# Non-linear oscillations

Oscillations in physical systems described by non-linear systems of ordinary differential equations

$$\tag{1 } \dot{\mathbf x} = \mathbf A ( t) \mathbf x + \mu \mathbf X ( t , \mathbf x , \mu ) + \mathbf f ( t) ,$$

where $\mathbf x \in \mathbf R ^ {n}$, $\mathbf X$ contains terms of at least the second degree in the components of the vector $\mathbf x$, $\mathbf f$ is a vector function of the time $t$, and $\mu > 0$ is a small parameter (or $\mu = 1$ and $\mathbf X = \mathbf X ( t , \mathbf x )$). Possible generalizations are connected with the discussion of discontinuous systems, actions with discontinuous characteristic (for example, of hysteresis type), delay and random actions, integro-differential and differential-operator equations, oscillating systems with distributed parameters described by partial differential equations, and also with the use of methods of optimal control of non-linear oscillating systems. The basic general problems of non-linear oscillations are: the search for equilibrium positions, for stationary regimes (in particular, for periodic motions, auto-oscillations), and the investigation of their stability, and problems of synchronization and stabilization of non-linear oscillations.

Strictly speaking, all physical systems are non-linear. One of the most characteristic features of non-linear oscillations is the violation of the principle of superposition of oscillations: The result of every action in the presence of a second turns out to be different from the case when the second action is absent. Quasi-linear systems are systems (1) with $\mu > 0$. A basic method for studying them is the method of the small parameter (cf. Small parameter, method of the). Above all there is the Poincaré–Lindstedt method for determining periodic solutions of quasi-linear systems that are analytic in the parameter for sufficiently small values of it, either in the form of power series in $\mu$( see [1], Chapt. IX) or in the form of power series in $\mu$ with $\beta _ {1} \dots \beta _ {n}$ added to the initial values of the components of $\mathbf x$( see [1], Chapt. III). For the subsequent development of this method see, for example, [2][4].

Another method of a small parameter is that of averaging. At the same time, in the study of quasi-linear systems new methods emerged: asymptotic methods (see [5], [6]), the method of $V$- functions (see [7]), which is based on the fundamental results of A.M. Lyapunov and N.G. Chetaev, and others.

Essentially non-linear system are those in which the small parameter prescribed in advance is absent ( $\mu = 1$ and $\mathbf X = \mathbf X ( t , \mathbf x )$ in (1)). For the Lyapunov system

$$\tag{2 } \dot{\mathbf x} = \mathbf A \mathbf x + \mathbf X ( \mathbf x ) ,$$

where

$$\mathbf A = \lambda \mathbf J _ {2} + \mathbf P ,\ \ \mathbf J _ {2} = \ \left \| \begin{array}{cr} 0 &- 1 \\ 1 & 1 \\ \end{array} \right \| ,$$

and where there are no multiple roots $\pm \lambda i$ among the eigen values of the $( k \times k )$- matrix $\mathbf P$, $\mathbf X ( \mathbf x )$ is an analytic vector function of $\mathbf x$ the expansion of which begins with terms of at least second order, and there is an analytic first integral of special form. Lyapunov (see [8], Sect. 42) has proposed a method of search for periodic solutions in the form of a series in powers of an arbitrary constant $c$( for which one can use the initial value of one of the two critical variables $x _ {1}$ or $x _ {2}$).

For systems close to Lyapunov systems,

$$\dot{\mathbf x} = \mathbf A \mathbf x + \mathbf X ( \mathbf x ) + \mu \mathbf F ( t , \mathbf x , \nu ) ,$$

where $\mathbf A$ and $\mathbf X ( \mathbf x )$ are as in (2) and $\mathbf F$ is an analytic vector function of $\mathbf x$ and a small parameter $\nu$, continuous and $2 \pi$- periodic in $t$, also a method has been proposed for determining periodic solutions (see [4], Chapt. VIII). Systems of Lyapunov type (2) in which the matrix $\mathbf A$ has $l$ zero eigen values with simple elementary divisors, a pair of purely imaginary eigen values $\pm \lambda i$ and no multiple eigen values $\pm \lambda i$, and where $\mathbf X ( \mathbf x )$ is as in (2), may be reduced to Lyapunov systems (see [9], Chapt. IV. 2). A study has also been made of non-linear oscillation in Lyapunov systems and in so-called Lyapunov systems with damping, and the general problem of energy transfer in them has been solved (see [9], Chapts. I, III, IV).

