scholarly journals Dynamics and Stability of Non-Smooth Dynamical Systems with Two Switches

2021 ◽  
Author(s):  
Guilherme Tavares da Silva ◽  
Ricardo Miranda Martins
Keyword(s):  
Dynamical Systems ◽  
Singular Perturbation ◽  
Singular Part ◽  
Qualitative Behavior ◽  
Algebraic Manifold ◽  
Singular Case ◽  
Regular Surface ◽  
Fine Grained ◽  
Straight Line ◽  
Abrupt Changes

Abstract One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. At this work, we leverage a methodology that, through blow-ups and singular perturbation, allows the extension of Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, non-linear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto-like theorems characterizing semi-local structural stability.

ON NONLINEAR WAVE EQUATIONS WITH BREAKING LOOP-SOLUTIONS

2010 ◽  
Vol 20 (02) ◽  
pp. 519-537 ◽  
Author(s):  
JIBIN LI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Nonlinear Wave ◽  
Traveling Wave ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Level Curves ◽  
Straight Line ◽  
Fixed Parameter ◽  
Unstable Manifolds

Using analytic methods from the dynamical systems theory, some new nonlinear wave equations are investigated, which have exact explicit parametric representations of breaking loop-solutions under some fixed parameter conditions. It is shown that these parametric representations are associated with some families of open level-curves of traveling wave systems corresponding to such nonlinear wave equations, each of which lies in an area bounded by a singular straight line and the stable and the unstable manifolds of a saddle point of such a system.

Modal Interactions in Dynamical and Structural Systems

1989 ◽  
Vol 42 (11S) ◽  
pp. S175-S201 ◽  
Author(s):  
A. H. Nayfeh ◽  
B. Balachandran
Keyword(s):  
Dynamical Systems ◽  
Surface Waves ◽  
Nonlinear Response ◽  
Experimental Studies ◽  
Structural Systems ◽  
Qualitative Behavior ◽  
Modal Interactions ◽  
Chaotic Motions ◽  
Beam Structures ◽  
Quadratic Nonlinearities

We review theoretical and experimental studies of the influence of modal interactions on the nonlinear response of harmonically excited structural and dynamical systems. In particular, we discuss the response of pendulums, ships, rings, shells, arches, beam structures, surface waves, and the similarities in the qualitative behavior of these systems. The systems are characterized by quadratic nonlinearities which may lead to two-to-one and combination autoparametric resonances. These resonances give rise to a coupling between the modes involved in the resonance leading to nonlinear periodic, quasi-periodic, and chaotic motions.

Quantitative Determination of Abrupt Changes in Dynamical Systems: Illustration Via Identification of Seizure Termination in Generalized Tonic–Clonic Seizure EEG Data

2003 ◽  
Vol 13 (09) ◽  
pp. 2641-2655
Author(s):  
C. J. Cellucci ◽  
A. M. Albano ◽  
P. E. Rapp ◽  
A. D. Krystal
Keyword(s):  
Dynamical Systems ◽  
Clonic Seizure ◽  
Seizure Prediction ◽  
Seizure Onset ◽  
End Point ◽  
Seizure Termination ◽  
Eeg Data ◽  
Abrupt Changes ◽  
Clonic Seizures

We discuss key theoretical and practical issues related to the identification of transitions in dynamical systems in real-time. We focus on the choice of candidate measures and optimization of decision thresholds for candidate measures. To illustrate these issues we develop and test a procedure for identifying one particular transition, the end-point of seizures in two-channel electro-encephalographic data recorded during generalized tonic–clonic seizures. Data from twenty-eight seizures were available and used to develop and test the procedure in terms of the agreement between the computationally identified seizure end-point compared against the ratings of an expert clinical electroencephalographer. Generalizations to multivariate seizure onset detection and to seizure prediction are described.

