Rayleigh equation

A non-linear ordinary differential equation of order two:

$$ \tag{* } \dot{x} dot + F ( \dot{x} ) + x = 0 ,\ \dot{x} = \frac{dx}{dt} , $$

where the function $ F ( u) $ satisfies the assumption:

$$ u F ( u) < 0 \ \textrm{ for small } | u | , $$

$$ u F ( u) > 0 \ \textrm{ for large } | u | . $$

The Rayleigh equation describes a typical non-linear system with one degree of freedom which admits auto-oscillations (cf. Auto-oscillation). This equation was named after Lord Rayleigh, who studied equations of this type related to problems in acoustics [1].

If one differentiates equation (*) and then puts $ y = \dot{x} $, one obtains the Liénard equation

$$ \dot{y} dot + f ( y) \dot{y} + y = 0 ,\ \ f ( u) = F ^ { \prime } ( u) . $$

The special case of the Rayleigh equation for

$$ F ( u) = - \lambda \left ( u - \frac{u ^ {2} }{3} \right ) ,\ \ \lambda = \textrm{ const } , $$

is the van der Pol equation. Sometimes the following special case of equation (*) is called the Rayleigh equation:

$$ \dot{x} dot - ( a - b \dot{x} ^ {2} ) \dot{c} + x = \ 0 ,\ a , b > 0 . $$

There is a large number of studies concerned with the existence and uniqueness conditions for a limit cycle of the Rayleigh equation, that is, conditions under which auto-oscillations occur. The question of periodic solutions was studied also for different generalizations of the Rayleigh equation, e.g. for

$$ \dot{x} dot + F ( x , \dot{x} ) \dot{x} + g ( x) = e ( t) , $$

where $ e ( t) $ is a periodic function.

The following equation is often called a Rayleigh-type system:

$$ \dot{x} dot + F ( \dot{x} ) + G ( x) = H ( t , x , \dot{x} ) , $$

$$ x \in \mathbf R ^ {n} ,\ F : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} ,\ G : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} ; $$

moreover, it is usually assumed that

$$ F = \mathop{\rm grad} f ,\ f: \mathbf R ^ {n} \rightarrow \mathbf R ,\ \ f \in C ^ {1} , $$

$$ G = \mathop{\rm grad} g ,\ g : \mathbf R ^ {n} \rightarrow \mathbf R ,\ g \in C ^ {2} , $$

and $ H $ is a bounded vector function that is periodic in $ t $. The problem of obtaining sufficient conditions for the existence of periodic solutions of such systems is of considerable interest.

See also the references to Liénard equation.

References

[1]	J.W. [Lord Rayleigh] Strutt, "Theory of sound" , 1 , Dover, reprint (1945)
[2]	L. Cesari, "Asymptotic behavior and stability problems in ordinary differential equations" , Springer (1959)

References

[a1]	J.J. Stoker, "Nonlinear vibrations in mechanical and electrical systems" , Interscience (1950)