Invariant manifolds of periodic orbits for piecewise linear three-dimensional systems

2004 ◽  
Vol 69 (1) ◽  
pp. 71-91 ◽  
Author(s):  
V. Carmona
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Piecewise Linear ◽  
Three Dimensional

On Discontinuous Piecewise Linear Models for Memristor Oscillators

2017 ◽  
Vol 27 (06) ◽  
pp. 1730022 ◽  
Author(s):  
Andrés Amador ◽  
Emilio Freire ◽  
Enrique Ponce ◽  
Javier Ros
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Linear Models ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Infinite Family ◽  
Closed Surfaces ◽  
The Rich ◽  
Piecewise Linear Models

In this paper, we provide for the first time rigorous mathematical results regarding the rich dynamics of piecewise linear memristor oscillators. In particular, for each nonlinear oscillator given in [Itoh & Chua, 2008], we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible to justify the periodic behavior exhibited by three-dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. The analysis developed not only confirms the numerical results contained in previous works [Messias et al., 2010; Scarabello & Messias, 2014] but also goes much further by showing the existence of closed surfaces in the state space which are foliated by periodic orbits. The important role of initial conditions that justify the infinite number of periodic orbits exhibited by these models, is stressed. The possibility of unsuspected bistable regimes under specific configurations of parameters is also emphasized.

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

Existence of Homoclinic Cycles and Periodic Orbits in a Class of Three-Dimensional Piecewise Affine Systems

2018 ◽  
Vol 28 (02) ◽  
pp. 1850024 ◽  
Author(s):  
Lei Wang ◽  
Xiao-Song Yang
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Spatial Location ◽  
Homoclinic Bifurcation ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine

For a class of three-dimensional piecewise affine systems, this paper focuses on the existence of homoclinic cycles and the phenomena of homoclinic bifurcation leading to periodic orbits. Based on the spatial location relation between the invariant manifolds of subsystems and the switching manifold, a concise necessary and sufficient condition for the existence of homoclinic cycles is obtained. Then the homoclinic bifurcation is studied and the sufficient conditions for the birth of a periodic orbit are obtained. Furthermore, the sufficient conditions are obtained for the periodic orbit to be a sink, a source or a saddle. As illustrations, several concrete examples are presented.

HOPF BIFURCATION FROM LINES OF EQUILIBRIA WITHOUT PARAMETERS IN MEMRISTOR OSCILLATORS

2010 ◽  
Vol 20 (02) ◽  
pp. 437-450 ◽  
Author(s):  
MARCELO MESSIAS ◽  
CRISTIANE NESPOLI ◽  
VANESSA A. BOTTA
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Fixed Set ◽  
Electronic Element ◽  
Stable Periodic ◽  
Parameter Values

The memristor is supposed to be the fourth fundamental electronic element in addition to the well-known resistor, inductor and capacitor. Named as a contraction for memory resistor, its theoretical existence was postulated in 1971 by L. O. Chua, based on symmetrical and logical properties observed in some electronic circuits. On the other hand its physical realization was announced only recently in a paper published on May 2008 issue of Nature by a research team from Hewlett–Packard Company. In this work, we present the bifurcation analysis of two memristor oscillators mathematical models, given by three-dimensional five-parameter piecewise-linear and cubic systems of ordinary differential equations. We show that depending on the parameter values, the systems may present the coexistence of both infinitely many stable periodic orbits and stable equilibrium points. The periodic orbits arise from the change in local stability of equilibrium points on a line of equilibria, for a fixed set of parameter values. This phenomenon is a kind of Hopf bifurcation without parameters. We have numerical evidences that such stable periodic orbits form an invariant surface, which is an attractor of the systems solutions. The results obtained imply that even for a fixed set of parameters the two systems studied may or may not present oscillations, depending on the initial condition considered in the phase space. Moreover, when they exist, the amplitude of the oscillations also depends on the initial conditions.

