Class Of Piecewise Smooth Systems Research Articles

Bifurcation of Limit Cycles from a Class of Piecewise Smooth Systems with Two Vertical Straight Lines of Singularity

Perturbing the following systems [Formula: see text] having a linear center with two vertical straight lines ([Formula: see text]) of singularity inside the class of all piecewise polynomials of degree [Formula: see text], we give the estimate of the maximum number [Formula: see text] of limit cycles bifurcating from the period annulus around the center based on the argument principle and the first order averaging theory. It shows that [Formula: see text] is at least [Formula: see text] bigger than the maximum number of limit cycles bifurcating from the period annulus around the linear center with two parallel straight lines of singularity in [Li & Liu, 2015].

Observer design for piecewise smooth and switched systems via contraction theory

The aim of this paper is to present the application of an approach to study contraction theory recently developed for piecewise smooth and switched systems. The approach that can be used to analyze incremental stability properties of so-called Filippov systems (or variable structure systems) is based on the use of regularization, a procedure to make the vector field of interest differentiable before analyzing its properties. We show that by using this extension of contraction theory to nondifferentiable vector fields, it is possible to design observers for a large class of piecewise smooth systems using not only Euclidean norms, as also done in previous literature, but also non-Euclidean norms. This allows greater flexibility in the design and encompasses the case of both piecewise-linear and piecewise-smooth (nonlinear) systems. The theoretical methodology is illustrated via a set of representative examples.

Stability and Perturbations of Homoclinic Loops in a Class of Piecewise Smooth Systems

Like for smooth systems, a typical method to produce multiple limit cycles for a given piecewise smooth planar system is via homoclinic bifurcation. Previous works only focused on limit cycles that bifurcate from homoclinic orbits of piecewise-linear systems. In this paper, we consider for the first time the same problem for a class of general nonlinear piecewise smooth systems. By introducing the Dulac map in a small neighborhood of the hyperbolic saddle, we obtain the approximation of the Poincaré map for the nonsmooth homoclinic orbit. Then, we give conditions for the stability of the homoclinic orbit and conditions under which one or two limit cycles bifurcate from it. As an example, we construct a nonlinear piecewise smooth system with two limit cycles that bifurcate from a homoclinic orbit.

Limit cycle bifurcations in a class of piecewise smooth systems with a double homoclinic loop

In this paper we consider a class of perturbed piecewise smooth systems. Applying the method of first order Melnikov function we give a lower bound for the maximal number of limit cycles bifurcated from a double homoclinic loop. As an application we construct a piecewise quadratic system with quartic perturbation, which has 11 limit cycles bifurcated from such loop.

Non-smooth plant disease models with economic thresholds

In order to control plant diseases and eventually maintain the number of infected plants below an economic threshold, a specific management strategy called the threshold policy is proposed, resulting in Filippov systems. These are a class of piecewise smooth systems of differential equations with a discontinuous right-hand side. The aim of this work is to investigate the global dynamic behavior including sliding dynamics of one Filippov plant disease model with cultural control strategy. We examine a Lotka–Volterra Filippov plant disease model with proportional planting rate, which is globally studied in terms of five types of equilibria. For one type of equilibrium, the global structure is discussed by the iterative equations for initial numbers of plants. For the other four types of equilibria, the bounded global attractor of each type is obtained by constructing appropriate Lyapunov functions. The ideas of constructing Lyapunov functions for Filippov systems, the methods of analyzing such systems and the main results presented here provide scientific support for completing control regimens on plant diseases in integrated disease management.

An event-driven method to simulate Filippov systems with accurate computing of sliding motions

This article describes how to use smooth solvers for simulation of a class of piecewise smooth systems of ordinary differential equations, called Filippov systems, with discontinuous vector fields. In these systems constrained motion along a discontinuity surface (so-called sliding) is possible and requires special treatment numerically. The introduced algorithms are based on an extension to Filippov's method to stabilise the sliding flow together with accurate detection of the entrance and exit of sliding regions. The methods are implemented in a general way in MATLAB and sufficient details are given to enable users to modify the code to run on arbitrary examples. Here, the method is used to compute the dynamics of three example systems, a dry-friction oscillator, a relay feedback system and a model of an oil well drill-string.

Analysis of bifurcation and chaos in a class of piecewise smooth systems based on symbolic sequence

Based on the sequence of circuit topologies, a symbolic sequence method is proposed for analyzing the bifurcations and chaos in a class of piecewise smooth systems like DC/DC converters. The largest subsequence(LS) is used to distinguish the type of bifurcation and detect border collision bifurcation e. g., when a perioddoubling bifurcation occurs, the LS is unchanged; when border collision bifurcation occurs, the LS is changed; and when chaos occurs,the LS does not exist.It is shown that the duty ratio is an essential variable for describing the nonlinear dynamics of a class of piecewise smooth systems like DC/DC converters， and saturating nonlinearity is the root cause of border collision bifurcation.

Suppression of chaotic motion in a class of piecewise-smooth systems by using sine periodic force

This paper studies how to suppress the abnormal vibration in a dynamic system wi th clearance. The abnormal vibration contains the bifurcations or chaotic motio n due to the change of some parameters of the nonlinear system. This paper prop oses a method to suppress chaotic motion by applying an external periodical for ce to the system. The periodical force is adjusted according to some real-time a lgorism related to an error between real-time states and the states when the sy stem is stable. This method has been applied to a two-degree-freedom impact vibration system. With numerical simulation, the effectiveness of the method is demonstrated.

Border collision bifurcations and chaos in a class of piecewise smooth systems w ith two boundaries

A class of piecewise smooth maps with three zones is derived to describe the dynamics of a current-programmed Buck-Boost converter operating in a discontinuous mode. The numerical simulation is carried out and the bifurcation diagrams w ith the input voltage as a parameter are obtained. It is shown that, when a bifu rcation occurs, some eigenvalues of the Jacobian matrix jumps over the unit circ le in a discontinuous way, and there are always some orbit points lying on the b oundaries which separate different regions in the phase plane. It is concluded t hat border collision bifurcations could occur when the input voltage varies, for example, a bifurcation from a periodic orbit to another periodic orbit or chaot ic orbit.