Smooth Differential Equations Research Articles

Bifurcations of degenerate homoclinic solutions in discontinuous systems under non-autonomous perturbations.

The main aim of this paper is to study bifurcations of bounded solutions from a degenerate homoclinic solution for discontinuous systems under non-autonomous perturbations. We use Lyapunov-Schmidt reduction to give bifurcation equations and prove that a single parameter is enough to unfold two distinct homoclinic solutions bifurcated from the unperturbed degenerate homoclinic solution. Furthermore, we give an example of a periodically perturbed piecewise smooth differential equation in R4 to support our conclusions.

Bifurcating Limit Cycles with a Perturbation of Systems Composed of Piecewise Smooth Differential Equations Consisting of Four Regions

Systems composed of piecewise smooth differential (PSD) mappings have quantitatively been searched for answers to a substantial issue of limit cycle (LC) bifurcations. In this paper, LC numbers (LCNs) of a PSD system (PSDS) consisting of four regions are dealt with. A Melnikov mapping whose order is one is implicitly obtained by finding its originators when the system is perturbed under any nth degree of real polynomials. Then, the approach employing the Picard–Fuchs mapping is utilized to attain a higher boundary of bifurcation LCNs of systems composed of PSD functions with a global center. The method we used could be implemented to examine the problems related to the LC of other PSDS.

Blowup Analysis of a Hysteresis Model Based Upon Singular Perturbations

In this paper, we provide a geometric analysis of a new hysteresis model that is based upon singular perturbations. Here hysteresis refers to a type of regularization of piecewise smooth differential equations where the past of a trajectory, in a small neighborhood of the discontinuity set, determines the vector-field at present. In fact, in the limit where the neighborhood of the discontinuity vanishes, hysteresis converges in an appropriate sense to Filippov’s sliding vector-field. Recently (2022), however, Bonet and Seara showed that hysteresis, in contrast to regularization through smoothing, leads to chaos in the regularization of grazing bifurcations, even in two dimensions. The hysteresis model we analyze in the present paper—which was developed by Bonet et al in a paper from 2017 as an attempt to unify different regularizations of piecewise smooth systems—involves two singular perturbation parameters and includes a combination of slow–fast and nonsmooth effects. The description of this model is therefore—from the perspective of singular perturbation theory—challenging, even in two dimensions. Using blowup as our main technical tool, we prove existence of an invariant cylinder carrying fast dynamics in the azimuthal direction and a slow drift in the axial direction. We find that the slow drift is given by Filippov’s sliding vector-field to leading order. Moreover, in the case of grazing, we identify two important parameter regimes that relate the model to smoothing (through a saddle-node bifurcation of limit cycles) and hysteresis (through chaotic dynamics, due to a folded saddle and a novel return mechanism).

The Number of Limit Cycles Bifurcating from a Degenerate Center of Piecewise Smooth Differential Systems

For two families of planar piecewise smooth polynomial differential systems, whose unperturbed system has a degenerate center at the origin, we study the biggest lower bound for the maximum number of limit cycles bifurcating from the periodic orbits of the center. These results are extensions of the known ones on unperturbed nondegenerate [Formula: see text]-center, derived from a nonsmooth harmonic oscillator model, to degenerate [Formula: see text]-center. Our study involves some new computational treatments. The main tools are the generalized polar coordinate change and the generalized Lyapunov polar coordinate change together with an averaging theory for one-dimensional piecewise smooth differential equations. Finally, we present two Maple programs for computing the averaging functions and consequently the biggest lower bound on the maximum number of limit cycles of degenerate [Formula: see text]-center under general polynomial perturbations of degree [Formula: see text].

Higher Order Analysis on the Existence of Periodic Solutions in Continuous Differential Equations via Degree Theory

Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of Brouwer degree theory. Between the Lipschitz and the discontinuous piecewise smooth differential equations, there is a huge class of differential equations lacking in a higher order analysis on the existence of periodic solutions, namely the class of continuous non-Lipschitz differential equations. In this paper, based on the coincidence degree theory for nonlinear operator equations, we perform a higher order analysis of continuous (non-Lipschitz) perturbed differential equations and derive sufficient conditions for the existence of periodic solutions for such systems. We apply our results to study continuous (non-Lipschitz) higher order perturbations of a harmonic oscillator.

Auxiliary Sliding Motions of Vibro-Impact Systems

In motion control systems, modes with more and more frequent switching are known. For systems with impacts (a certain class of impulse systems), an analog of these modes is motions with an infinite number of impact interactions occurring in a finite time. For such motions, a description in terms of smooth differential equations is proposed. Some applications of such a description are presented.

