Difference between revisions of "Liénard equation"
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A non-linear second-order ordinary differential equation | A non-linear second-order ordinary differential equation | ||
| − | + | $$ \tag{* } | |
| + | x ^ {\prime\prime} + f ( x) x ^ \prime + x = 0 . | ||
| + | $$ | ||
| − | This equation describes the dynamics of a system with one degree of freedom in the presence of a linear restoring force and non-linear damping. If the function | + | This equation describes the dynamics of a system with one degree of freedom in the presence of a linear restoring force and non-linear damping. If the function $ f $ |
| + | has the property | ||
| − | < | + | $$ |
| + | f ( x) < 0 \ \ | ||
| + | \textrm{ for small } | x | , | ||
| + | $$ | ||
| − | + | $$ | |
| + | f ( x) > 0 \ \textrm{ for large } | x | , | ||
| + | $$ | ||
that is, if for small amplitudes the system absorbs energy and for large amplitudes dissipation occurs, then in the system one can expect self-exciting oscillations (the appearance of auto-oscillations, cf. [[Auto-oscillation|Auto-oscillation]]). Sufficient conditions for the appearance of auto-oscillations in the system (*) were first proved by A. Liénard [[#References|[1]]]. | that is, if for small amplitudes the system absorbs energy and for large amplitudes dissipation occurs, then in the system one can expect self-exciting oscillations (the appearance of auto-oscillations, cf. [[Auto-oscillation|Auto-oscillation]]). Sufficient conditions for the appearance of auto-oscillations in the system (*) were first proved by A. Liénard [[#References|[1]]]. | ||
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The Liénard equation is closely connected with the [[Rayleigh equation|Rayleigh equation]]. An important special case of it is the [[Van der Pol equation|van der Pol equation]]. Instead of equation (*) it is often convenient to consider the system | The Liénard equation is closely connected with the [[Rayleigh equation|Rayleigh equation]]. An important special case of it is the [[Van der Pol equation|van der Pol equation]]. Instead of equation (*) it is often convenient to consider the system | ||
| − | + | $$ | |
| + | x ^ \prime = v ,\ \ | ||
| + | v ^ \prime = - x - f ( x) v | ||
| + | $$ | ||
| + | |||
| + | (a stable limit cycle on the phase plane $ x , v $ | ||
| + | is adequate for an auto-oscillating process in the system (*)), or the equivalent equation | ||
| − | + | $$ | |
| − | + | \frac{dv}{dx} | |
| + | = | ||
| + | \frac{- x - f ( x) v }{v} | ||
| + | . | ||
| + | $$ | ||
| − | If one introduces a new variable | + | If one introduces a new variable $ y = x ^ \prime + F ( x) $, |
| + | where $ F ( x) = \int _ {0} ^ {x} f ( \xi ) d \xi $, | ||
| + | then (*) goes into the system | ||
| − | + | $$ | |
| + | x ^ \prime = y - F ( x) ,\ \ | ||
| + | y ^ \prime = - x . | ||
| + | $$ | ||
More general than the Liénard equation are the equations | More general than the Liénard equation are the equations | ||
| − | + | $$ | |
| + | x ^ {\prime\prime} + f ( x) x ^ \prime + g ( x) = 0 , | ||
| + | $$ | ||
| − | + | $$ | |
| + | x ^ {\prime\prime} + \phi ( x , x ^ \prime ) x ^ \prime + g ( x) = 0 . | ||
| + | $$ | ||
The main interest is in the determination of possibly more general sufficient conditions under which these equations have a unique stable periodic solution. The non-homogeneous Liénard equation | The main interest is in the determination of possibly more general sufficient conditions under which these equations have a unique stable periodic solution. The non-homogeneous Liénard equation | ||
| − | + | $$ | |
| + | x ^ {\prime\prime} + f ( x) x ^ \prime + x = e ( t) | ||
| + | $$ | ||
and generalizations of it have also been studied in detail. | and generalizations of it have also been studied in detail. | ||
Latest revision as of 22:17, 5 June 2020
A non-linear second-order ordinary differential equation
$$ \tag{* } x ^ {\prime\prime} + f ( x) x ^ \prime + x = 0 . $$
This equation describes the dynamics of a system with one degree of freedom in the presence of a linear restoring force and non-linear damping. If the function $ f $ has the property
$$ f ( x) < 0 \ \ \textrm{ for small } | x | , $$
$$ f ( x) > 0 \ \textrm{ for large } | x | , $$
that is, if for small amplitudes the system absorbs energy and for large amplitudes dissipation occurs, then in the system one can expect self-exciting oscillations (the appearance of auto-oscillations, cf. Auto-oscillation). Sufficient conditions for the appearance of auto-oscillations in the system (*) were first proved by A. Liénard [1].
The Liénard equation is closely connected with the Rayleigh equation. An important special case of it is the van der Pol equation. Instead of equation (*) it is often convenient to consider the system
$$ x ^ \prime = v ,\ \ v ^ \prime = - x - f ( x) v $$
(a stable limit cycle on the phase plane $ x , v $ is adequate for an auto-oscillating process in the system (*)), or the equivalent equation
$$ \frac{dv}{dx} = \frac{- x - f ( x) v }{v} . $$
If one introduces a new variable $ y = x ^ \prime + F ( x) $, where $ F ( x) = \int _ {0} ^ {x} f ( \xi ) d \xi $, then (*) goes into the system
$$ x ^ \prime = y - F ( x) ,\ \ y ^ \prime = - x . $$
More general than the Liénard equation are the equations
$$ x ^ {\prime\prime} + f ( x) x ^ \prime + g ( x) = 0 , $$
$$ x ^ {\prime\prime} + \phi ( x , x ^ \prime ) x ^ \prime + g ( x) = 0 . $$
The main interest is in the determination of possibly more general sufficient conditions under which these equations have a unique stable periodic solution. The non-homogeneous Liénard equation
$$ x ^ {\prime\prime} + f ( x) x ^ \prime + x = e ( t) $$
and generalizations of it have also been studied in detail.
References
| [1] | A. Liénard, Rev. Gen. Electr. , 23 (1928) pp. 901–912; 946–954 |
| [2] | A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian) |
| [3] | G. Sansone, "Ordinary differential equations" , 2 , Zanichelli (1948) (In Italian) |
| [4] | S. Lefschetz, "Differential equations: geometric theory" , Interscience (1957) |
| [5] | R. Reissig, G. Sansone, R. Conti, "Nichtlineare Differentialgleichungen höherer Ordnung" , Cremonese (1969) |
Liénard equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Li%C3%A9nard_equation&oldid=23384