dynamical system inspired by plankton observations and constructed for one predator switching its diet between two different types of prey. We then discuss two smooth formulations of the piecewise-smooth model obtained by using a hyperbolic tangent function and adding a dimension to the system. We compare model behaviour of the three systems and show an example case where the steepness of the switch is determined from a comparison with data on freshwater plankton.

A possible origin of the frictional travelling waves usually occuring between sliding interfaces is discussed: various solutions, including propagating wavetrains, pulses and fronts, can appear under rate-and-state friction from homoclinic or heteroclinic bifurcations.

The Mathematics and Climate Research Network (MCRN) was invited to run an informal Climate Modeling Workshop as part of the Intensive Research Programme on Advances on Nonsmooth Dynamics hosted by the Centre de Recerca Matemàtica (CRM). The workshop was attended by a core group of about 10 participants with a nice mix of junior and senior researchers. A summary of the proceedings of the workshop is presented here.

The collision of a fixed point with a switching manifold in a piecewise-smooth continuous map, known as a border-collision bifurcation, can give rise to a seemingly endless zoo of complicated dynamics. An understanding of these dynamics, which are described merely by piecewise-linear continuous maps, is one of the most fundamental problems in nonsmooth bifurcation theory. This extended abstract recalls some aspects of border-collision bifurcations and provides a list of pertinent open problems for future research.

Sleep-wake regulation is an example of a system with multiple timescales, with switching between sleep and wake states occurring in minutes but the states of wake or sleep usually existing for some hours. Here, we discuss some general features of models of sleep-wake regulation. We show that some typical models of sleep-wake regulation can be reduced to one-dimensional maps with discontinuities, and show that this reduction is useful in understanding some of the dynamical behaviour seen in sleep-wake models.

In part I, the Two-Process model for sleep-wake regulation was discussed and it was shown that it could usefully be represented as a one-dimensional map with discontinuities. Here, we discuss some recent, more physiological, models of sleep-wake dynamics. We describe how their fast-slow structure means that one can expect them to inherit many of the dynamical features of the Two-Process model.

The conventional existence-uniqueness theorems are not applicable for differential equations with right hand sides as discontinuous state functions. This is the case for the systems with discontinuous controls and sliding modes, when state trajectories belong to discontinuity surfaces. Many authors offered their methods of deriving sliding mode equations, or solution continuations on the discontinuity surfaces. Due to uncertainties of right hand sides, the proposed methods led to different solutions. These methods are compared, the reasons of ambiguity are discussed in the paper. It is assumed that any solution is under the umbrella of the method proposed by A.F. Filippov.

In the second part of this article, solution methods for scalar or vector control are considered.

This short paper proposes a general economics model for the supply and demand of a commodity in a domestic market, when investments are required for supporting it. Starting from System Dynamics, we recover a well-known model. Then, we improve the mathematical equations in order to be precise at the simulation level.