Quasiperiodic oscillations

Post-publication activity

Curator: Anatoly M. Samoilenko

Quasiperiodic oscillation is an oscillation that can be described by a quasiperiodic function, i.e., a function $$F$$ of real variable $$t$$ such that $F(t)= f(\omega _{1} t, \ldots, \omega_{m} t)$ for some continuous function $$f(\varphi _{1}, \ldots, \varphi _{m})$$ of $$m$$ variables $$(m\geq 2),$$ periodic on $$\varphi _{1}, \ldots, \varphi _{m}$$ with the period $$2\pi,$$ and some set of positive frequencies $$\omega _{1}, \ldots, \omega _{m}\ ,$$ rationally linearly independent, which is equivalent to the condition $(k, \omega)=k_{1}\omega _{1}+ \ldots + k_{m}\omega _{m}\neq 0$ for any non-zero integer-valued vector $$k=(k_{1}, \ldots, k_{m})\ .$$ The frequency vector $$\omega =(\omega _{1}, \ldots, \omega _{m})$$ is often called the frequency basis of a quasiperiodic function.

Properties

Properties of quasiperiodic oscillations depend on the properties of quasiperiodic functions:

• The sum or product of quasiperiodic functions, as well as the uniform limit on $$\mathbb{R}$$ of a sequence of quasiperiodic functions with the same frequency basis, is also a quasiperiodic function;
• A quasiperiodic function has mean value

$\bar F= \lim \limits_{T\rightarrow \infty} \frac{1}{T}\int\limits_{0}^{T} F(t)dt\ ,$

it can be decomposed into Fourier series

$F(t)\simeq \sum \limits_{k=1}^{m} a_{k_{1}\ldots \,k_{m}} e^{i(k_{1}\omega _{1}+ \ldots \,+k_{m}\omega _{m})t}\ ,$ $a_{k_{1}\ldots \,k_{m}}= \lim \limits_{T\rightarrow \infty} \frac{1}{T}\int\limits_{0}^{T} F(t) e^{-i(k_{1}\omega _{1}+ \ldots \,+k_{m}\omega _{m})t}dt\ ,$

converging in mean-square:

$\sum \limits_{k=1}^{m} |\,a_{k_{1}\ldots \,k_{m}}|^{2}= \lim \limits_{T\rightarrow \infty} \frac{1}{T}\int\limits_{0}^{T} |F(t)|^{2}dt.$

• Values of a quasiperiodic function $$F$$ are everywhere dense in the set of values of its defining function $$f\ ,$$ that is, for any $$\varphi _{0}\in \mathbb{R}^m$$

$\lim \limits_{n \rightarrow \infty} F(t_{n})=f(\varphi _{0}) \ :$ for some sequence $$t_{n}, \, n=1, 2, \ldots\, .$$

Theory of Oscillations

Quasiperiodic oscillations could appear when an oscillator is forced by a time-dependent input; then they are called forced quasiperiodic oscillations. They can also be generated intrinsically by a non-linear system without any external forcing; they are often called quasiperiodic auto-oscillations in this case. The frequency basis is imposed by the external forces in the former case, and by the intrinsic properties of the system in the latter case.

Theory of oscillations is a subfield of the applied theory of differential equations (dynamical systems) devoted to studies of oscillations in nature and engineering. The main goal is to prove the existence and determine the properties of oscillatory motions. For quasiperiodic oscillations, this typically reduces to studies of either the system of non-autonomous ordinary differential equations $\tag{1} \frac{dx}{dt}=X(t, x)$

with quasiperiodic (with respect to $$t$$) right-hand side, or to the autonomous system of differential equations $\tag{2} \frac{dx}{dt}=X(x) \ ,$

where variables $$x$$ and $$X$$ are $$n$$-dimensional vectors from $$\mathbb{R}^{n}\ ,$$ and $$t$$ is time.

Phase Coordinates

Using the definition of quasiperiodic function, the non-autonomous system of differential equations (1) can be written in phase variables $$\varphi = (\varphi_{1}, \ldots ,\varphi_{m})$$ as $\tag{3} \frac{d \varphi}{dt}=\omega, \qquad \frac{dx}{dt}=F(\varphi, x) \ ,$

where $$\omega = (\omega_{1}, \ldots , \omega_{m} )$$ is the frequency basis of the right-hand side of (1) as a function of $$t\ ,$$ and $$F$$ is a function of variables $$\varphi = (\varphi_{1}, \ldots , \varphi_{m} )\ ,$$ $$2\pi$$-periodic in each variable $$\varphi_{\nu}, \, \nu=\overline{1,\,m},\ ,$$ such that $\tag{4} X(t, x)=F(\omega t, x)\ .$

Except some general linear results, the theory of quasiperiodic oscillations, founded by Poincaré, was most successful within the framework of perturbation theory, which concerns the existence, construction, and properties of solutions of perturbed quasiperiodic systems of the form (1), (2).

