Quasiperiodic motion

Last updated November 27, 2024

In mathematics and theoretical physics, quasiperiodic motion is motion on a torus that never comes back to the same point. This behavior can also be called quasiperiodic evolution, dynamics, or flow. The torus may be a generalized torus so that the neighborhood of any point is more than two-dimensional. At each point of the torus there is a direction of motion that remains on the torus. Once a flow on a torus is defined or fixed, it determines trajectories. If the trajectories come back to a given point after a certain time then the motion is periodic with that period, otherwise it is quasiperiodic.

The quasiperiodic motion is characterized by a finite set of frequencies which can be thought of as the frequencies at which the motion goes around the torus in different directions. For instance, if the torus is the surface of a doughnut, then there is the frequency at which the motion goes around the doughnut and the frequency at which it goes inside and out. But the set of frequencies is not unique –by redefining the way position on the torus is parametrized another set of the same size can be generated. These frequencies will be integer combinations of the former frequencies (in such a way that the backward transformation is also an integer combination). To be quasiperiodic, the ratios of the frequencies must be irrational numbers.^[1]^[2]^[3]^[4]

In Hamiltonian mechanics with $n$ position variables and associated rates of change it is sometimes possible to find a set of $n$ conserved quantities. This is called the fully integrable case. One then has new position variables called action-angle coordinates, one for each conserved quantity, and these action angles simply increase linearly with time. This gives motion on "level sets" of the conserved quantities, resulting in a torus that is an $n$ -manifold –locally having the topology of $n$ -dimensional space.^[5] The concept is closely connected to the basic facts about linear flow on the torus. These essentially linear systems and their behaviour under perturbation play a significant role in the general theory of non-linear dynamic systems.^[6] Quasiperiodic motion does not exhibit the butterfly effect characteristic of chaotic systems. In other words, starting from a slightly different initial point on the torus results in a trajectory that is always just slightly different from the original trajectory, rather than the deviation becoming large.^[4]

Rectilinear motion

Rectilinear motion along a line in a Euclidean space gives rise to a quasiperiodic motion if the space is turned into a torus (a compact space) by making every point equivalent to any other point situated in the same way with respect to the integer lattice (the points with integer coordinates), so long as the direction cosines of the rectilinear motion form irrational ratios. When the dimension is 2, this means the direction cosines are incommensurable. In higher dimensions it means the direction cosines must be linearly independent over the field of rational numbers.^[5]

Torus model

If we imagine that the phase space is modelled by a torus T (that is, the variables are periodic, like angles), the trajectory of the quasiperiodic system is modelled by a curve on T that wraps around the torus without ever exactly coming back on itself. Assuming the dimension of T is at least two, these can be thought of as one-parameter subgroups of the torus given group structure (by specifying a certain point as the identity element).

Quasiperiodic functions

A quasiperiodic motion can be expressed as a function of time whose value is a vector of "quasiperiodic functions". A quasiperiodic function $f$ on the real line is a function obtained from a function $F$ on a standard torus T (defined by $n$ angles), by means of a trajectory in the torus in which each angle increases at a constant rate.^[7]There are $n$ "internal frequencies", being the rates at which the $n$ angles progress, but as mentioned above the set is not uniquely determined. In many cases the function in the torus can be expressed as a multiple Fourier series. For $n$ equal to 2 this is:

F(\theta _{1},\theta _{2})=\sum _{j=-\infty }^{\infty }\sum _{k=-\infty }^{\infty }C_{jk}\exp(ij\theta _{1})\exp(ik\theta _{2})

If the trajectory is

\theta _{1}=a_{1}+\omega _{1}t

\theta _{2}=a_{2}+\omega _{2}t

then the quasiperiodic function is:

f(t)=\sum _{j=-\infty }^{\infty }\sum _{k=-\infty }^{\infty }C_{jk}\exp(ija_{1}+ika_{2}+i(j\omega _{1}+k\omega _{2})t)

This shows that there may be an infinite number of frequencies in the expansion, not multiples of a finite number of frequencies. Depending on which coefficients $C_{jk}$ are non-zero the "internal frequencies" $\omega _{1}$ and $\omega _{2}$ themselves may not contribute terms in this expansion, even if one uses an alternative set of internal frequencies such as $\omega _{1}$ and $\omega _{1}+\omega _{2}.$ ^[8] If the $C_{jk}$ are non-zero only when the ratio $i/j$ is some specific constant, then the function is actually periodic rather than quasiperiodic.

