Difference between revisions of "Van der Pol equation"
(Importing text file) |
m (fixing superscript) |
||
(2 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
+ | <!-- | ||
+ | v0960301.png | ||
+ | $#A+1 = 22 n = 0 | ||
+ | $#C+1 = 22 : ~/encyclopedia/old_files/data/V096/V.0906030 van der Pol equation | ||
+ | Automatically converted into TeX, above some diagnostics. | ||
+ | Please remove this comment and the {{TEX|auto}} line below, | ||
+ | if TeX found to be correct. | ||
+ | --> | ||
+ | |||
+ | {{TEX|auto}} | ||
+ | {{TEX|done}} | ||
+ | |||
The non-linear second-order ordinary differential equation | The non-linear second-order ordinary differential equation | ||
− | + | $$ \tag{1 } | |
+ | \ddot{x} - \mu ( 1 - x ^ {2} ) | ||
+ | \dot{x} + x = 0,\ \ | ||
+ | \mu = \textrm{ const } > 0,\ \ | ||
+ | \dot{x} ( t) \equiv { | ||
+ | \frac{dx}{dt} | ||
+ | } , | ||
+ | $$ | ||
which is an important special case of the [[Liénard equation|Liénard equation]]. Van der Pol's equation describes the auto-oscillations (cf. [[Auto-oscillation|Auto-oscillation]]) of one of the simplest oscillating systems (the van der Pol oscillator). In particular, equation (1) serves — after making several simplifying assumptions — as a mathematical model of a generator on a triode for a tube with a cubic characteristic. The character of the solutions of equation (1) was first studied in detail by B. van der Pol . | which is an important special case of the [[Liénard equation|Liénard equation]]. Van der Pol's equation describes the auto-oscillations (cf. [[Auto-oscillation|Auto-oscillation]]) of one of the simplest oscillating systems (the van der Pol oscillator). In particular, equation (1) serves — after making several simplifying assumptions — as a mathematical model of a generator on a triode for a tube with a cubic characteristic. The character of the solutions of equation (1) was first studied in detail by B. van der Pol . | ||
− | Equation (1) is equivalent to the following system of two equations in two phase variables | + | Equation (1) is equivalent to the following system of two equations in two phase variables $ x, v $: |
− | + | $$ \tag{2 } | |
+ | \dot{x} = v,\ \ | ||
+ | \dot{v} = - x + \mu ( 1 - x ^ {2} ) v. | ||
+ | $$ | ||
− | It is sometimes convenient to replace the variable | + | It is sometimes convenient to replace the variable $ x $ |
+ | by the variable $ z( t) = {\int _ {0} ^ {t} } x ( \tau ) d \tau $; | ||
+ | equation (1) then becomes | ||
− | + | $$ | |
+ | \ddot{z} - \mu \left ( | ||
+ | \dot{z} - | ||
+ | \frac{\dot{z} ^ {3} }{3} | ||
− | + | \right ) + z = 0, | |
+ | $$ | ||
− | + | which is a special case of the [[Rayleigh equation|Rayleigh equation]]. If, together with $ x $, | |
+ | one also considers the variable $ y = - x + ( x ^ {3} /3) + ( \dot{x} / \mu ) $, | ||
+ | introduces a new time $ \tau = t / \mu $ | ||
+ | and puts $ \epsilon = \mu ^ {- 2} $, | ||
+ | one obtains the system | ||
− | + | $$ \tag{3 } | |
+ | \epsilon x ^ \prime = \ | ||
+ | y - x + | ||
+ | \frac{x ^ {3} }{3} | ||
+ | ,\ \ | ||
+ | y ^ \prime = - x,\ \ | ||
+ | {} ^ \prime = | ||
+ | \frac{d}{d \tau } | ||
+ | , | ||
+ | $$ | ||
− | For | + | instead of equation (1). For any $ \mu > 0 $ |
+ | there exists a unique stable limit cycle in the phase plane of the system (2) to which all other trajectories (except for the equilibrium position at the coordinate origin) converge as $ t \rightarrow \infty $; | ||
+ | this limit cycle describes the oscillations of the van der Pol oscillator [[#References|[2]]], [[#References|[3]]], [[#References|[4]]]. | ||
+ | |||
+ | For small $ \mu $ | ||
+ | the auto-oscillations of the oscillator (1) are close to simple harmonic oscillations (cf. [[Non-linear oscillations|Non-linear oscillations]]) with period $ 2 \pi $ | ||
+ | and specified amplitude. In order to calculate the oscillation process more accurately, asymptotic methods are employed. As $ \mu $ | ||
+ | increases, the auto-oscillations of the oscillator (1) deviate more and more from harmonic oscillations. If $ \mu $ | ||
+ | is large, equation (1) describes [[Relaxation oscillation|relaxation oscillation]] with period $ 1.614 \mu $ (to a first approximation). More accurate asymptotic expansions of magnitudes characterizing relaxation oscillations [[#References|[5]]] are known: The study of these oscillations is equivalent to the study of the solutions of the system (3) with a small coefficient $ \epsilon $ | ||
+ | in front of the derivative [[#References|[6]]]. | ||
The equation | The equation | ||
− | + | $$ | |
+ | \ddot{x} - \mu ( 1 - x ^ {2} ) | ||
+ | \dot{x} + x = E _ {0} + E \sin \omega t | ||
+ | $$ | ||
