Liénard equation

Liénard equation (Redirected from Liénard's theorem) Vai alla navigazione Vai alla ricerca In matematica, more specifically in the study of dynamical systems and differential equations, a Liénard equation[1] is a second order differential equation, named after the French physicist Alfred-Marie Liénard.

During the development of radio and vacuum tube technology, Liénard equations were intensely studied as they can be used to model oscillating circuits. Under certain additional assumptions Liénard's theorem guarantees the uniqueness and existence of a limit cycle for such a system. A Liénard system with piecewise-linear functions can also contain homoclinic orbits [2].

Contenuti 1 Definizione 2 Liénard system 3 Esempio 4 Liénard's theorem 5 Guarda anche 6 Note a piè di pagina 7 External links Definition Let f and g be two continuously differentiable functions on R, with g an odd function and f an even function. Then the second order ordinary differential equation of the form {stile di visualizzazione {d^{2}x over dt^{2}}+f(X){dx over dt}+g(X)=0} is called the Liénard equation.

Liénard system The equation can be transformed into an equivalent two-dimensional system of ordinary differential equations. We define {stile di visualizzazione F(X):=int _{0}^{X}f(xi )dxi } {stile di visualizzazione x_{1}:=x} {stile di visualizzazione x_{2}:={dx over dt}+F(X)} poi {stile di visualizzazione {inizio{bmatrice}{punto {X}}_{1}\{punto {X}}_{2}fine{bmatrice}}=mathbf {h} (X_{1},X_{2}):={inizio{bmatrice}X_{2}-F(X_{1})\-g(X_{1})fine{bmatrice}}} is called a Liénard system.

In alternativa, since the Liénard equation itself is also an autonomous differential equation, the substitution {displaystyle v={dx over dt}} leads the Liénard equation to become a first order differential equation: {stile di visualizzazione v{dv over dx}+f(X)v+g(X)=0} which belongs to Abel equation of the second kind.[3][4] Example The Van der Pol oscillator {stile di visualizzazione {d^{2}x over dt^{2}}-in (1-x^{2}){dx over dt}+x=0} is a Liénard equation. The solution of a Van der Pol oscillator has a limit cycle. Such cycle has a solution of a Liénard equation with negative {stile di visualizzazione f(X)} at small {stile di visualizzazione |X|} and positive {stile di visualizzazione f(X)} altrimenti. The Van der Pol equation has no exact, analytic solution. Such solution for a limit cycle exists if {stile di visualizzazione f(X)} is a constant piece-wise function.[5] Liénard's theorem A Liénard system has a unique and stable limit cycle surrounding the origin if it satisfies the following additional properties:[6] g(X) > 0 for all x > 0; {displaystyle lim _{xto infty }F(X):=lim _{xto infty }int _{0}^{X}f(xi )dxi =infty ;} F(X) has exactly one positive root at some value p, where F(X) < 0 for 0 < x < p and F(x) > 0 and monotonic for x > p. See also Autonomous differential equation Abel equation of the second kind Biryukov equation Footnotes ^ Liénard, UN. (1928) "Etude des oscillations entretenues," Revue générale de l'électricité 23, pp. 901–912 and 946–954. ^ Phase Curve And VectorField For Piecewise Linear Lienard Systems ^ Liénard equation at eqworld. ^ Abel equation of the second kind at eqworld. ^ Pilipenko A. M., and Biryukov V. N. «Investigation of Modern Numerical Analysis Methods of Self-Oscillatory Circuits Efficiency», Journal of Radio Electronics, No 9, (2013). http://jre.cplire.ru/jre/aug13/9/text-engl.html ^ For a proof, see Perko, Lawrence (1991). Differential Equations and Dynamical Systems (Terza ed.). New York: Springer. pp. 254–257. ISBN 0-387-97443-1. link esterno "Liénard equation", Enciclopedia della matematica, EMS Press, 2001 [1994] LienardSystem at PlanetMath. Categorie: Dynamical systemsDifferential equationsTheorems in dynamical systems