Complicated Poincaré Half-Maps in a Linear System

1983 ◽  
Vol 38 (10) ◽  
pp. 1107-1113 ◽  
Author(s):  
Bernhard Uehleke ◽  
Otto E. Rössler
Keyword(s):  
Dynamical Systems ◽  
Linear System ◽  
Linear Systems ◽  
Hormonal Regulation ◽  
Topological Structure ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Feedback System ◽  
Loop Feedback ◽  
Simple Behavior

Abstract Poincaré half-maps can be used to characterize the behavior of recurrent dynamical systems. Their usefulness is demonstrated for a linear three-dimensional single-loop feedback system. In this example everything can be calculated analytically. The resulting half-maps are "benign" endomorphic maps with a complicated topological structure. This is surprising since the combination of two such half-maps (yielding an ordinary Poincaré map) always implies simple behavior in a linear system. The method has a direct bearing on the theory of piecewise linear systems -like the well-known Danziger-Elmergreen system of hormonal regulation.

scholarly journals DYNAMICAL SYSTEMS ON THREE MANIFOLDS PART II: THREE-MANIFOLDS, HEEGAARD SPLITTINGS AND THREE-DIMENSIONAL SYSTEMS

2007 ◽  
Vol 17 (06) ◽  
pp. 2085-2095 ◽  
Author(s):  
YI SONG ◽  
STEPHEN P. BANKS
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Systems Theory ◽  
Topological Structure ◽  
Three Dimensional ◽  
Local System ◽  
Operating Point ◽  
Global Behavior ◽  
Heegaard Splittings

The global behavior of nonlinear systems is extremely important in control and systems theory since the usual local theories will only provide information about a system in some neighborhood of an operating point. Away from that point, the system may have totally different behavior and so the theory developed for the local system will be useless for the global one. In this paper we shall consider the analytical and topological structure of systems on two- and three-manifolds and show that it is possible to obtain systems with "arbitrarily strange" behavior, i.e. arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We shall do this by considering Heegaard Splittings of these manifolds and the resulting systems defined on the boundaries.

BIMODAL PIECEWISE LINEAR DYNAMICAL SYSTEMS: REDUCED FORMS

2010 ◽  
Vol 20 (09) ◽  
pp. 2795-2808 ◽  
Author(s):  
JOSEP FERRER ◽  
M. DOLORS MAGRET ◽  
MARTA PEÑA
Keyword(s):  
Dynamical Systems ◽  
Linear Systems ◽  
Piecewise Linear ◽  
Equivalence Classes ◽  
State Variables ◽  
Linear Dynamical Systems ◽  
Piecewise Linear Systems ◽  
Wide Range ◽  
General Systems ◽  
Reduced Forms

Piecewise linear systems constitute a class of nonlinear systems which have recently attracted the interest of researchers because of their interesting properties and the wide range of applications from which they arise. Different authors have used reduced forms when studying these systems, mostly in the case where they are observable. In this work, we focus on bimodal continuous dynamical systems (those consisting of two linear systems on each side of a given hyperplane, having continuous dynamics along that hyperplane) depending on two or three state variables, which are the most common piecewise linear systems found in practice. Reduced forms are obtained for general systems, not necessarily observable. As an application, we calculate the dimension of the equivalence classes.

scholarly journals Hidden attractors from the switching linear systems

2020 ◽  
Vol 66 (5 Sept-Oct) ◽  
pp. 683
Author(s):  
F. Delgado-Aranda ◽  
I. Campos-Cantón ◽  
E. Tristán-Hernández ◽  
P. Salas-Castro
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Stable Equilibrium ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Switching Systems ◽  
Hidden Attractors ◽  
Frobenius Theorem ◽  
Quadratic Nonlinearities ◽  
Number Of Equilibria

Recently, chaotic behavior has been studied in dynamical systems that generates hidden attractors. Most of these systems have quadratic nonlinearities. This paper introduces a new methodology to develop a family of three-dimensional hidden attractors from the switching of linear systems. This methodology allows to obtain strange attractors with only one stable equilibrium, attractors with an infinite number of equilibria or attractors without equilibrium. The main matrix and the augmented matrix of every linear system are considered in Rouché-Frobenius theorem to analyze the equilibrium of the switching systems. Also, a systematic search assisted by a computer is used to find the chaotic behavior. Basic chaotic properties of the attractors are verified by the Lyapunov exponents.

