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| In [[mathematics]], more specifically in the study of [[dynamical system]]s and [[differential equation]]s, a '''Liénard equation'''<ref>Liénard, A. (1928) "Etude des oscillations entretenues," ''Revue générale de l'électricité'' '''23''', pp. 901–912 and 946–954.</ref> is a second order differential equation, named after the French physicist [[Alfred-Marie Liénard]].
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| During the development of [[radio]] and [[vacuum tube]] technology, Liénard equations were intensely studied as they can be used to model [[oscillating circuit]]s. Under certain additional assumptions '''Liénard's theorem''' guarantees the uniqueness and existence of a [[limit cycle]] for such a system.
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| ==Definition==
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| 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
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| :<math>{d^2x \over dt^2}+f(x){dx \over dt}+g(x)=0</math>
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| is called the '''Liénard equation'''. | |
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| ==Liénard system==
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| The equation can be transformed into an equivalent two-dimensional [[system of ordinary differential equation]]s. We define
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| :<math>F(x) := \int_0^x f(\xi) d\xi</math>
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| :<math>x_1:= x\,</math>
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| :<math>x_2:={dx \over dt} + F(x)</math>
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| then
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| :<math>
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| \begin{bmatrix}
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| \dot{x}_1 \\
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| \dot{x}_2
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| \end{bmatrix}
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| =
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| \mathbf{h}(x_1, x_2)
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| :=
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| \begin{bmatrix}
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| x_2 - F(x_1) \\
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| -g(x_1)
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| \end{bmatrix}
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| </math>
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| is called a '''Liénard system'''. | |
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| Alternatively, since Liénard equation itself is also an [[autonomous differential equation]], the substitution <math>v = {dx \over dt}</math> leads the Liénard equation to become a [[first order differential equation]]:
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| :<math>v{dv \over dx}+f(x)v+g(x)=0</math> | |
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| which belongs to [[Abel equation of the second kind]].<ref>[http://eqworld.ipmnet.ru/en/solutions/ode/ode0317.pdf Liénard equation] at [[eqworld]].</ref><ref>[http://eqworld.ipmnet.ru/en/solutions/ode/ode0125.pdf Abel equation of the second kind] at [[eqworld]].</ref>
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| ==Example==
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| The [[Van der Pol oscillator]]
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| :<math>{d^2x \over dt^2}-\mu(1-x^2){dx \over dt} +x= 0</math>
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| is a Liénard equation.
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| ==Liénard's theorem==
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| A Liénard system has a unique and [[Stability theory|stable]] [[limit cycle]] surrounding the origin if it satisfies the following additional properties:
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| * ''g''(''x'') > 0 for all ''x'' > 0;
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| * <math>\lim_{x \to \infty} F(x) := \lim_{x \to \infty} \int_0^x f(\xi) d\xi\ = \infty;</math>
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| * ''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''.
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| ==See also==
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| *[[Autonomous differential equation]]
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| *[[Abel equation of the second kind]]
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| ==Footnotes==
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| {{reflist}}
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| ==External links==
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| * {{springer|title=Liénard equation|id=p/l058790}}
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| * {{PlanetMath|title=LienardSystem|urlname=LienardSystem}}
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| {{DEFAULTSORT:Lienard equation}}
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| [[Category:Dynamical systems]]
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| [[Category:Differential equations]]
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| [[Category:Theorems in dynamical systems]]
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My weblog :: real psychic (additional reading)