Let $$C^{r}(\mathbb{T}^{m})$$ be a space of $$r$$-times continuously differentiable functions $$f$$ defined on the $$m$$-torus $$\mathbb{T}^{m}$$ with variables $$\varphi_{1}, \ldots , \varphi_{m}$$ ($$f$$ is $$2\pi$$periodic in each variable $$\varphi_{\nu}, \, \nu=\overline{1,\,m}$$). Let $\tag{5} x(t)=f(\omega t), \qquad f\in C^{r}(\mathbb{T}^{m}), \qquad r\geq 2,$

be a quasiperiodic solution of (2). Then, under the general assumptions $\tag{6} {\rm rank} \, \frac{\partial f(\varphi)}{\partial \varphi}=m<n, \qquad \varphi \in\mathbb{T}^{m},$

the set $\tag{7} x=f(\varphi), \qquad \varphi \in \mathbb{T}^{m}$

is a manifold in $$\mathbb{R}^{n}\ ,$$ which is diffeomorhic to the $$m$$-dimensional torus $$\mathbb{T}^{m}\ .$$ Actually, the system (2) has an $$m$$-parameter family of quasiperiodic solutions that is defined by the equation $\tag{8} x(t, \varphi)=f(\omega t+\varphi)$

for any value of $$\varphi \in\mathbb{T}^{m}\ .$$ The set (7) is called the $$C^{r}$$-smooth invariant toroidal manifold of system (2). It determines the property of quasiperiodic auto-oscillation of this system. The quasiperiodic trajectory (5) of system (2) is said to form an everywhere dense winding of this manifold.

Amplitude–Phase Coordinates

In the case $$n>2m$$ or $$n=m+1\ ,$$ the system can be reduced to the canonical form $\frac{d\varphi}{dt}= \omega +A(\varphi, h), \qquad \frac{dh}{dt}=P(\varphi, h)h$ using local coordinates $$\varphi, h$$ in a neighborhood of manifold (7) in the space $$\mathbb{R}^{n}\ .$$ Here, the function $$P$$ is periodic in $$\varphi=(\varphi _{1}, \ldots ,\varphi _{m})$$ with period $$2\pi$$ with respect to each variable $$\varphi_{\nu}, \, \nu=\overline{1, m}\ .$$ Systems with other relations among dimensions $$n$$ and $$m$$ can also be reduced to the canonical form, but, first, (2) must be embedded into a larger system of appropriate dimension consisting of (2) and, for example, $\frac{dy_\nu}{dt}=y_\nu, \qquad \nu=\overline{1, k} \ .$ In this case, the perturbed system (2) of the type $\tag{9} \frac{dx}{dt}=X(x)+\mu X_1(x, \mu)$

reduces to the system of equations $\frac{d\varphi}{dt}= \omega +\mu a(\varphi, h, \mu)\ ,$ $\tag{10} \frac{dh}{dt}=P(\varphi, h, \mu)h+\mu c(\varphi),$

where $$\mu$$ is a small (in absolute value) parameter, i.e., $$|\mu| \ll 1\ .$$ Note that the perturbed system (3) is a particular case of the system (10).

Studies of Quasiperiodic Solutions

Studies of quasiperiodic solutions of (10) consist of two problems:

• Prove the existence of invariant manifold for (10) of the form

$\tag{11} M(\mu)\colon h=u(\varphi, \mu), \ :$

where $$u$$ is a function periodic in $$\varphi=(\varphi _{1}, \ldots ,\varphi _{m})$$ with period $$2\pi$$ in each variable, that contracts as $$\mu \rightarrow 0$$ to the unperturbed manifold of the system (9) $M(0)\colon h=0 \ .$

• Prove the existence of such solutions of the system (10) on the invariant manifold (11), i.e., the solutions of the system

$\frac{d\varphi}{dt}= \omega +\mu a(\varphi, u(\varphi, \mu), \mu)\ :$ that correspond to quasiperiodic solutions of the system (10) with respect to coordinate $$h$$ $h(t)=u(\varphi(t), \mu)=g(\omega t, \mu)\ ,$

where $$g\in C^{s}(\mathbb{T}^{m})$$ for $$|\mu| \ll 1,$$ and $$\omega=(\omega_1,\, \ldots, \,\omega_m)$$ is the frequency basis.

The first problem could be solved using perturbation methods (Krylov and Bogolyubov 1934, Bogolyubov and Mitropolski 1961, Samoilenko 1991). The second problem could be solved using Poincaré–Denjoy theory and methods of accelerated convergence of Newton iterations (see Kolmogorov 1954, Arnold 1963, Moser 1968, Bogolyubov et al. 1969).

