An oscillatory system can have opposite senses of rotation, clockwise or anticlockwise. We present a general mathematical description how to obtain counter-rotating oscillators from the definition of a dynamical system. A type of mixed synchronization emerges in counter-rotating oscillators under diffusive scalar coupling when complete synchronization and antisynchronization coexist in different state variables. We present numerical examples of limit cycle van der Pol oscillator and, chaotic Rössler and Lorenz systems. Stability conditions of mixed synchronization are analytically obtained for both Rössler and Lorenz systems. Experimental evidences of counter-rotating limit cycle and chaotic oscillators and, mixed synchronization are given in electronic circuits.PACS numbers: 05.45.Xt, 05.45.GgCounter-rotating vortices coexist in a fluid medium, atmosphere and ocean, which destroy each other when they collide and reborn if the interaction is withdrawn. Attempts are found in literature to explain such spatiotemporal behaviors and their instabilities. On the other hand, counter-rotating vortices are created in physical systems, plasma flow and Bose-Einstein condensates, for useful purposes. The trajectory of nonlinear dynamical systems too shows opposite senses of rotation, clockwise and anticlockwise direction. A question naturally arises how counter-rotating vortices or oscillators are created and interact with each other. These issues are not well understood. We make an attempt to address the questions here, however, restricting our interest to the simpler cases of nonlinear dynamical systems. In studies of nonlinear oscillators, all oscillators are assumed to rotate in same direction, either clockwise or anticlockwise. These corotating oscillators when interact through coupling emerge into a synchronization regime where all the state variables follow one and unique correlation rule, complete synchronization or antisynchronization. In contrast, counter-rotating oscillators under diffusive coupling show emergence of an uncommon type of synchronization defined as mixed synchronization. In this synchronization regime, some of the state variables emerge into complete synchronization while others are in antisynchronization. We try to understand here the underlying principle how counter-rotating oscillators are created from a known dynamical system and then derive the stability condition of mixed synchronization. We construct counter-rotating oscillators in electronic circuits to evidence mixed synchronization.

We report the existence of diverse routes of transition from amplitude death to oscillation death in three different diffusively coupled systems, which are perturbed by a symmetry breaking repulsive coupling link. For limit-cycle systems the transition is through a pitchfork bifurcation, as has been noted before, but in chaotic systems it can be through a saddle-node or a transcritical bifurcation depending on the nature of the underlying dynamics of the individual systems. The diversity of the routes and their dependence on the complex dynamics of the coupled systems not only broadens our understanding of this important phenomenon but can lead to potentially new practical applications.

We report a design of delay coupling for lag synchronization in two unidirectionally coupled chaotic oscillators. A delay term is introduced in the definition of the coupling to target any desired lag between the driver and the response. The stability of the lag synchronization is ensured by using the Hurwitz matrix stability. We are able to scale up or down the size of a driver attractor at a response system in presence of a lag. This allows compensating the attenuation of the amplitude of a signal during transmission through a delay line. The delay coupling is illustrated with numerical examples of 3D systems, the Hindmarsh-Rose neuron model, the Rössler system, a Sprott system, and a 4D system. We implemented the coupling in electronic circuit to realize any desired lag synchronization in chaotic oscillators and scaling of attractors.

We report a transition from homogeneous steady state to inhomogeneous steady state in coupled oscillators, both limit cycle and chaotic, under cyclic coupling and diffusive coupling as well when an asymmetry is introduced in terms of a negative parameter mismatch. Such a transition appears in limit cycle systems via pitchfork bifurcation as usual. Especially, when we focus on chaotic systems, the transition follows a transcritical bifurcation for cyclic coupling while it is a pitchfork bifurcation for the conventional diffusive coupling. We use the paradigmatic Van der Pol oscillator as the limit cycle system and a Sprott system as a chaotic system. We verified our results analytically for cyclic coupling and numerically check all results including diffusive coupling for both the limit cycle and chaotic systems.