Suppose that an essentially non-linear autonomous system has been reduced to the Jordan form of its linear part:

$$\tag{3 } \dot{x} _ \nu = \lambda _ \nu x _ \nu + \delta _ \nu x _ {\nu + 1 } + x _ \nu \sum _ {Q \in \mathfrak M _ \nu } f _ {\nu Q } x _ {1} ^ {q _ {1} } \dots x _ {n} ^ {q _ {n} } ,$$

$$\nu = 1 \dots n ; \ \delta _ {n} = 0 ,$$

where the vector $\Lambda = ( \lambda _ {1} \dots \lambda _ {n} )$ has by assumption at least one non-zero component, $\delta _ {1} \dots \delta _ {n-} 1$ are zero or one according whether non-simple elementary divisors of the matrix of the linear part are absent or present, the $f _ {\nu Q }$ are coefficients, and $\mathfrak M _ \nu$ is the set of values of the vectors $Q = ( q _ {1} \dots q _ {n} )$ with integer components such that:

$$q _ {1} \dots q _ {\nu - 1 } ,\ q _ {\nu + 1 } \dots q _ {n} \geq 0 ,$$

$$q _ \nu \geq - 1 ,\ q _ {1} + \dots + q _ {n} \geq 1 .$$

Then there exists a normalizing transformation

$$\tag{4 } x _ \nu = y _ \nu + y _ \nu \sum _ {Q \in \mathfrak M _ \nu } h _ {\nu Q } y _ {1} ^ {q _ {1} } \dots y _ {n} ^ {q _ {n} } ,\ \ \nu = 1 \dots n ,$$

reducing (3) to the normal form of differential equations

$$\tag{5 } \dot{y} _ \nu = \ \lambda _ \nu y _ \nu + \delta _ \nu y _ {\nu + 1 } + y _ \nu \sum _ { \begin{array}{c} Q \in \mathfrak M _ \nu \\ ( \Lambda , Q ) = 0 \end{array} } g _ {\nu Q } y _ {1} ^ {q _ {1} } \dots y _ {n} ^ {q _ {n} } ,$$

$$\nu = 1 \dots n ; \ \delta _ {n} = 0 ,$$

such that $g _ {\nu Q } = 0$ when $( \Lambda , Q ) \neq 0$. Thus, the normal form (5) contains only resonance terms, that is, the coefficients $g _ {\nu Q }$ in (5) can be non-zero only for those $Q$ for which the resonance equation

$$( \Lambda , Q ) \equiv \lambda _ {1} q _ {1} + \dots + \lambda _ {n} q _ {n} = 0 ,$$

which plays an essential role in the theory of oscillations, holds. The convergence and divergence of the normalizing transformation (4) has been investigated (see , Part I, Chapts. II, III); the coefficients $h _ {\nu Q }$ have been computed (by means of their symmetrization) (see [9], Sect. 5.3). In a number of problems on non-linear oscillations of essentially non-linear autonomous systems, the method of normal forms has proved effective (see , [9], Chapts. VI–VIII).

Of the other methods for studying essentially non-linear systems the method of point mappings (see [2], [11]), the stroboscopic method [12] and methods of functional analysis [13] have been applied.

Qualitative methods for non-linear oscillations. The starting point here is the study of the form of the integral curves of non-linear ordinary differential equations, which was undertaken by H. Poincaré (see [14]). Applications to problems of non-linear oscillations that can be described by second-order autonomous systems can be found in [2] and [15]. Questions of the existence of periodic solutions and their stability in the large for many-dimensional systems have been studied [16]; almost-periodic non-linear oscillations are treated in [13]. For applications of the theory of ordinary differential equations with a small parameter in front of some of the derivatives to problems of non-linear relaxation oscillations (cf. Relaxation oscillation) see [17].

For other important aspects of non-linear oscillations and further references see Perturbation theory and Oscillations, theory of.

#### References

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