The influence of slow processes on the probability distribution of fast random processes is investigated. By reviewing four examples we show that such influence is apparently of a universal character and that, in some cases, this universality is of multifractal form. As our examples we consider theoretically stochastic resonance, turbulent jets with acoustic forcing, and two problems studied experimentally by Shnoll on the influence of the Earth’s slow rotation on the probability distribution for the velocities of model Brownian particles and on alpha decay. In the case of stochastic resonance, the slow process is a low frequency, harmonic, external force. In the case of turbulent jets, the slow process is acoustic forcing. In the models based on Shnoll’s experiments, the slow processes are inertial forces arising from the rotation of the Earth, both about its own axis and about the Sun. It is shown that all of these slow processes cause changes in the probability distributions for the velocities of fast processes interacting with them, and that these changes are similar in form.

By using small deviations from the stationary solution of the Navier-Stokes equation, the problem of linear instability of a plane submerged subsonic jet is considered. In the approximation of weak divergence of the jet, this problem reduces to a linear not self-adjoint boundary value problem with a given behavior of the variables at large values of the transversal coordinate. The solution of this boundary value problem allow us to calculate the gain factor and the phase velocity of hydrodynamical waves as functions of frequency and of distance from the nozzle. We have found that the dependence of the gain factor on the frequency has a resonant character. As the distance from the nozzle increases, the dependence of the gain factor on the frequency becomes more narrow and the maximum of that shifts into the small frequency region. Hence, the hydrodynamical waves become more and more coherent. The obtained results are in good agreement with experimental data.

Brief description of the properties of turbulent flows in submerged subsonic jets is given. Within the jet’s initial part, where turbulent pulsations are sufficiently small, the asymptotic Krylov-Bogolyubov method is used for calculating of the jet processes. It is shown that the results оf the calculations are in good coincidence with experimental data.

It shown that, in the case that the system’s variables can be separated into fast and slow ones, synchronization of only slow variables is possible whereas fast variables remain nonsynchronized. In particular, the phase shift for oscillations of two coupled self-oscillatory systems may be such a slow variable. Only in this case the notion of phase synchronization is physically meaningful.

The memory retrieval process of number problems with external noise is studied with the use of the Bonhoeffer-van der Pol oscillator model. Three cell assembly responses are simulated, coding one true number and two neighboring erroneous. The time of a correct response, Tc, was averaged over statistical assemblies of numerous trials. It is demonstrated that Tc takes a minimum value for a certain noise intensity. This result correlates well with experimental data by Usher and Feingold (2000). The location of the minimum as a function of the time delay between two consecutive simulation trials is investigated.

Each dynamical system can be characterized by a set of natural oscillations called normal oscillations. The number of normal oscillations depends on the number of degrees of freedom of the system. For linear systems the number of normal oscillations is exactly equal to the number of degrees of freedom, whereas for nonlinear systems this is not necessarily so. For continuous systems having an infinitely large number of degrees of freedom the number of normal oscillations is infinite, but in the case of bounded systems it is denumerable.

We study here different modes of self-oscillations in ring Toda chain with negative friction. Assuming that at small friction the shape of self-oscillations is close to one of the known Toda soliton-like solutions we use analytical methods in combination with numerical ones for study о the self-oscillations. We show that а Toda chain consisting of N elements possesses N+1 different modes of self-oscillations. Among them two modes correspond to left and right rotations of the chain as a whole with a constant velocity. Each of the other modes represents а combination of moving soliton and the rotation with a velocity depending on the mode number. Only for the mode corresponding to anti-phase oscillation of the chain neighboring elements (such oscillation are possible for an even N) the constant component of velocity is equal to zero.

Studies of the interaction of three or more self-oscillatory systems began many years ago [333, 263, 232, 221, 283]. In particular, interest in this phenomenon was caused by the elaborations of frequency standards based on many generators coupled in a unified synchronized system (see, e.g., [28]). The accuracy of such frequency standards was studied in [357, 150, 222, 158].

In this book the modern theory of both regular and chaotic nonlinear oscillations is set out, primarily, as applied to mechanical problems. The material is presented in a nontraditional manner with em

Studies of self-oscillatory systems with one degree of freedom are markedly simpler than these of systems with two or more degrees of freedom. Therefore, whenever feasible, investigators endeavor to describe any real system by a model with one degree of freedom. In many cases such a model adequately depicts the most essential features of the processes studied.

The simplest equation for a nonlinear oscillator under an external action varying its natural frequency is $$\ddot{x} + 2\delta \dot{x} + \omega _{0}^{2}\left( {1 + f(t)} \right)F(x) = 0,$$ (13.1) where f (t) is a function of time, and F(x) is a nonlinear function of x.

As an example of phenomena which lead to noise-induced ordering we consider noise-induced phase transitions resulting in the excitation of noise-induced oscillations. Our first example is a pendulum with the randomly vibrated suspension axis. We discuss a control of noise-induced oscillations in this system and an effect of the on-off intermittency. The second example is a model for the interaction between tropical atmosphere and ocean water to explain a generation of an annual signal in tropics. This model demonstrates noise-induced oscillations which closely resemble in its form chaotic oscillations.

A model for human vocal folds in the form of two spring-suspended plates, which can collide with one another, is considered. It is shown that due to air flow between the plates excitation of chaotic self-oscillations occurs. In this process the shape of oscillations of air flow volume velocity was found to be close to observed experimentally.

A model of human vocal folds in the form of two spring-loaded plates, which can collide with one another, is considered. It is shown that due to air flow between the plates excilation of chaotic self-oscillations occurs. In this process the shape of oscillations of flow volume velocity was found to be close to observed experimentally.

А number of examples of the transport of Brownian particles induced by nonequilibrium fluctuations is considered. The results of approximate analytical calculations for the averaged particle velocity in periodic ratchei—like potential are presented. An analogy between fluctnation—induced transport and well known in mechanics vibrational transport is discussed.