IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences
The analysis of the Takens-Bogdanov bifurcation of the equilibrium at the origin in the Chua's eq... more The analysis of the Takens-Bogdanov bifurcation of the equilibrium at the origin in the Chua's equation with a cubic nonlinearity is carried out. The local analysis provides, in first approximation, different bifurcation sets, where the presence of several dynamical behaviours (including periodic, homoclinic and heteroclinic orbits) is predicted. The local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Takens-Bogdanov bifurcation shows the presence of a degenerate (codimension-three) situation, which is analyzed in both homoclinic and heteroclinic cases.
A METHOD FOR HOMOCLINIC AND HETEROCLINIC CONTINU-ATION IN TWO AND THREE DIMENSIONS AJ RODRfGUEZ-L... more A METHOD FOR HOMOCLINIC AND HETEROCLINIC CONTINU-ATION IN TWO AND THREE DIMENSIONS AJ RODRfGUEZ-LUIS, E. FREIRE and E. PONCE Department of Applied Mathematics (University of Sevilla) Escuela Ttcnica Superior de Ingenieros Industriales Avda. Aetna ...
The bifurcation of limit cycles in single-input single-output control systems with saturation is ... more The bifurcation of limit cycles in single-input single-output control systems with saturation is considered. Under some non-degeneracy conditions, a theorem characterizing such bifurcation is stated for the cases of dimension two and three. In terms of the deviation from the critical value of the bifurcation parameter, expressions in form of power series for the period, amplitude and the characteristic multipliers of the bifurcating limit cycle are obtained. These results are similar to the Hopf bifurcation theorem for differentiable systems, but they show some differences coming from the non-smooth character of the saturation characteristic.
Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis
We analyze simplified normal forms for equilibria of vector fields with zero linear part in the p... more We analyze simplified normal forms for equilibria of vector fields with zero linear part in the presence of a symmetry. We perform both nearly identical transformations as well as reparametrizations of the time to determine the structure of the normal form. A degenerate case, in which a normal form coefficient vanishes, is also included.
For discontinuous switched linear systems, even when they are built by composing stable systems, ... more For discontinuous switched linear systems, even when they are built by composing stable systems, examples of unstable systems are known. Here, three-dimensional homogeneous continuous piecewise linear systems composed by two linear systems sharing a boundary plane are considered. If the two linearization matrices are non-singular, then the only equilibrium point is at the origin, which is in the separation plane of the linear regions. For the important case where both matrices have complex eigenvalues and their whole spectra are in the open-left half of the complex plane, the possible counter-intuitive instability of the origin is analytically proved. Thus, in this paper, examples of unstable continuous switched linear system with just two stable components are shown. Copyright c 2005 IFAC.
ABSTRACT In this Letter we consider a three parameter unfolding of a linear degeneracy correspond... more ABSTRACT In this Letter we consider a three parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue of an equilibrium point. Using blow-up techniques we obtain a system where an exact homoclinic connection is determined. The numerical continuation of this global connection shows that it exhibits three different kinds of codimension-two degeneracies. Finally, these same codimension-two homoclinic bifurcations are detected in the Rössler system, ensuring in this way the existence of chaotic dynamics.
IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences
The analysis of the Takens-Bogdanov bifurcation of the equilibrium at the origin in the Chua's eq... more The analysis of the Takens-Bogdanov bifurcation of the equilibrium at the origin in the Chua's equation with a cubic nonlinearity is carried out. The local analysis provides, in first approximation, different bifurcation sets, where the presence of several dynamical behaviours (including periodic, homoclinic and heteroclinic orbits) is predicted. The local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Takens-Bogdanov bifurcation shows the presence of a degenerate (codimension-three) situation, which is analyzed in both homoclinic and heteroclinic cases.
A METHOD FOR HOMOCLINIC AND HETEROCLINIC CONTINU-ATION IN TWO AND THREE DIMENSIONS AJ RODRfGUEZ-L... more A METHOD FOR HOMOCLINIC AND HETEROCLINIC CONTINU-ATION IN TWO AND THREE DIMENSIONS AJ RODRfGUEZ-LUIS, E. FREIRE and E. PONCE Department of Applied Mathematics (University of Sevilla) Escuela Ttcnica Superior de Ingenieros Industriales Avda. Aetna ...
The bifurcation of limit cycles in single-input single-output control systems with saturation is ... more The bifurcation of limit cycles in single-input single-output control systems with saturation is considered. Under some non-degeneracy conditions, a theorem characterizing such bifurcation is stated for the cases of dimension two and three. In terms of the deviation from the critical value of the bifurcation parameter, expressions in form of power series for the period, amplitude and the characteristic multipliers of the bifurcating limit cycle are obtained. These results are similar to the Hopf bifurcation theorem for differentiable systems, but they show some differences coming from the non-smooth character of the saturation characteristic.
Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis
We analyze simplified normal forms for equilibria of vector fields with zero linear part in the p... more We analyze simplified normal forms for equilibria of vector fields with zero linear part in the presence of a symmetry. We perform both nearly identical transformations as well as reparametrizations of the time to determine the structure of the normal form. A degenerate case, in which a normal form coefficient vanishes, is also included.
For discontinuous switched linear systems, even when they are built by composing stable systems, ... more For discontinuous switched linear systems, even when they are built by composing stable systems, examples of unstable systems are known. Here, three-dimensional homogeneous continuous piecewise linear systems composed by two linear systems sharing a boundary plane are considered. If the two linearization matrices are non-singular, then the only equilibrium point is at the origin, which is in the separation plane of the linear regions. For the important case where both matrices have complex eigenvalues and their whole spectra are in the open-left half of the complex plane, the possible counter-intuitive instability of the origin is analytically proved. Thus, in this paper, examples of unstable continuous switched linear system with just two stable components are shown. Copyright c 2005 IFAC.
ABSTRACT In this Letter we consider a three parameter unfolding of a linear degeneracy correspond... more ABSTRACT In this Letter we consider a three parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue of an equilibrium point. Using blow-up techniques we obtain a system where an exact homoclinic connection is determined. The numerical continuation of this global connection shows that it exhibits three different kinds of codimension-two degeneracies. Finally, these same codimension-two homoclinic bifurcations are detected in the Rössler system, ensuring in this way the existence of chaotic dynamics.
Uploads
Papers by Emilio Freire