Global limit cycle bifurcations of a polynomial Euler–Lagrange–Liénard system

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In this paper, using our bifurcation-geometric approach, we study global dynamics and solve the problem of the maximum number and distribution of limit cycles (self-oscillating regimes corresponding to states of dynamical equilibrium) in a planar polynomial mechanical system of the Euler–Lagrange–Liйnard type. Such systems are also used to model electrical, ecological, biomedical and other systems, which greatly facilitates the study of the corresponding real processes and systems with complex internal dynamics. They are used, in particular, in mechanical systems with damping and stiffness. There are a number of examples of technical systems that are described using quadratic damping in second-order dynamical models. In robotics, for example, quadratic damping appears in direct-coupled control and in nonlinear devices, such as variable impedance (resistance) actuators. Variable impedance actuators are of particular interest to collaborative robotics. To study the character and location of singular points in the phase plane of the Euler–Lagrange–Liйnard polynomial system, we use our method the meaning of which is to obtain the simplest (well-known) system by vanishing some parameters (usually, field rotation parameters) of the original system and then to enter sequentially these parameters studying the dynamics of singular points in the phase plane. To study the singular points of the system, we use the classical Poincarй index theorems, as well as our original geometric approach based on the application of the Erugin twoisocline method which is especially effective in the study of infinite singularities. Using the obtained information on the singular points and applying canonical systems with field rotation parameters, as well as using the geometric properties of the spirals filling the internal and external regions of the limit cycles and applying our geometric approach to qualitative analysis, we study limit cycle bifurcations of the system under consideration.

Keywords: Euler–Lagrange–Liénard equation, mechanical system, planar polynomial dynamical system, bifurcation, field rotation parameter, singular point, limit cycle
Citation in English: Gaiko V.A., Savin S.I., Klimchik A.S. Global limit cycle bifurcations of a polynomial Euler–Lagrange–Liénard system // Computer Research and Modeling, 2020, vol. 12, no. 4, pp. 693-705
Citation in English: Gaiko V.A., Savin S.I., Klimchik A.S. Global limit cycle bifurcations of a polynomial Euler–Lagrange–Liénard system // Computer Research and Modeling, 2020, vol. 12, no. 4, pp. 693-705
DOI: 10.20537/2076-7633-2020-12-4-693-705

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