See Kronecker's theorem for the geometric and Fourier theory attached to the number of modes. The closure of (the image of) any one-parameter subgroup in T is a subtorus of some dimension d. In that subtorus the result of Kronecker applies: there are d real numbers, linearly independent over the rational numbers, that are the corresponding frequencies.

In the quasiperiodic case, where the image is dense, a result can be proved on the ergodicity of the motion: for any measurable subset A of T (for the usual probability measure), the average proportion of time spent by the motion in A is equal to the measure of A.^[9]

Terminology and history

The theory of almost periodic functions is, roughly speaking, for the same situation but allowing T to be a torus with an infinite number of dimensions. The early discussion of quasi-periodic functions, by Ernest Esclangon following the work of Piers Bohl, in fact led to a definition of almost-periodic function, the terminology of Harald Bohr.^[10] Ian Stewart wrote that the default position of classical celestial mechanics, at this period, was that motions that could be described as quasiperiodic were the most complex that occurred.^[11] For the Solar System, that would apparently be the case if the gravitational attractions of the planets to each other could be neglected: but that assumption turned out to be the starting point of complex mathematics.^[12] The research direction begun by Andrei Kolmogorov in the 1950s led to the understanding that quasiperiodic flow on phase space tori could survive perturbation.^[13]

NB: The concept of quasiperiodic function, for example the sense in which theta functions and the Weierstrass zeta function in complex analysis are said to have quasi-periods with respect to a period lattice, is something distinct from this topic.

Related Research Articles

The Kolmogorov–Arnold–Moser (KAM) theorem is a result in dynamical systems about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory of classical mechanics.

In mechanics and physics, simple harmonic motion is a special type of periodic motion an object experiences by means of a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in an oscillation that is described by a sinusoid which continues indefinitely.

Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or superposition, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.

In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.

Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections. The mathematical basis for tomographic imaging was laid down by Johann Radon. A notable example of applications is the reconstruction of computed tomography (CT) where cross-sectional images of patients are obtained in non-invasive manner. Recent developments have seen the Radon transform and its inverse used for tasks related to realistic object insertion required for testing and evaluating computed tomography use in airport security.

In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation: $where \nabla 2 is the Laplace operator, k 2 is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number. The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle.$

The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was instead a set of heuristic corrections to classical mechanics. The theory has come to be understood as the semi-classical approximation to modern quantum mechanics. The main and final accomplishments of the old quantum theory were the determination of the modern form of the periodic table by Edmund Stoner and the Pauli exclusion principle, both of which were premised on Arnold Sommerfeld's enhancements to the Bohr model of the atom.

In signal processing, linear phase is a property of a filter where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time by the same constant amount, which is referred to as the group delay. Consequently, there is no phase distortion due to the time delay of frequencies relative to one another.

In quantum field theory, the theta vacuum is the semi-classical vacuum state of non-abelian Yang–Mills theories specified by the vacuum angleθ that arises when the state is written as a superposition of an infinite set of topologically distinct vacuum states. The dynamical effects of the vacuum are captured in the Lagrangian formalism through the presence of a θ-term which in quantum chromodynamics leads to the fine tuning problem known as the strong CP problem. It was discovered in 1976 by Curtis Callan, Roger Dashen, and David Gross, and independently by Roman Jackiw and Claudio Rebbi.

In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function. A function $is quasiperiodic with quasiperiod if, where is a " simpler " function than . What it means to be " simpler " is vague.$

A cyclostationary process is a signal having statistical properties that vary cyclically with time. A cyclostationary process can be viewed as multiple interleaved stationary processes. For example, the maximum daily temperature in New York City can be modeled as a cyclostationary process: the maximum temperature on July 21 is statistically different from the temperature on December 20; however, it is a reasonable approximation that the temperature on December 20 of different years has identical statistics. Thus, we can view the random process composed of daily maximum temperatures as 365 interleaved stationary processes, each of which takes on a new value once per year.

In mathematics, particularly in dynamical systems, Arnold tongues are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamical system, or other related invariant property thereof, changes according to two or more of its parameters. The regions of constant rotation number have been observed, for some dynamical systems, to form geometric shapes that resemble tongues, in which case they are called Arnold tongues.

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle $, the sine and cosine functions are denoted as and .$

In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector $e$ indicating the direction of an axis of rotation, and an angle of rotation $θ$ describing the magnitude and sense of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector $e$ rooted at the origin because the magnitude of $e$ is constrained. For example, the elevation and azimuth angles of $e$ suffice to locate it in any particular Cartesian coordinate frame.

