Homoclinic orbits in a piecewise system and their relation with invariant sets

Medrano-T, Rene O.; Baptista, Murilo S.; Caldas, Iberê L.

doi:10.1016/j.physd.2003.08.002

Cited by 23 publications

(10 citation statements)

References 8 publications

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“…In this work, we have studied a forced chaotic system that is a modified version of the Chua's circuit [8][9][10][11][12] with a very simple forcing mechanism, i.e., a series dc voltage source with the Chua's diode (see Fig. 1(a)) [12].…”

Section: Introductionmentioning

confidence: 99%

Periodicity detection on the parameter-space of a forced Chua’s circuit

et al. 2011

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We report a Periodicity-Detection algorithm, implemented in a LabVIEW routine for realtime data analysis on experimental chaos, to evaluate the periodicity P of experimental time series. The Periodicity-Detector (PD) algorithm was applied to the forced Chua's circuit with the aim to build the Periodicity-parameter-space (P -parameter-space). As results of the P -parameter-space, we could observe very complex dynamical behaviors, as regions of periodic structures, a new sequence of accumulation boundary, and the periodic structures organizing themselves in a period-adding bifurcation cascade. Those results agree with the maximal Lyapunov exponent and the bifurcation diagram analysis, presented in a previous work.

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Section: Introductionmentioning

confidence: 99%

Periodicity detection on the parameter-space of a forced Chua’s circuit

et al. 2011

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“…Each homoclinic bifurcation curve describes a family of homoclinic orbits in the parameter space. 19 Our present work deals with the numerical identification of basic structures formed by the bifurcation curves of the Chua circuit 20 in the parameter space. 15 Then, given a primary Shilnikov homoclinic orbit, the existence of other Shilnikov subsidiary orbits of arbitrary order was demonstrated ͑see Ref.…”

Section: Introductionmentioning

confidence: 99%

Basic structures of the Shilnikov homoclinic bifurcation scenario

Medrano-T

Baptista

Caldas

2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We find numerically small scale basic structures of homoclinic bifurcation curves in the parameter space of the Chua circuit. The distribution of these basic structures in the parameter space and their geometrical properties constitute a complete homoclinic bifurcation scenario of this system. Furthermore, these structures and the scenario are theoretically demonstrated to be generic to a large class of dynamical systems that presents, as the Chua circuit, Shilnikov homoclinic orbits. We classify the complexity of primary and subsidiary homoclinic orbits by their order given by the number of their returning loops. Our results confirm previous predictions of structures of homoclinic bifurcation curves and extend this study to high order primary orbits. Furthermore, we identify accumulations of bifurcation curves of subsidiary homoclinic orbits into bifurcation curves of both primary and subsidiary orbits.

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“…Following full integration method described in [20] tends to be very time consuming and, as a part of an optimization routine, it can diverge even in the case if homoclinic orbit exists. Our problem should be addressed separately for dynamical system cases A and B .…”

Section: Strategic Trajectoriesmentioning

confidence: 99%

Dynamical Tangles in Third-Order Oscillator with Single Jump Function

Petržela

Götthans

Guzan

2014

The Scientific World Journal

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This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity. The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems. The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

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Homoclinic orbits in a piecewise system and their relation with invariant sets

Cited by 23 publications

References 8 publications

Periodicity detection on the parameter-space of a forced Chua’s circuit

Periodicity detection on the parameter-space of a forced Chua’s circuit

Basic structures of the Shilnikov homoclinic bifurcation scenario

Dynamical Tangles in Third-Order Oscillator with Single Jump Function

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