ON PERIODIC ORBITS OF 3D SYMMETRIC PIECEWISE LINEAR SYSTEMS WITH REAL TRIPLE EIGENVALUES

2009 ◽  
Vol 19 (07) ◽  
pp. 2391-2399 ◽  
Author(s):  
E. PONCE ◽  
J. ROS
Keyword(s):  
Linear Systems ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Global Bifurcation ◽  
Nonlinear Behavior ◽  
High Gain ◽  
Approximate Methods ◽  
Saturated Control ◽  
Piecewise Linear Systems

The counter-intuitive appearance of stable periodic orbits in three-dimensional piecewise linear systems has been recently reported for stable saturated control systems with high gain and real triple eigenvalues [Moreno & Suárez 2004]. In this letter using several mathematical tools, the reported sudden phenomenon is explained, and the analysis is completed by providing a global bifurcation diagram of the symmetric periodic orbits. Approximate methods are used only to illustrate the nonlinear behavior with respect to the bifurcation parameter. Analytical methods are employed to rigorously prove the main result. The employed techniques are useful not only for the family studied but also for generic three-dimensional symmetric piecewise linear systems.

ON PERIODIC ORBITS AND HOMOCLINIC BIFURCATIONS IN CHUA’S CIRCUIT WITH A SMOOTH NONLINEARITY

1993 ◽  
Vol 03 (02) ◽  
pp. 363-384 ◽  
Author(s):  
ALEXANDER I. KHIBNIK ◽  
DIRK ROOSE ◽  
LEON O. CHUA
Keyword(s):  
Periodic Orbits ◽  
Bifurcation Theory ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Homoclinic Bifurcation ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Homoclinic Bifurcations ◽  
Cubic Polynomial

We present the bifurcation analysis of Chua’s circuit equations with a smooth nonlinearity, described by a cubic polynomial. Our study focuses on phenomena that can be observed directly in the numerical simulation of the model, and on phenomena which are revealed by a more elaborate analysis based on continuation techniques and bifurcation theory. We emphasize how a combination of these approaches actually works in practice. We compare the dynamics of Chua’s circuit equations with piecewise-linear and with smooth nonlinearity. The dynamics of these two variants are similar, but we also present some differences. We conjecture that this similarity is due to the central role of homoclinicity in this model. We describe different ways in which the type of a homoclinic bifurcation influences the behavior of branches of periodic orbits. We present an overview of codimension 1 bifurcation diagrams for principal periodic orbits near homoclinicity for three-dimensional systems, both in the generic case and in the case of odd symmetry. Most of these diagrams actually occurs in the model. We found several homoclinic bifurcations of codimension 2, related to the so called resonant conditions. We study one of these bifurcations, a double neutral saddle loop.

On the Number of Invariant Cones and Existence of Periodic Orbits in 3-dim Discontinuous Piecewise Linear Systems

2016 ◽  
Vol 26 (03) ◽  
pp. 1650043 ◽  
Author(s):  
Song-Mei Huan ◽  
Xiao-Song Yang
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Piecewise Linear ◽  
Spatial Relationship ◽  
Quadratic Equation ◽  
Linear Dynamical Systems ◽  
Real Roots ◽  
Invariant Cones ◽  
Straight Lines ◽  
Discontinuous Piecewise Linear Systems

For a family of discontinuous 3-dim homogeneous piecewise linear dynamical systems with two zones, we investigate the number of invariant cones and the existence of periodic orbits as a spatial relationship between the invariant manifolds of the subsystem changes. By studying the number of real roots of a quadratic equation induced by slopes of half straight lines starting from the origin in required domain, we obtain complete results on the number and stability of invariant cones. Especially, we prove that the maximum number of invariant cones is two, and obtain complete parameter regions on which there exist one or two invariant cones, on which one or two fake cones (corresponding to real roots of the quadratic equation that are not in the required domain) appear and on which an invariant cone will be foliated by periodic orbits.

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