Multistability, transient chaos and hyperchaos, synchronization, and chimera states in wireless magnetically coupled VDPCL oscillators

Abstract

Systems in nature are for the most part rich in connection, interaction, and communication in various ways that are generally complex. Synchronization is one of the main phenomenons associated with couple systems. Synchronization may be understood as a process where several systems adjust time scales of their oscillations or a given property of their motion over time due to their interactions. In compact designs, unconnected circuits may interact through the magnetic field. This interaction can be avoided or exploited in some cases. This work investigated the dynamics and the synchronization study of a set of magnetically coupled van der Pol oscillators coupled to a linear circuit (the VDPCL oscillator) in wireless interaction. The coupling coefficient between the coupled but unconnected oscillators is used as the only control parameter allowing investigating the dynamics and the synchronization of the oscillators. We first derive a mathematical model of the ring of the magnetically coupled oscillators. After this, some basic properties of the model are derived. The dynamical study of the model is carried out in the case of two and three sub-circuits including the phase synchronization, by using the well-known classical tools, that is, the bifurcation diagrams, the Lyapunov exponents, time series, and phase portraits. It is found that the system displays a rich catalog of behaviors such as regular, chaotic, and hyperchaotic (for a weak coupling) behaviors in terms of the control parameter. Moreover, it is found that the systems exhibit the coexistence of many attractors. The system is shown to exhibit global and partial synchronization (chimera state) in the case of three sub-circuits. Transient chaos/hyperchaos are also highlighted in the two sub-circuits cases. PSPICE based simulations and real laboratory measurements are included and are consistent with the theoretical/numerical results.