Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

In this tutorial, we discuss self-excited and hidden attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited attractor and a hidden attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden attractors were obtained analytically.Comment: submitted to EP

Experimental Analysis of Emergent Dynamics in Complex Networks of Nonlinear Oscillators

The aim of this thesis is to explore and investigate the emergent dynamics of complex networks through a novel and insightful experimental setup realized as a configurable network of chaotic Chua's circuits. In particular part of our work has been devoted to the implementation and characterization of a "2.0 hardware version" of it, where the interconnection network has improved greatly in its main features. In this way the setup has been fully automatized in providing control on network structure and coupling strength. A large set of experiments has been carried out in networks with proportional coupling and arbitrary topology, showing, emergent dynamics encompassing synchronization, patterns and traveling waves, clusters formation. Also, the case of dynamic coupling has been experimentally addressed. The experimental observations have been compared with theoretical results by carrying out a local stability analysis of networks with static and dynamic links. Here we use the Master Stability approach (MSF) and its extensions to the case where the links are of dynamic nature (Proportional Derivative-MSF). Last part of the work has been devoted to the experimental study of cluster synchronization, stimulated by novel theoretical advances based on group theory and network symmetries. A novel network structure referred as "Multiplexed Network" has been experimentally examined, resulting in a great enhancement in synchronization, for which no theoretical models are yet available

Stability analysis of digital microwave power amplifiers

This work presents a stability analysis of microwave power amplifiers (PAs) driven by binary pulse trains, as in the case of class-S PAs. First, using a simplified digital PA test bench in class-D configuration, different qualitative behaviors are obtained when varying the pulsewidth, including subharmonic and incommensurable oscillations. The mechanisms affecting the stability properties are studied with a harmonic balance-based formulation, by means of pole-zero identification and bifurcation detection. A sufficiently high number of harmonic components must be considered, together with the Krylov decomposition for an efficient computation of the inverse of the Jacobian matrix. It is demonstrated that, when varying the pulsewidth, the distinct pairs of complex-conjugate poles may shift to the right-hand side of the complex plane and, therefore, lead to different kinds of unstable behavior. This phenomenon is related to the dependence of the critical poles on the average value of the input signal. Boundaries of the various types of unstable behavior are traced in the plane defined by the pulse repetition rate and pulsewidth, using bifurcation detection techniques. All the predicted phenomena have been confirmed experimentally. In a second step, the algorithms derived from the simple class-D circuit are transferred to study the stability of a more complex tri-band class-S amplifier. It has been analyzed versus the input bitrate, obtaining a fully stable behavior that has been validated experimentally.This work has been supported by the Spanish Government under contract TEC2014-60283-C3-1-R, the European Regional Development Fund (ERDF/FEDER) and the Parliament of Cantabria (12.JP02.64069)

On Andronov-Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies

The bifurcation of the birth of a closed invariant curve in the two-parameter unfolding of a two-dimensional diffeomorphism with a homoclinic tangency of invariant manifolds of a hyperbolic fixed point of neutral type (i.e. such that the Jacobian at the fixed point equals to 1) is studied. The existence of periodic orbits with multipliers e±iψ (0 < ψ < π) is proved and the first Lyapunov value is computed. It is shown that, generically, the first Lyapunov value is non-zero and its sign coincides with the sign of some separatrix value (i.e. a function of coefficients of the return map near the global piece of the homoclinic orbit)

Oscillation modes in multiresonant oscillator circuits

An in-depth analysis of the oscillation modes in free-running oscillators loaded with multiresonance networks is presented. The analysis illustrates the mechanisms for the generation and stabilization of the various periodic modes and establishes the conditions for existence of a single stable periodic mode in distinct regions of the parameter plane. The mechanisms for the generation and stabilization of quasi-periodic regimes, with two concurrent oscillations, are also analyzed, considering different situations in terms of two relevant poles. The stability analysis of quasi-periodic solutions, derived in terms of admittance functions, can be applied to circuits simulated with harmonic balance, under the assumption of high quality factor resonators. The impact of the transistor biasing on the stability properties of the quasi-periodic regimes has been analyzed, demonstrating that it can be used to isolate the quasi-periodic solution from the periodic ones. The analysis procedures have been applied to a practical oscillator based on two cross-coupled transistors at the two frequencies 900 MHz and 2.5 GHz. The case of two independent oscillations operating in a synchronized regime is also analyzed, as well as its impact on the phase-noise behavior.This work has been funded by the Spanish Government under contract TEC2014-60283-C3-1-R, the European Regional Development Fund (ERDF/FEDER) and the Parliament of Cantabria (12.JP02.64069)

