We use Elie Cartan's method of equivalence to give a complete classification, in terms of differential invariants, of second-order ordinary differential equations admitting Lie groups of fibre-preserving point symmetries. We then apply our results to the determination of all second-order equations which are equivalent, under fibre-preserving transformations, to the free particle equation. In ad...

In this work, some general forms for forced and damped complex Duffing oscillators (FDCDOs), including two different models, which are known as the oscillator (I) (FDCDO (I)) FDCDO (II), investigated by using effective analytical numerical approaches. For approximation, models of FDCDOs reduced to decoupled standard (FDDOs). After that, both ansatz method Krylov–Bogoliubov–Mitropolsky (KBM) app...

We have studied experimentally the inverse system approach, which is used when we want to synchronize two chaotic systems (master – slave). We apply this method to a second order nonlinear circuit, which is described by a Duffing equation. We present two different implementations of the slave circuit with the inverse method and we demonstrate the possibility of synchronization when the two circ...

This paper focuses on the design of robust internal model-based regulators for a class of nonlinear systems and exosystems such that the desired steady state control law fulfills a nonlinear regression formula affine in possible uncertainties. The design methodology builds upon the ideas presented in [9] in the context of linear adaptive output regulation. The design methodology relies upon an ...

Synchronization properties of two identical mutually coupled Duffing oscillators with parametric modulation in one of them are studied. Intermittent lag synchronization is observed in the vicinity of saddle-node bifurcations where the system changes its dynamical state. This phenomenon is seen as intermittent jumps from phase to lag synchronization, during which the chaotic trajectory visits cl...

The problem of nonlinear Hamiltonian oscillators is one of the classical questions in physics. When an analytic solution is not possible, one can resort to obtaining a numerical solution or using perturbation theory around the linear problem. We apply the Fourier series expansion to find approximate solutions to the oscillator position as a function of time as well as the period-amplitude relat...