Generalized Liouville Equation for Nonlinear Dynamic Circuits

Journal of Computational Physics, 1996

͵ proj͕⌬⌫[t],1͖ dq 1 dp 1 ϩ и и и ϩ ͵ proj͕⌬⌫[t],N͖ dq N dp N ϭ const, (3a)

Communications in Nonlinear Science and Numerical Simulation, 2005

This paper presents a bifurcation analysis of a simple electronic device, with only one nonlinearity, exhibiting complex dynamics. We focus on the study of a Takens-Bogdanov bifurcation, which is degenerate for some parameter values. In this way, we detect several types of periodic and homoclinic dynamical behaviors.

IAEME PUBLICATION, 2020

Van der pol is a classic example of self-oscillatory system of logistic differential equation. It is second order ordinary differential equation with two dimension phase space. It can be applied to physics, electronics, biology, neurology, sociology and economics so we can say that the model also approaches to chaotic behavior due to its nature of nonlinearity. It was first chaotic oscillator Van der Pol circuit is originally an electronic circuit. We have taken this classical problem in our work because this was the first electronic circuit, which shows results related with our work chaotic behavior of the system of logistic equations, like bifurcation, limit cycles and relaxation oscillations. It is useful to explore the concept of nonlinear dynamics and serves as a starting point for nonlinear and chaotic systems of logistic equations. Here we have also taken the MATLAB output of the oscillator which gave us the attractive results of the Limit cycle which is similar to those attractors that came as the output of the Lorenz attractor in the form of Butterfly (Lorenz 1963).

Nonlinear Analysis: Theory, Methods & Applications, 2015

1999

This paper presents a spectral approach, based on the harmonic-balance technique, for detecting limit-cycle bifurcations in complex nonlinear circuits. The key step of the proposed approach is a method for a simple and effective computation of the Floquet multipliers (FM's) that yield stability and bifurcation conditions. As a case-study, a quite complex system, Chua's circuit, is considered. It is shown that the spectral approach is able to accurately evaluate the most significant bifurcation curves.

A general class of prototype dynamical systems is introduced, which allows to study the generation of complex bifurcation cascades of limit cycles, including bifurcations breaking spontaneously a symmetry of the system, period doubling and homoclinic bifurcations, and transitions to chaos induced by sequences of limit cycle bifurcations. The prototype systems are adaptive, with friction forces f(V(x)) being functionally dependent exclusively on the mechanical potential V(x), characterized in turn by a finite number of local minima. We discuss several low-dimensional systems, with friction forces f(V) which are linear, quadratic or cubic polynomials in the potential V. We point out that the zeros of f(V) regulate both the relative importance of energy uptake and dissipation respectively, serving at the same time as bifurcation parameters, hence allowing for an intuitive interpretation of the overall dynamical behavior. Starting from simple Hopf-and homoclinic bifurcations, complex sequences of limit cycle bifurcations are observed when the energy uptake gains progressively in importance. The term 'prototype dynamical system' is employed for generic, but otherwise reduced systems, allowing to study and to understand a certain relevant phenomenon (like dynamical behavior and/or bifurcation scenario). For this, the dynamical behavior of the system should be dominated by the prime phenomenon of interest, with the system being otherwise simple enough to allow for straightforward numerical and (at least partial) analytic investigations 1–4. Additionally, their dynamical behavior can often be understood in terms of general concepts, such as energy balance, symmetry breaking, etc. Examples of prototype systems are the normal forms of standard bifurcation analysis 5,6 and classical systems, like the Van der Pol oscillator 5 , or the Lorenz model 7 , which have been of central importance for the development of dynamical systems (systems) theory. As an example we consider the Liénard equation, + () + () = , ()  ̈ x f x x g x 0 1 a generic adaptive mechanical system, which includes the Van der Pol oscillator and the Takens-Bogdanov system 8,9. The periodically forced extended Liénard systems with a double-well potential have also been studied by many authors (see e.g. the double-well Duffing oscillator 10–12). In this paper we propose a new class of autonomous Liénard-type systems, which allow to study cascades of limit cycle bifurcations, using a bifurcation parameter controlling directly the balance between energy dissipation and uptake, and hence the underlying physical driving mechanism. Though there are a range of alternative construction methods for dynamical systems in the literature (see e.g. 13–15), they generally involve abstract concepts, such as implicitly defined manifolds, or mathematical tools accessible only to researchers with an in-depth math training. In contrast to these methods, we provide here a mechanistic design procedure, based on the construction of attractors through the interaction of generalized friction forces with potential forces, an intuitive concept especially suitable for interdiscipli-nary investigations (e.g. in modeling cardiovascular systems 16 or for solving optimization problems 17),

2010

Bifurcation theory is the mathematical investigation of changes in the qualitative or topological structure of a studied family. In this paper, we numerically investigate the qualitative behavior of nonlinear RLC circuit excited by sinusoidal voltage source based on the bifurcation analysis. Poincare mapping and bifurcation methods are applied to study both dynamics and qualitative properties of the periodic responses of such oscillator. As numerically illustrated here, a small variation of amplitude or frequency of the driver sinusoidal voltage may involve qualitative changes for witch the system exhibits fold, period doubling and pitchfork bifurcations. In fact, the presence of these kinds of bifurcation necessitates an examination of the role of these singularities in the dynamical behavior of circuit. Particularly, we numerically study the qualitative changes may affect number and stability of the periodic solutions and the shapes of its basins of attraction associated while app...

International Journal of Non-Linear Mechanics, 1985

paper concerns the dynamics of a class of non-linear oscillators of the form: x" + x -.sx'(l -ax'bx'2) = 0. The non-linear term contains two parameters a and b which may be varied to give the Rayleigh and Van der Pal differential equations as special cases. The existence and approximation of limit cycles in this system are investigated using the Poincare-Bendixson theorem and the Lindstedt perturbation method. Analysis of the system at infinity is used to study the global bifurcation through which the limit cycle is created from four saddle-saddle connections between equilibrium points at infinity. Center manifold theory is used to determine the stability of the equilibrium points at infinity. Numerical integration is used to verify the analytical results. It is shown that an arbitrarily small perturbation to the damping term of the Rayleigh equation results in points close to the stable limit cycle escaping to infinity.

Nonlinear Dynamics, 2010

Oscillators control many functions of electronic devices, but are subject to uncontrollable perturbations induced by the environment. As a consequence, the influence of perturbations on oscillators is a question of both theoretical and practical importance. In this paper, a method based on Abelian integrals is applied to determine the emergence of limit cycles from centers, in strongly nonlinear oscillators subject to weak dissipative perturbations. It is shown how Abelian integrals can be used to determine which terms of the perturbation are influent. An upper bound to the number of limit cycles is given as a function of the degree of a polynomial perturbation, and the stability of the emerging limit cycles is discussed. Formulas to determine numerically the exact number of limit cycles, their stability, shape and position are given.

Proceedings of the Edinburgh Mathematical Society, 2000

We consider three classes of polynomial differential equations of the form ẋ = y + establish Pn (x, y), ẏ = x + Qn (x, y), where establish Pn and Qn are homogeneous polynomials of degree n, having a non-Hamiltonian centre at the origin. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centres when we perturb them inside the class of all polynomial differential systems of the above form. A more detailed study is made for the particular cases of degree n = 2 and n = 3.