Non-Smooth Dynamics in Energy Market Models: A Complex Approximation From System Dynamics and Dynamical Systems Approach

This paper reports a general model that describes the supply and demand of electricity in a national market based on the system dynamics (SD) approach. From the resulting SD model, it derives piecewise smooth (non-smooth) differential equations from the nonlinear functions and feedback cycles of the corresponding stock-flow structure. Subsequently, the stability of the equilibrium points and non-smooth dynamics of the SD model are investigated using the dynamical systems theory. Filippov systems are found in the proposed SD model and non-smooth vector fields associated with generators investment decisions are accumulated. Under this combining methodology, the non-smooth dynamics of energy markets that are governed by the supply and demand laws are uncovered mathematically and deeply described. In fact, we extend our investigation results to any energy market model attached to various investment decisions, confirming the generalizability of our research.

Bifurcation phenomena and statistical regularities in dynamics of forced impacting oscillator

The paper is devoted to the study of harmonically forced impacting oscillator. The physical model for oscillator is a cart on a guide connected to the support with springs and excited by the stepper motor. The support also is provided with limiter of motion. The mathematical model for this system is defined with the second-order piecewise smooth differential equation. Model’s nonlinearity is connected with the incorporation of dry friction and generalized Hertz contact law. Analyzing the classical Poincare sections and inter-impact sequences obtained experimentally and numerically, the bifurcations and statistical properties of periodic, multi-periodic, and chaotic regimes were examined. The development of impact-adding regime as a new nonlinear phenomenon when the forcing frequency varies was observed.

Non-smooth analysis in a national electricity market model: a complex approach

In this paper, we show the preliminary results in a proposed a model for the supply and demand of electricity in a domestic market based on system dynamics. Additionally, the model indicates piecewise smooth differential equations arising from the diagram of flows and levels, using dynamical systems theory for the study of the stability of the equilibrium points that have such a system. A bifurcation analysis approach is proposed to define and understand the complex behavior. Until now, no work has been reported related to this topic using bifurcations criteria. The growing interest in personal ways of self-generation using renewable sources can lead the national grid to a standstill and low investment in the system. However, it is essential to preserve the national network as a power supply support to domestic and enterprise demand. To understand this scenario, we include an analysis of zero-rate demand growth. Under this hypothesis, a none smooth bifurcation appears related to a policy which involves the variation of the capacity charge. As a first significant result, we found that it is possible to preserve the investments in the market since, through the capacity charge parameter, the system dynamics can be controlled. Then, from a business approach, it is necessary to know the effects of the capacity charge as the strategic policy in the system generation price scheme.

The chaotic behaviour of piecewise smooth differential equations on two-dimensional torus and sphere

ABSTRACTThis paper studies the global dynamics of piecewise smooth differential equations defined in the two-dimensional torus and sphere in the case when the switching manifold breaks the manifold into two connected components. Over the switching manifold, we consider the Filippov's convention for discontinuous differential equations. The study of piecewise smooth dynamical systems over torus and sphere is common for maps and up to where we know this is the first characterization for piecewise smooth flows arising from solutions of differential equations. We provide conditions under generic families of piecewise smooth equations to get periodic and dense trajectories. Considering these generic families of piecewise differential equations, we prove that a non-deterministic chaotic behaviour appears. Global bifurcations are also classified.

Limit Cycles of Piecewise Smooth Differential Equations on Two Dimensional Torus

In this paper we study the limit cycles of some classes of piecewise smooth vector fields defined in the two dimensional torus. The piecewise smooth vector fields that we consider are composed by linear, Ricatti with constant coefficients and perturbations of these one, which are given in (3). Considering these piecewise smooth vector fields we characterize the global dynamics, studying the upper bound of number of limit cycles, the existence of non-trivial recurrence and a continuum of periodic orbits. We also present a family of piecewise smooth vector fields that posses a finite number of fold points and, for this family we prove that for any 2k number of limit cycles there exists a piecewise smooth vector fields in this family that presents k number of limit cycles and prove that some classes of piecewise smooth vector fields presents a non-trivial recurrence or a continuum of periodic orbits.

Solutions leaving a codimension- $$\varvec{2}$$ 2 sliding

Piece-wise smooth differential equations (their regularization, numerical integration, and classification of solutions) is the topic of the present work. The behaviour close to one discontinuity surface and also the entering into the intersection of two discontinuity surfaces is well understood. Here, we study the solutions that exit a codimension-2 sliding mode. Some results are expected, others come as a surprise. We are able to explain situations, where difficulties in numerical computations are reported in the recent literature. The analysis is based on asymptotic expansions for singularly perturbed problems and on the study of a time-parameterized two-dimensional dynamical system (hidden dynamics). Various situations are illustrated by examples.