BEHAVIOR PATTERNS IN MULTIPARAMETRIC DYNAMICAL SYSTEMS: LORENZ MODEL

2012 ◽  
Vol 22 (06) ◽  
pp. 1230019 ◽  
Author(s):  
ROBERTO BARRIO ◽  
FERNANDO BLESA ◽  
SERGIO SERRANO
Keyword(s):  
Dynamical Systems ◽  
Chaotic Attractors ◽  
Behavior Patterns ◽  
Theoretical Studies ◽  
Careful Selection ◽  
Lorenz Model ◽  
Straight Line ◽  
Global Study ◽  
Experimental And Theoretical Studies ◽  
Selection Of

In experimental and theoretical studies of Dynamical Systems, there are usually several parameters that govern the models. Thus, a detailed study of the global parametric phase space is not easy and normally unachievable. In this paper, we show that a careful selection of one straight line (or other 1D manifold) permits us to obtain a global idea of the evolution of the system in some circumstances. We illustrate this fact with the paradigmatic example of the Lorenz model, based on a global study, changing all three parameters. Besides, searching in other regions, for all the detected behavior patterns in one straight line, we have been able to see that missing topological structures of the chaotic attractors may be found on the chaotic-saddles.

scholarly journals Periodic Peakon and Smooth Periodic Solutions for KP-MEW(3,2) Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Liping He
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Phase Portrait ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Wave System ◽  
Integral Constant ◽  
Straight Line

In this paper, we consider the KP-MEW(3,2) equation by the bifurcation theory of dynamical systems when integral constant is considered. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. The phase portrait for c < 0 , 0 < c < 1 , and c > 1 is drawn. Exact parametric representations of periodic peakon solutions and smooth periodic solution are presented.

Phase Space Reconstruction

2018 ◽  
Author(s):  
Ray Huffaker ◽  
Marco Bittelli ◽  
Rodolfo Rosa
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Dynamical Systems ◽  
Statistical Mechanics ◽  
State Space ◽  
Phase Plane ◽  
Material Point ◽  
The Other ◽  
Straight Line ◽  
Mathematical Construction

In this chapter we introduce an important concept concerning the study of both discrete and continuous dynamical systems, the concept of phase space or “state space”. It is an abstract mathematical construction with important applications in statistical mechanics, to represent the time evolution of a dynamical system in geometric shape. This space has as many dimensions as the number of variables needed to define the instantaneous state of the system. For instance, the state of a material point moving on a straight line is defined by its position and velocity at each instant, so that the phase space for this system is a plane in which one axis is the position and the other one the velocity. In this case, the phase space is also called “phase plane”. It is later applied in many chapters of the book.

scholarly journals A Contribution to Avalanche Dynamics: A Formal and an Exact Solution of the Voellmy/Perla Two-Parametric Equation of Motion (Abstract only)

1983 ◽  
Vol 4 ◽  
pp. 304
Author(s):  
Bonsak Schieldrop
Keyword(s):  
Exact Solution ◽  
Velocity Profile ◽  
Average Velocity ◽  
Equation Of Motion ◽  
Centrifugal Effect ◽  
Circular Arcs ◽  
Straight Line ◽  
Abrupt Changes ◽  
Friction Parameters ◽  
Straight Lines

The two-parameter equation of motion for snow avalanches proposed by Voellmy in 1955 was later formally derived by Perla in 1979. It has been the object of numerous investigations, mainly to its applications. It has been solved for tracks approximated by straight lines, and this solution has, in some countries, been used extensively with a two-segment approximation. Perla and Cheng programmed such a solution for digital computation by matching an arbitrary number of straight line segments. This solution can also include impact losses due to abrupt changes in the track. In the first part of this paper a formal integration of the Voellmy/Perla equation is carried out for the general case of a track. The averaged values of the different terms are discussed and evaluated as to their relative orders of magnitude. It is shown that the “centrifugal” effect, which is, of course, automatically omitted in the straight-line solution, can be neglected in most cases. As a conclusion it is shown that all avalanche motions governed by the Voellmy/Perla equation will have the same average velocity on all tracks having the same vertical drop H, the same horizontal extension L, and the same set of “friction” parameters, as long as the length S of the track is the same, regardless of the shape of the tracks. The shape will only determine the velocity profile along the track. The second part of the paper shows the exact solution of the equation for the special case of tracks with constant curvature, i.e. circular arcs. If the conclusion of the first part of the paper holds true, this solution can be used to determine the average velocity on other shaped tracks of the same length, etc. It is finally shown that a number of well-known avalanches described in the literature can well be approximated by a circular arc. In these cases even the velocity profile is determined by the exact solution.

Sign in / Sign up

Export Citation Format

Share Document