Limit Cycles in Discontinuous Planar Piecewise Linear Systems Separated by a Nonregular Line of Center–Center Type

2021 ◽  
Vol 31 (09) ◽  
pp. 2150136
Author(s):  
Qianqian Zhao ◽  
Cheng Wang ◽  
Jiang Yu
Keyword(s):  
Dynamical Systems ◽  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
The Other ◽  
Natural Phenomena ◽  
Separation Line ◽  
Piecewise Linear Systems

Many natural phenomena can be modeled as discontinuous dynamical systems separated by a nonregular line. The number and distribution of limit cycles in discontinuous linear systems are important topics for research. In this paper, we focus on the limit cycles created by discontinuous planar piecewise linear systems separated by a nonregular line of center–center type, and prove that such systems have at most two limit cycles, which can be reached. Furthermore, the two limit cycles are nested and intersect the separation line at two points or four points, that is, either both intersect the separation line at two points or one intersects the separation line at two points and the other one at four points.

scholarly journals Periodic orbits and invariant cones in three-dimensional piecewise linear systems

2015 ◽  
Vol 35 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Victoriano Carmona ◽  
◽  
Emilio Freire ◽  
Soledad Fernández-García ◽  
Keyword(s):  
Linear Systems ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Piecewise Linear Systems ◽  
Invariant Cones

LIMIT CYCLE BIFURCATION IN 3D CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH TWO ZONES: APPLICATION TO CHUA'S CIRCUIT

2005 ◽  
Vol 15 (10) ◽  
pp. 3153-3164 ◽  
Author(s):  
V. CARMONA ◽  
E. FREIRE ◽  
E. PONCE ◽  
J. ROS ◽  
F. TORRES
Keyword(s):  
Linear Systems ◽  
Limit Cycle ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Periodic Oscillations ◽  
Bounded Linear ◽  
Piecewise Linear Systems ◽  
Chua’S Oscillator ◽  
Center Configuration ◽  
Limit Cycle Bifurcation

The generic case of three-dimensional continuous piecewise linear systems with two zones is analyzed. From a bounded linear center configuration we prove that the periodic orbit which is tangent to the separation plane becomes a limit cycle under generic conditions. Expressions for the amplitude, period and characteristic multipliers of the bifurcating limit cycle are given. The obtained results are applied to the study of the onset of asymmetric periodic oscillations in Chua's oscillator.

Grazing and Chaos in a Periodically Forced, Piecewise Linear System

2004 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Linear System ◽  
Strange Attractor ◽  
Piecewise Linear ◽  
Strange Attractors ◽  
Poincaré Mapping ◽  
Grazing Bifurcation ◽  
Piecewise Linear System ◽  
Poincare Mapping

The criteria for the grazing bifurcation of a periodically forced, piecewise linear system are developed and the initial grazing manifolds are obtained. The grazing flows are illustrated. The mechanism for the fragmentation of the strange attractors caused by the grazing is discussed and the strange attractor fragmentized by grazing is illustrated through the Poincare mapping. This fragmentation phenomenon extensively exists in non-smooth dynamical systems.

Grazing and Chaos in a Periodically Forced, Piecewise Linear System

2005 ◽  
Vol 128 (1) ◽  
pp. 28-34 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Linear System ◽  
Strange Attractor ◽  
Piecewise Linear ◽  
Mathematical Structure ◽  
Strange Attractors ◽  
Grazing Bifurcation ◽  
Piecewise Linear System ◽  
Nonsmooth Dynamical Systems ◽  
Analytic Prediction

The criteria for the grazing bifurcation of a periodically forced, piecewise linear system are developed and the initial grazing manifolds are obtained. The initial grazing manifold is invariant. The grazing flows are illustrated to verify the analytic prediction of grazing. The mechanism of the strange attractors fragmentation caused by the grazing is discussed, and an illustration of the fragmentized strange attractor is given through the Poincaré mapping. This fragmentation phenomenon exists extensively in nonsmooth dynamical systems. The mathematical structure of the fragmentized strange attractors should be further developed.

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