The Routh array is a tabular method permitting one to establish the stability of a system using only the coefficients of the characteristic polynomial. Central to the field of control systems design, the Routh–Hurwitz theorem and Routh array emerge by using the Euclidean algorithm and Sturm's theorem in evaluating Cauchy indices.

In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus $which is represented by the following differential equations with respect to the standard angular coordinates$

The kicked rotator, also spelled as kicked rotor, is a paradigmatic model for both Hamiltonian chaos and quantum chaos. It describes a free rotating stick in an inhomogeneous "gravitation like" field that is periodically switched on in short pulses. The model is described by the Hamiltonian

The spectrum of a chirp pulse describes its characteristics in terms of its frequency components. This frequency-domain representation is an alternative to the more familiar time-domain waveform, and the two versions are mathematically related by the Fourier transform. The spectrum is of particular interest when pulses are subject to signal processing. For example, when a chirp pulse is compressed by its matched filter, the resulting waveform contains not only a main narrow pulse but, also, a variety of unwanted artifacts many of which are directly attributable to features in the chirp's spectral characteristics.

In fluid dynamics, a trochoidal wave or Gerstner wave is an exact solution of the Euler equations for periodic surface gravity waves. It describes a progressive wave of permanent form on the surface of an incompressible fluid of infinite depth. The free surface of this wave solution is an inverted (upside-down) trochoid – with sharper crests and flat troughs. This wave solution was discovered by Gerstner in 1802, and rediscovered independently by Rankine in 1863.

<span class="mw-page-title-main">Stokes problem</span>

In fluid dynamics, Stokes problem also known as Stokes second problem or sometimes referred to as Stokes boundary layer or Oscillating boundary layer is a problem of determining the flow created by an oscillating solid surface, named after Sir George Stokes. This is considered one of the simplest unsteady problems that has an exact solution for the Navier–Stokes equations. In turbulent flow, this is still named a Stokes boundary layer, but now one has to rely on experiments, numerical simulations or approximate methods in order to obtain useful information on the flow.

References

↑ Sergey Vasilevich Sidorov; Nikolai Alexandrovich Magnitskii. New Methods For Chaotic Dynamics. World Scientific. pp. 23–24. ISBN 9789814477918.
↑ Weisstein, Eric W. (12 December 2002). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 2447. ISBN 978-1-4200-3522-3.
↑ Ruelle, David (7 September 1989). Chaotic Evolution and Strange Attractors. Cambridge University Press. p. 4. ISBN 978-0-521-36830-8.
1 2 Broer, Hendrik W.; Huitema, George B.; Sevryuk, Mikhail B. (25 January 2009). Quasi-Periodic Motions in Families of Dynamical Systems: Order amidst Chaos. Springer. p. 2. ISBN 978-3-540-49613-7.
1 2 "Quasi-periodic motion", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
↑ Broer, Hendrik W.; Huitema, George B.; Sevryuk, Mikhail B. (25 January 2009). Quasi-Periodic Motions in Families of Dynamical Systems: Order amidst Chaos. Springer. pp. 1–4. ISBN 978-3-540-49613-7.
↑ Komlenko, Yu. V.; Tonkov, E. L. (2001) [1994], "Quasi-periodic function", Encyclopedia of Mathematics , EMS Press
↑ For instance, if only $C_{1,2},\,C_{2,1},$ and $C_{2,2}$ are non-zero.
↑ Giorgilli, Antonio (5 May 2022). Notes on Hamiltonian Dynamical Systems. Cambridge University Press. p. 131. ISBN 978-1-009-15114-6.
↑ Ginoux, Jean-Marc (18 April 2017). History of Nonlinear Oscillations Theory in France (1880-1940). Springer. pp. 311–312. ISBN 978-3-319-55239-2.
↑ Howe, Leo; Wain, Alan (25 March 1993). Predicting the Future. Cambridge University Press. p. 30. ISBN 978-0-521-41323-7.
↑ Broer, Henk; Takens, Floris (20 October 2010). Dynamical Systems and Chaos. Springer Science & Business Media. pp. 89–90. ISBN 978-1-4419-6870-8.
↑ Dumas, H. Scott (28 February 2014). Kam Story, The: A Friendly Introduction To The Content, History, And Significance Of Classical Kolmogorov-arnold-moser Theory. World Scientific Publishing Company. p. 67. ISBN 978-981-4556-60-6.