Feigenbaum scenario without parameters

Typically, the period-doubling bifurcations exhibited by nonlinear dissipative systems are observed when varying systems' parameters. In contrast, the period-doubling bifurcations considered in the current research are induced by changing the initial conditions whereas parameter values are fixed. Thus, the studied bifurcations can be classified as the period-doubling bifurcations without parameters. Moreover, we show a cascade of the period-doubling bifurcations without parameters resulting in transition to deterministic chaos. The explored effects are demonstrated by means of numerical modelling on an example of a modified Anishchenko-Astakhov self-oscillator where the ability to exhibit bifurcations without parameters is associated with the properties of a memristor. Finally, we compare the dynamics of the ideal-memristor-based oscillator with the behaviour of a model taking into account the memristor forgetting effect.Comment: 9 pages, 5 figure

Some contributions to the analysis of piecewise linear systems.

This thesis consists of two parts, with contributions to the analysis of dynamical systems in continuous time and in discrete time, respectively. In the first part, we study several models of memristor oscillators of dimension three and four, providing for the first time rigorous mathematical results regarding the rich dynamics of such memristor oscillators, both in the case of piecewise linear models and polynomial models. Thus, for some families of discontinuous 3D piecewise linear memristor oscillators, we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible so to justify the periodic behavior exhibited by such three dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. By using the first-order Melnikov theory, we derive the bifurcation set for a three-parametric family of Bogdanov-Takens systems with symmetry and deformation. As an applications of these results, we study a family of 3D memristor oscillators where the characteristic function of the memristor is a cubic polynomial. In this family we also show the existence of an infinity number of invariant manifolds. Also, we clarify some misconceptions that arise from the numerical simulations of these systems, emphasizing the important role of invariant manifolds in these models. In a similar way than for the 3D case, we study some discontinuous 4D piecewise linear memristor oscillators, and we show that the dynamics in each stratum is topologically equivalent to a continuous 3D piecewise linear dynamical system. Some previous results on bifurcations in such reduced systems, allow us to detect rigorously for the first time a multiple focus-center-cycle bifurcation in a three-parameter space, leading to the appearance of a topological sphere in the original model, completely foliated by stable periodic orbits. In the second part of this thesis, we show that the two-dimensional stroboscopic map defined by a second order system with a relay based control and a linear switching surface is topologically equivalent to a canonical form for discontinuous piecewise linear systems. Studying the main properties of the stroboscopic map defined by such a canonical form, the orbits of period two are completely characterized. At last, we give a conjecture about the occurrence of the big bang bifurcation in the previous map

Forced synchronization of an oscillator with a line of equilibria

The model of a non-autonomous memristor-based oscillator with a line of equilibria is studied. A numerical simulation of the system driven by a periodical force is combined with a theoretical analysis by means of the quasi-harmonic reduction. Both two mechanisms of synchronization are demonstrated: capture of the phase and frequency of oscillations and suppression by an external signal. Classification of undamped oscillations in an autonomous system with a line of equilibria as a special kind of self-sustained oscillations is concluded due to the possibility to observe the effect of frequency-phase locking in the same system in the presence of an external influence. It is established that the occurrence of phase locking in the considered system continuously depends both on parameter values and initial conditions. The simultaneous dependence of synchronization area boundaries on the initial conditions and the parameter values is also shown.Comment: 10 pages, 9 figure

Saddle-Node bifurcations in classical and memristive circuits

This paper addresses a systematic characterization of saddle-node bifurcations in nonlinear electrical and electronic circuits. Our approach is a circuit-theoretic one, meaning that the bifurcation is analyzed in terms of the devices’ characteristics and the graph-theoretic properties of the digraph underlying the circuit. The analysis is based on a reformulation of independent interest of the saddle-node theorem of Sotomayor for semiexplicit index one differential-algebraic equations (DAEs), which define the natural context to set up nonlinear circuit models. The bifurcation is addressed not only for classical circuits, but also for circuits with memristors. The presence of this device systematically leads to nonisolated equilibria, and in this context the saddle-node bifurcation is shown to yield a bifurcation of manifolds of equilibria; in cases with a single memristor, this phenomenon describes the splitting of a line of equilibria into two, with different stability properties