describes the behaviour of the van der Pol oscillator when acted upon by a periodic external disturbance. The most important in this context is the study of frequency capture (the existence of periodic oscillations), beats (the possibility of almost-periodic oscillations) and chaotic behaviour [[#References|[2]]], [[#References|[4]]]. | describes the behaviour of the van der Pol oscillator when acted upon by a periodic external disturbance. The most important in this context is the study of frequency capture (the existence of periodic oscillations), beats (the possibility of almost-periodic oscillations) and chaotic behaviour [[#References|[2]]], [[#References|[4]]]. | ||
Line 29: | Line 82: | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> B. van der Pol, "On oscillation hysteresis in a triode generator with two degrees of freedom" ''Philos. Mag. (6)'' , '''43''' (1922) pp. 700–719</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> B. van der Pol, ''Philos. Mag. (7)'' , '''2''' (1926) pp. 978–992</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lefschetz, "Differential equations: geometric theory" , Interscience (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.J. Stoker, "Nonlinear vibrations in mechanical and electrical systems" , Interscience (1950)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.A. Dorodnitsyn, "Asymptotic solution of the van der Pol equation" ''Priklad. Mat. Mekh.'' , '''11''' (1947) pp. 313–328 (In Russian) (English abstract)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> E.F. Mishchenko, N.Kh. Rozov, "Differential equations with small parameters and relaxation oscillations" , Plenum (1980) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1a]</TD> <TD valign="top"> B. van der Pol, "On oscillation hysteresis in a triode generator with two degrees of freedom" ''Philos. Mag. (6)'' , '''43''' (1922) pp. 700–719</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> B. van der Pol, ''Philos. Mag. (7)'' , '''2''' (1926) pp. 978–992</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lefschetz, "Differential equations: geometric theory" , Interscience (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.J. Stoker, "Nonlinear vibrations in mechanical and electrical systems" , Interscience (1950)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.A. Dorodnitsyn, "Asymptotic solution of the van der Pol equation" ''Priklad. Mat. Mekh.'' , '''11''' (1947) pp. 313–328 (In Russian) (English abstract)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> E.F. Mishchenko, N.Kh. Rozov, "Differential equations with small parameters and relaxation oscillations" , Plenum (1980) (Translated from Russian)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | For small | + | For small $ \mu $ |
+ | the first $ 164 $ | ||
+ | terms of the series for amplitude and period have been computed by symbolic calculation, see [[#References|[a1]]]. The computation of [[#References|[5]]] has been refined in [[#References|[a2]]]. For a recent survey of the free and forced van der Pol oscillator, see [[#References|[a3]]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.B. Dadfar, J. Geer, C.M. Andersen, "Perturbation analysis of the limit cycle of the free Van der Pol equation" ''SIAM J. Appl. Math.'' , '''44''' (1984) pp. 881–895</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Bavinck, J. Grasman, "The method of matched asymptotic expansions for the periodic solution of the Van der Pol equation" ''Int. J. Nonlin. Mech.'' , '''9''' (1974) pp. 421–434</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Grasman, "Asymptotic methods for relaxation oscillations and applications" , Springer (1987)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.B. Dadfar, J. Geer, C.M. Andersen, "Perturbation analysis of the limit cycle of the free Van der Pol equation" ''SIAM J. Appl. Math.'' , '''44''' (1984) pp. 881–895</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Bavinck, J. Grasman, "The method of matched asymptotic expansions for the periodic solution of the Van der Pol equation" ''Int. J. Nonlin. Mech.'' , '''9''' (1974) pp. 421–434</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Grasman, "Asymptotic methods for relaxation oscillations and applications" , Springer (1987)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983)</TD></TR></table> |
Latest revision as of 08:41, 13 May 2022
The non-linear second-order ordinary differential equation
$$ \tag{1 } \ddot{x} - \mu ( 1 - x ^ {2} ) \dot{x} + x = 0,\ \ \mu = \textrm{ const } > 0,\ \ \dot{x} ( t) \equiv { \frac{dx}{dt} } , $$
which is an important special case of the Liénard equation. Van der Pol's equation describes the auto-oscillations (cf. Auto-oscillation) of one of the simplest oscillating systems (the van der Pol oscillator). In particular, equation (1) serves — after making several simplifying assumptions — as a mathematical model of a generator on a triode for a tube with a cubic characteristic. The character of the solutions of equation (1) was first studied in detail by B. van der Pol .
