As known, in the linear approximation the string oscillations are described by a partial differential equation

where y(x,t) is the transverse displacement of the string at the point x at time t, r is the linear density of the string (mass per unit length), G is the tension force. For a homogeneous string at constant tension the value of c=(G/r)^1/2 is the velocity of propagation of waves.

A string of length L fixed at the ends possesses a set of natural modes with frequencies f_n=nc/L. In classic Melde’s experiment, in a case of periodic variation of the string tension in time with a frequency 2f₀, where f₀ corresponds to one of the modes, parametric oscillations excite of this mode; in the equation of the string the factor G in this case is a function of time of period 1/2f₀, and the a linear wave equation describes the initial stage of development of the parametric instability. Oscillations of the tension force represent the pump that is a source of energy, providing the parametric excitation.

If you apply the pump alternately at some low and tripled higher frequency, it is possible to provide a parametric excitation of short-wave and long-wave patterns turn by turn with the transfer of the spatial phase from one pattern to another, so that for the complete period of the pump modulation the phase variable will be tripled [1, 2]. In the presence of dissipation, compression in other directions in the state space of the system will ensure the presence of the attractor of Smale - Williams type. Conceptually, the easiest way to implement this idea relates for a system with periodic boundary conditions, i.e., closed in a ring (although it is not the easiest option to be arranged in an experiment).

Let the tension of the string varies in time according to the relation

where the coefficients a₂, a₆ slowly oscillate in time with some period T, being turn by turn relatively large or small. Concretely, we set

So, the pump contains the components of frequencies 2w₀ and 6w₀ with amplitudes modulated in time. The distribution of the mass over the string is supposed to be slightly non-uniform, depending on the spatial coordinate as

where k₀=w₀/c₀, c₀=(G₀/r₀)^1/2 In addition, we introduce the linear and nonlinear dissipation, adding a nonlinear term in the right-hand part of the equation proportional to the time derivative. The presence of nonlinear dissipation will stabilize the parametric instability, but for the system considered there is another essential circumstance that the cubic nonlinearity provides the emergence of third-harmonic component of the oscillatory-wave motions. Finally, we add a linear term responsible for damping of zero wave number oscillations. Using appropriate normalization of variables and parameters, we set c₀=1, k₀=w₀, and arrive at a system described by a partial differential equation

Let us impose periodic boundary conditions

The system operation take place as follows.

Suppose that by pumping at the frequency of 2w₀ in the systen the standing wave is excited at a frequency w₀ with the wave number k₀, in which the location of the nodes and anti-nodes are characterized by a constant Ф, so that in a rough approximation y~cosw₀t sin(k₀x+Ф). The amplitude is stabilized at some final level by the nonlinear dissipation. Additionally, due to its presence, the oscillatory-wave motion will have a component at the third harmonic of the form sin 3w₀t sin(3k₀x+3Ф).

Next, we have a stage, when the pump amplitude at the frequency 2w₀ becomes small, and the oscillations at the frequency w₀ decay. But now, the amplitude of pumping at the frequency 6w₀ becomes large and sufficient for parametric excitation of the standing wave of frequency 3w₀ and wave number 3k₀. This wave is born from the perturbation remaining from the previous stage of the process, and hence is characterized by the spatial phase 3Ф.

Then, agqin the stage comes of pumping at frequency 2w₀, and the short-wave pattern decays. The initial oscillatory-wave perturbation for parametric excitation of frequency w₀ and wave number k₀ is provided due to the component sin 3w₀t sin(3k₀x+3Ф) remaining from the previous stage in combination with the component of frequency 2w₀ and wave number4k₀, which presents because of the pumping and the spatial variation of the density of the string (with fixed spatial phase). As a result, the updated spatial phase of the arisen pattern Ф' appears as transformed by triple expanding circle map Ф'=-3Ф+const. This is a map with chaotic dynamics characterized by the positive Lyapunov exponent L=ln3=1.0986...

In other directions in the state space of the system compression of the phase volume will take place; for a map describing evolution on a period of pump modulation, attractor of Smale-Williams will occur.

With parameters

the results of numerical integration of the partial differential equation are illustrated by the animation of the evolution of the standing wave patterns on time and by the 3D diagram. In this diagram the spatial distributions are drawn with the time step equal to the period of high-frequency oscillations (so, the high-frequency component is not visible). Observe that the long-wave and shoet-wave patterns arise turn by turn, and their spatial phases vary from one period of pump modulation to another one chaotically.

One can verify that for a period of pomp modulation the spatial phase indeed undergoes the trple expanding transformation. To do this, in the process of computations on many periods of modulation we evaluate the spatial phases at time instants t_n=nT. Presenting the data graphically in coordinates Ф_n, Ф_n+1 observe that the disposition of the branches is of such nature that one turn around circle for pre-image implies three turns for the image in backward direction. The neighbor diagram shows a stroboscopic portrait of the attractor in projection on the plane of variables of the string displacements at two points separated from one another by a quarter of the basic wavelength. It is a Smale-Williams solenoid although its transversal fractal structure is indistinguishable because of strong transversal compression. The object looks like a closed curve, but motion of the representative point on it occurs as jumps according to the triple expanding circle map.

Accounting that the parametric excitation takes place at the wave numbers k₀ and 3k₀, one can compose a reduced finite-dimensional model; the dynamical variables will be the time-dependent coefficients of expansion of the string form over the respective spatial modes. This is a set of ordinary differential equations of 12-th order, which is cumbersome enough, and we do not show it here. It may be found in Ref. [3], and it is shown there that the numerical simulations based on the partial differential equation and on the reduced model agree well, at least in a certain parameter domain of interest. Besides, in [3] some other situations of parametric excitation of patterns are considered, with different frequency ratios, when the expanding factors for the spatial phase are determined by odd numbers from 3 to 11.

In Refs. [1, 2] it is shown that the hyperbolic attractor may be obtained in a system with fixed ends of the string. It may be done by increase of the system length and with spatial dependence of the linear dissipation according to certain profile with increase of the dissipation at te ends of the system.

The page is elaborated under support of RSF grant No 15-12-20035, in the Udmurt State University (Izhevsk).

REFERENCES

[1] Isaeva O.B., Kuznetsov A.S., Kuznetsov S.P. Hyperbolic chaos of standing wave patterns generated parametrically by a modulated pump source. Phys. Rev. E. 2013. V. 87, 040901.

[2] Isaeva O.B., Kuznetsov A.S. Kuznetsov S.P. Hyperbolic chaos in parametric oscillations of a string. Rus. J. Nonlin. Dyn. 2013. V. 9. No. 1, 3-10.

[3] Kuznetsov A.S., Kuznetsov S.P., Kruglov V.P. Hyperbolic chaos in systems with parametrically excited patterns of standing waves. Rus. J. Nonlin. Dyn. 2014. V. 10. No. 3, 265-277.