TY - JOUR A1 - Medrano-T., R. O. A1 - Baptista, Murilo da Silva A1 - Caldas, Ibere Luiz T1 - Shilnikov homoclinic orbit bifurcations in the Chua’s circuit JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We analytically describe the complex scenario of homoclinic bifurcations in the Chua’s circuit. We obtain a general scaling law that gives the ratio between bifurcation parameters of different nearby homoclinic orbits. As an application of this theoretical approach, we estimate the number of higher order subsidiary homoclinic orbits that appear between two consecutive lower order subsidiary orbits. Our analytical finds might be valid for a large class of dynamical systems and are numerically confirmed in the parameter space of the Chua’s circuit. Shilnikov homoclinic orbits are trajectories that depart from a fixed saddle-focus point, with specific eigenvalues, and return to it after an infinite amount of time (that is also true to time reversal evolution). That results in an orbit that is unstable and has an infinite period. These two main characteristics contribute in the hardness for its observation in a dynamical system as well as in nature. However, its presence reveals fundamental characteristics of the system involved, as the existence of unstable periodic orbits embedded in a chaotic set. Once the unstable periodic orbits give invariants quantities of this set,1 the Shilnikov homoclinic orbits are also related to the characteristics of the chaotic set. Their connection with the fundamental dynamical properties is verified in a wide variety of systems. A series of numerical and experimental investigations reveal how Shilnikov homoclinic orbits, in the vicinity of a chaotic attractor, determine its dynamical and topological properties.4 Thus, the Shilnikov orbits are related to the returning time of the trajectory of a CO2 laser,5 also to the topology of a glow-discharge system.6 Moreover, some class of spiking neurons are modeled by chaos governed by such orbits,7,8 and their presence are connected to the intermittence present in rabbit arteries.9 These orbits are shown to be behind the mechanism of noise-induced phenomena,10 and they are also responsible for the dynamics of an electrochemical oscillator.11 In this work, we contribute to the understanding of how Shilnikov homoclinic orbits appear on the parameter space of systems as the ones above mentioned, by showing that these orbits are not only distributed following an universal rule but also exist for large parameter variations. We then confirm our previsions in the Chua’s circuit system Y1 - 2006 U6 - https://doi.org/10.1063/1.2401060 SN - 1054-1500 VL - 16 IS - 4 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Pereira, Tiago A1 - Baptista, Murilo da Silva A1 - Reyes, Marcelo B. A1 - Caldas, Ibere Luiz A1 - Sartorelli, José Carlos A1 - Kurths, Jürgen T1 - A scenario for torus T-2 destruction via a global bifurcation N2 - We show a scenario of a two-frequeney torus breakdown, in which a global bifurcation occurs due to the collision of a quasi-periodic torus T-2 with saddle points, creating a heteroclinic saddle connection. We analyze the geometry of this torus-saddle collision by showing the local dynamics and the invariant manifolds (global dynamics) of the saddle points. Moreover, we present detailed evidences of a heteroclinic saddle-focus orbit responsible for the type- if intermittency induced by this global bifurcation. We also characterize this transition to chaos by measuring the Lyapunov exponents and the scaling laws. Y1 - 2009 UR - http://www.sciencedirect.com/science/journal/09600779 U6 - https://doi.org/10.1016/j.chaos.2007.06.115 SN - 0960-0779 ER - TY - JOUR A1 - Pereira, Tiago A1 - Baptista, Murilo da Silva A1 - Reyes, Marcelo Bussotti A1 - Caldas, Ibere Luiz A1 - Sartorelli, José Carlos A1 - Kurths, Jürgen T1 - Global bifurcation destroying the experimental torus T-2 N2 - We show experimentally the scenario of a two-frequency torus T-2 breakdown, in which a global bifurcation occurs due to the collision of a torus with an unstable periodic orbit, creating a heteroclinic saddle connection, followed by an intermittent behavior Y1 - 2006 UR - http://pre.aps.org/ U6 - https://doi.org/10.1103/Physreve.73.017201 ER - TY - JOUR A1 - Baptista, Murilo da Silva A1 - Pereira, Tiago A1 - Sartorelli, J. C. A1 - Caldas, Ibere Luiz A1 - Kurths, Jürgen T1 - Non-transitive maps in phase synchronization N2 - Concepts from Ergodic Theory are used to describe the existence of special non-transitive maps in attractors of phase synchronous chaotic oscillators. In particular, it is shown that, for a class of phase-coherent oscillators, these special maps imply phase synchronization. We illustrate these ideas in the sinusoidally forced Chua's circuit and two coupled Rossler oscillators. Furthermore, these results are extended to other coupled chaotic systems. In addition, a phase for a chaotic attractor is defined from the tangent vector of the flow. Finally, it is discussed how these maps can be used for the real-time detection of phase synchronization in experimental systems. (c) 2005 Elsevier B.V. All rights reserved Y1 - 2005 ER -