Computational Complexity of Smooth Differential Equations

The computational complexity of the solutions $h$ to the ordinary differential equation $h(0)=0$, $h'(t) = g(t, h(t))$ under various assumptions on the function $g$ has been investigated. Kawamura showed in 2010 that the solution $h$ can be PSPACE-hard even if $g$ is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of $g$ and obtain the following results: the solution $h$ can still be PSPACE-hard if $g$ is assumed to be of class $C^1$; for each $k\ge2$, the solution $h$ can be hard for the counting hierarchy even if $g$ is of class $C^k$.

On the topological structure of attraction basins for differential inclusions

We show that when a compact set is globally asymptotically stable under the action of a differential inclusion satisfying certain regularity properties, there exists a smooth differential equation rendering the same compact set globally asymptotically stable. The regularity properties assumed in this work stem from the consideration of Krasovskii/Filippov solutions to discontinuous differential equations and the robustness of asymptotic stability under perturbation. In particular, the results in this work show that when a compact set cannot be globally asymptotically stabilized by continuous feedback due to topological obstructions, it cannot be robustly globally asymptotically stabilized by discontinuous feedback either. The results follow from converse Lyapunov theory and parallel what is known for the local stabilization problem.

Nonsmooth bifurcations of equilibria in planar continuous systems

In this paper we present a procedure to find all limit sets near bifurcating equilibria in a class of hybrid systems described by continuous, piecewise smooth differential equations. For this purpose, the dynamics near the bifurcating equilibrium is locally approximated as a piecewise affine systems defined on a conic partition of the plane. To guarantee that all limit sets are identified, conditions for the existence or absence of limit cycles are presented. Combining these results with the study of return maps, a procedure is presented for a local bifurcation analysis of bifurcating equilibria in continuous, piecewise smooth systems. With this procedure, all limit sets that are created or destroyed by the bifurcation are identified in a computationally feasible manner.

On torus Burth bifurcations in the pulse-width system

The dynamics of the pulse-width system described by nonautonomous piecewise smooth differential equations is studied. In the parameter plane domains of different dynamic behaviour are determined both analytically and numerically. A special type of C-bifurcation resulting in the invariant torus birth from a stable equilibrium is discovered. It is shown that the periodic motion due to this bifurcation arises as an unstable focal cycle surrounded by a resonance or ergodic torus.

Coordinated Gradient Descent: A Case Study of Lagrangian Dynamics with Projected Gradient Information

The paper studies gradient descent algorithms for vehicle networks. Each vehicle within the network is modeled as a double integrator in the plane. For each individual vehicle, the control input enabling coordinated gradient descent consists of a gradient descent control term and additional inter-vehicle forcing terms. When each vehicle has enough sensors to measure the full gradient at its current position, then the closed-loop system becomes Lagrangian. We focus in the present paper upon the more practical situation where each vehicle has only one sensor with which to sample the environment. We take this into account by replacing the full gradient in the closed-loop equations by its projection on the direction of motion for each individual vehicle. This gives rise to a differential equation with discontinuous right-hand side. In order to avoid the (practical and theoretical) complications that arise as a consequence of these discontinuities, we modify the inter-vehicle forcing terms and represent the velocity of each vehicle by a magnitude and an angle, resulting in a set of smooth differential equations. We demonstrate our approach with simulations

Compositionality issues in discrete, continuous, and hybrid systems

AbstractModels of complex dynamical systems are often built by connecting submodels of smaller parts. The key to this method is the operation of ‘interconnection’ or ‘composition’ which serves to define the whole in terms of its parts. In the setting of smooth differential equations the composition operation has often been regarded as trivial, but a quite different attitude is found in the discrete domain where several definitions of composition have been proposed and different semantics have been developed. The non‐triviality of composition carries over from discrete systems to hybrid systems. The paper discusses the compositionality issue in the context of discrete, continuous, and hybrid systems, mainly on the basis of a number of examples. Copyright © 2001 John Wiley & Sons, Ltd.

Topological Equivalence of Local Singularities of Particular Type for Dynamical Systems with Shock Interactions

A theorem on topological equivalence is proved for local singularities of particular type for dynamical systems with shock interactions. The proof is based on a previously established result concerning the description of motion in the neighborhood of the specified local singularity in terms of smooth differential equations.

Center-focus and isochronous center problems for discontinuous differential equations

The study of the center focus problem and the isochronicity problem for differential equations with a line of discontinuities is usually done bycomputing the whole return map as the composition of the two maps associated to the two smooth differential equations. This leads to large formulaswhich usually are treated with algebraic manipulators. In this paper we approach to this problem from a more theoretical point of view. The results thatwe obtain relate the order of degeneracy of the critical point of the discontinuous differential equations with the order of degeneracy of the two smoothcomponent differential equations. Finally we apply them to some families ofexamples.