Equation (1) is equivalent to the following system of two equations in two phase variables $ x, v $:
$$ \tag{2 } \dot{x} = v,\ \ \dot{v} = - x + \mu ( 1 - x ^ {2} ) v. $$
It is sometimes convenient to replace the variable $ x $ by the variable $ z( t) = {\int _ {0} ^ {t} } x ( \tau ) d \tau $; equation (1) then becomes
$$ \ddot{z} - \mu \left ( \dot{z} - \frac{\dot{z} ^ {3} }{3} \right ) + z = 0, $$
which is a special case of the Rayleigh equation. If, together with $ x $, one also considers the variable $ y = - x + ( x ^ {3} /3) + ( \dot{x} / \mu ) $, introduces a new time $ \tau = t / \mu $ and puts $ \epsilon = \mu ^ {- 2} $, one obtains the system
$$ \tag{3 } \epsilon x ^ \prime = \ y - x + \frac{x ^ {3} }{3} ,\ \ y ^ \prime = - x,\ \ {} ^ \prime = \frac{d}{d \tau } , $$
instead of equation (1). For any $ \mu > 0 $ there exists a unique stable limit cycle in the phase plane of the system (2) to which all other trajectories (except for the equilibrium position at the coordinate origin) converge as $ t \rightarrow \infty $; this limit cycle describes the oscillations of the van der Pol oscillator [2], [3], [4].
For small $ \mu $ the auto-oscillations of the oscillator (1) are close to simple harmonic oscillations (cf. Non-linear oscillations) with period $ 2 \pi $ and specified amplitude. In order to calculate the oscillation process more accurately, asymptotic methods are employed. As $ \mu $ increases, the auto-oscillations of the oscillator (1) deviate more and more from harmonic oscillations. If $ \mu $ is large, equation (1) describes relaxation oscillation with period $ 1.614 \mu $ (to a first approximation). More accurate asymptotic expansions of magnitudes characterizing relaxation oscillations [5] are known: The study of these oscillations is equivalent to the study of the solutions of the system (3) with a small coefficient $ \epsilon $ in front of the derivative [6].
The equation
$$ \ddot{x} - \mu ( 1 - x ^ {2} ) \dot{x} + x = E _ {0} + E \sin \omega t $$
describes the behaviour of the van der Pol oscillator when acted upon by a periodic external disturbance. The most important in this context is the study of frequency capture (the existence of periodic oscillations), beats (the possibility of almost-periodic oscillations) and chaotic behaviour [2], [4].
References
[1a] | B. van der Pol, "On oscillation hysteresis in a triode generator with two degrees of freedom" Philos. Mag. (6) , 43 (1922) pp. 700–719 |
[1b] | B. van der Pol, Philos. Mag. (7) , 2 (1926) pp. 978–992 |
[2] | A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian) |
[3] | S. Lefschetz, "Differential equations: geometric theory" , Interscience (1957) |
[4] | J.J. Stoker, "Nonlinear vibrations in mechanical and electrical systems" , Interscience (1950) |
[5] | A.A. Dorodnitsyn, "Asymptotic solution of the van der Pol equation" Priklad. Mat. Mekh. , 11 (1947) pp. 313–328 (In Russian) (English abstract) |
[6] | E.F. Mishchenko, N.Kh. Rozov, "Differential equations with small parameters and relaxation oscillations" , Plenum (1980) (Translated from Russian) |
Comments
For small $ \mu $ the first $ 164 $ terms of the series for amplitude and period have been computed by symbolic calculation, see [a1]. The computation of [5] has been refined in [a2]. For a recent survey of the free and forced van der Pol oscillator, see [a3].
References
[a1] | M.B. Dadfar, J. Geer, C.M. Andersen, "Perturbation analysis of the limit cycle of the free Van der Pol equation" SIAM J. Appl. Math. , 44 (1984) pp. 881–895 |
[a2] | H. Bavinck, J. Grasman, "The method of matched asymptotic expansions for the periodic solution of the Van der Pol equation" Int. J. Nonlin. Mech. , 9 (1974) pp. 421–434 |
[a3] | J. Grasman, "Asymptotic methods for relaxation oscillations and applications" , Springer (1987) |
[a4] | J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983) |
Van der Pol equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Van_der_Pol_equation&oldid=12868