Among perturbation methods used in the theory of quasiperiodic oscillations the most notable is the method of Krylov and Bogolyubov (1934; see also Bogolyubov and Mitropolski 1961 and Bogolyubov et al. 1969). In particular, this method allows one to study quasiperiodic auto-oscillations of systems of the form $\tag{12} \frac{d^{2} x_{\nu}}{dt^{2}}+\omega^{2}_{\nu}x_{\nu}= \varepsilon f_{\nu}(x_{1}, \ldots , x_{n}; \frac{dx_1}{dt}, \ldots , \frac{dx_n}{dt}), \quad \nu=\overline{1, n} \ ,$

where $$\varepsilon$$ is a small positive parameter. The amplitude–phase coordinates $$a$$ and $$\varphi$$ and the formulas $x_{\nu}=a_\nu \cos \varphi _\nu, \qquad \frac{dx_\nu}{dt}=-a_\nu \omega_\nu \sin \varphi _\nu$ reduce (12) to a system of the form (10) $\frac{da}{dt}=\varepsilon A(a, \varphi, \varepsilon)\ ,$ $\tag{13} \frac{d\varphi}{dt}=\omega +\varepsilon B(a, \varphi, \varepsilon)$

whose right-hand side is $$2\pi$$-periodic with respect to $$\varphi=(\varphi_1, \ldots, \varphi_n)$$ in each $$\varphi _\nu, \, \nu=\overline{1, n}\ .$$

Under general assumptions, (13) has an invariant manifold $\tag{14} a=a(\varphi, \varepsilon)\ ,$

where $$a\in C^{s}(\mathbb{T}^m)$$ for $$\varepsilon \ll 1$$ whenever the averaged system of amplitude equations $\frac{db}{dt}=\varepsilon A(b)\ ,$ $A(b)=\frac{1}{(2\pi)^m} \int\limits^{2\pi}_0 ... \int\limits^{2\pi}_0 A(\varphi, 0)d\varphi_1 ... d\varphi_n$ has “rough” equilibrium at $$b=b^0\ :$$ $A(b^0)=0$ and all eigenvalues of $$\frac{\partial A(b^0)}{\partial a}$$ have nonzero real parts. The system (13) on the manifold (14) can be reduced to the system of equations $\tag{15} \frac{d\varphi}{dt}=\omega + \varepsilon B(a(\varphi, \varepsilon),\, \varphi, \, \varepsilon) \ .$

If $$n=2\ ,$$ then we can apply the Poincaré–Denjoy theory and its generalizations. If $$n=3\ ,$$ then we can apply the Arnold–Moser theorem (Bogolyubov et al. 1969, Arnold et al. 1993). These results provide existence conditions for solutions of (15) that guarantee the quasiperiodicity of auto-oscillations of system (12), which are close to the oscillations $x_\nu=b_\nu^0\cos(\omega_\nu t+\varphi_\nu), \qquad \nu=\overline{1, n}\ ,$ $\frac{dx_\nu}{dt}=-b_\nu^0 \omega_\nu \sin(\omega_\nu t+\varphi_\nu), \qquad \varphi_\nu=const \ .$

Hamiltonian Systems

The theory of quasiperiodic oscillations was extensively developed in the context of perturbation theory of Hamiltonian systems with the Hamiltonian $H(p, q, \varepsilon)=H_0(p)+\varepsilon H_1(p, q)+ \ldots \ ,$ where $$H$$ is a function $$2\pi$$-periodic in $$q=(q_1, \, \ldots,\, q_n)$$ with respect to each variable $$q_\nu\ ,$$ $$\nu=\overline{1, n}\ ,$$ and $$\varepsilon$$ is a small parameter. There is a separate theory for such systems, namely, KAM-theory (Kolmogorov–Arnold–Moser Theory; see also Arnold et al. 2002).

It should be noted that quasiperiodic oscillations are studied not only in models described by ordinary differential equations. Many researchers study such oscillations as solutions of systems of evolution equations with deviating argument, partial differential equations, and equations in infinite-dimensional spaces.

References

Arnold V.I. (1963) Small denominators and problems of the stability of motion in classical and celestial mechanics (in Russian). Usp. Mat. Nauk. 18, 91--192 (English transl. in Russ. Math. Surv. 18 (1963), 85–193).

Arnold V.I., Kozlov V.V., Neishtadt A.I. (1993) Mathematical aspects of classical and celestial mechanics. Translated from the 1985 Russian original by A. Iacob. Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Dynamical systems. III, Encyclopaedia Math. Sci., 3, Springer, Berlin, 1993]. Springer-Verlag, Berlin, 1997. xiv+291 pp.

Bogoliubov N.N., Mitropolsky Yu.A. (1961) Asymptotic methods in the theory of nonlinear oscillations. New York: Gordon and Breach Sci. Publ. (Delhi Hindustan "Publishing Corp. India")

Bogoliubov N.N., Mitropolsky Yu.A., Samoilenko A.M. (1976) Methods of accelerated convergence in nonlinear mechanics. Springer-Verlag, Berlin–New York.

Krylov N. M. and Bogolyubov N. N. (1934) New Methods of Nonlinear Mechanics. Moscow–Leningrad (in Russian).

Kolmogorov A. N. (1954) On conservation of conditionally periodic motions for a small change in Hamilton's function. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 98, 527–530.

Moser J. (1968) Lectures on Hamiltonian Systems. Mem. AMS, no. 81. P. 1–60.

Poincaré H. (1880–1890) Mémoire sur les courbes définies par les équations différentielles I–VI, Oeuvre I. Gauthier-Villar: Paris

Samoilenko A.M. (1991) Elements of the Mathematical Theory of Multi-Frequency Oscillations. Mathematics and Its Applications, V.71.). Kluwer Academic Publishers Group, Dordrecht. – Netherlands.

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