Cone-like Invariant Manifolds for Nonsmooth Systems

This thesis deals with rigorous mathematical techniques for higher-dimensional nonsmooth systems and their applications. The dynamical behaviour of these systems is a nonlocal problem due to the lack of smoothness. Motivated by various examples of nonsmooth systems in applications, we propose to explore the concept of invariant surfaces in the phase space which is separated by a discontinuity hypersurface. For such systems the corresponding Poincaré map can be determined; it turns out that under suitable conditions an invariant cone occurs which is characterized by a fixed point of the Poincaré map. The invariant cone seems to serve in a similar way as a generalisation of the classical center manifold for smooth differential systems. Hence, the stability of the whole system can be reduced to investigate the stability on the two-dimensional surface of the cone. Motivated to study the generation of invariant cones out of smooth systems, a numerical procedure to establish invariant cones and their stability is presented. It has been found that the flat degenerate cone in a smooth system develops under nonsmooth perturbations into a cone-like configuration. Also a simple example is used to explain a paradoxical situation concerning stability. Theoretical results concerning the existence of invariant cones and possible mechanisms responsible for the observed behavior for general three dimensional nonsmooth systems are discussed. These investigations reveal that the system possesses a rich dynamic behavior and new phenomena such as, for instance, the existence of multiple invariant cones for such system. Our approach is developed to include the case when sliding motion takes place on the manifold. Sliding dynamical equations are formulated by using Filippov's method. Existence of invariant cones containing a segment of sliding orbits are given as well as stability on these cones. Different sliding bifurcation scenarios are treated by theoretical analysis and simulation. As an application we have investigated the dynamics of an automotive brake system model under the excitation of dry friction force which has served as a motivating example to develop our concepts. This model belongs to the class of nonsmooth systems of Filippov type which is investigated from direct crossing and a sliding motion point of view. Existence of invariant cones and different types of bifurcation phenomena such as sliding periodic doubling and multiple periodic orbits are observed. Finally, extensions to nonlinear perturbations of nonsmooth linear systems have been obtained by using the nonsmooth linear system as basic system. If the basic system possesses an attractive invariant cone without sliding motion, we have shown that locally the Poincaré map contains the necessary information with regard to attractivity of the invariant cone. The existence of a generalized center manifold reduction of nonlinear system has been proven by using Hadamard graph transformation approach. A class of nonlinear systems having a cone-like invariant "manifold" is presented to illustrate the center manifold reduction and associated bifurcation. The scientific contributions of parts of this thesis are presented in [32,39,66]

Second order mem-circuits

This paper presents a comprehensive taxonomy of so-called second order memory devices, which include charge-controlled memcapacitors and flux-controlled meminductors, among other novel circuit elements. These devices, which are classified according to their differential and state orders, are necessary to get a complete extension of the family of classical nonlinear circuit elements (resistors, capacitors, inductors) for all possible controlling variables. Using a fully nonlinear formalism, we obtain nondegeneracy conditions for a broad class of second order mem-circuits. This class of circuits is expected to yield a rich dynamic behavior; in this regard we explore certain bifurcation phenomena exhibited by a family of circuits including a charge-controlled memcapacitor and a flux-controlled meminductor, providing some directions for future research

Hybrid analysis of nonlinear circuits: DAE models with indices zero and one

We extend in this paper some previous results concerning the differential-algebraic index of hybrid models of electrical and electronic circuits. Specifically, we present a comprehensive index characterization which holds without passivity requirements, in contrast to previous approaches, and which applies to nonlinear circuits composed of uncoupled, one-port devices. The index conditions, which are stated in terms of the forest structure of certain digraph minors, do not depend on the specific tree chosen in the formulation of the hybrid equations. Additionally, we show how to include memristors in hybrid circuit models; in this direction, we extend the index analysis to circuits including active memristors, which have been recently used in the design of nonlinear oscillators and chaotic circuits. We also discuss the extension of these results to circuits with controlled sources, making our framework of interest in the analysis of circuits with transistors, amplifiers, and other multiterminal devices