Chaotic Dynamics

An improved mass-spring model for a dripping faucet is presented. The model is constructed based on the numerical results which we recently obtained from fluid dynamical calculations. Both the fluid dynamical calculations and the present mass-spring model exhibit a variety of complex behavior including transition to chaos in good agreement with experiments. Further, the mass-spring model reveals fundamental dynamics inherent in the dripping faucet system.

Bifurcations associated with stability of the saddle fixed point of the Poincaré map, arising from the unstable equilibrium point of the potential, are investigated in a forced Duffing oscillator with a double-well potential. One interesting behavior is the dynamic stabilization of the saddle fixed point. When the driving amplitude is increased through a threshold value, the saddle fixed point becomes stabilized via a pitchfork bifurcation. We note that this dynamic stabilization is similar to that of the inverted pendulum with a vertically oscillating suspension point. After the dynamic stabilization, the double-well Duffing oscillator behaves as the single-well Duffing oscillator, because the effect of the central potential barrier on the dynamics of the system becomes negligible.

We review some properties of dynamical systems with slowly varying parameters, when a parameter is moved through a bifurcation point of the static system. Bifurcations with a single zero eigenvalue may create hysteresis cycles, whose area scales in a nontrivial way with the adiabatic parameter. Hopf bifurcations lead to the delayed appearance of oscillations. Feedback control theory motivates the study of a bifurcation with double zero eigenvalue, in which this delay is suppressed.

Dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively. Since for such systems the KS-entropy diverges we analyse the difference between the total entropy of a noisy system and the entropy of the noise itself. We show that this quantity is non negative and in the weak noise limit is conjectured to tend to the KS-entropy of the deterministic system. In particular, we consider one-dimensional systems with noise described by a finite-dimensional kernel, for which the Frobenius-Perron operator can be represented by a finite matrix.

An approximate Poincare map near equally strong multiple resonances is reduced by means the method of averaging. Near the resonance junction of three degrees of freedom, we find that some homoclinic orbits ``whiskers'' in single resonance lines survive and form nearly periodic orbits, each of which looks like a pair of homoclinic orbits.

The Burridge-Knopoff model of earthquake faults with viscous friction is equivalent to a van der Pol-FitzHugh-Nagumo model for excitable media with elastic coupling. The lubricated creep-slip friction law we use in the Burridge-Knopoff model describes the frictional sliding dynamics of a range of real materials. Low-dimensional structures including synchronized oscillations and propagating fronts are dominant, in agreement with the results of laboratory friction experiments. Here we explore the dynamics of fronts in elastic excitable media.

We study the dynamics of "finger" formation in Laplacian growth without surface tension in a channel geometry (the Saffman-Taylor problem). Carefully determining the role of boundary geometry, we construct field equations of motion, these central to the analytic power we can here exercise. We consider an explicit analytic class of maps to the physical space, a basis of solutions for infinite fluid in an infinitely long channel, characterized by meromorphic derivatives. We verify that these maps never lose analyticity in the course of temporal evolution, thus justifying the underlying machinery. However, the great bulk of these solutions can lose conformality in time, this the circumstance of finite-time singularities. By considerations of the nature of the analyticity of all these solutions, we show that those free of such singularities inevitably result in a {\em single} asymptotic "finger". This is purely nonlinear behavior: the very early "finger" actually already has a waist, this having signalled the end of any linear regime. The single "finger" has nevertheless an arbitrary width determined by initial conditions. This is in contradiction with the experimental results that indicate selection of a finger of width 1/2. In the last part of this paper we motivate that such a solution can be determined by the boundary conditions when the fluid is finite. This is a strong signal that {\em finiteness} is determinative of pattern selection.

We study the effects of time delayed linear and nonlinear feedbacks on the dynamics of a single Hopf bifurcation oscillator. Our numerical and analytic investigations reveal a host of complex temporal phenomena such as phase slips, frequency suppression, multiple periodic states and chaos. Such phenomena are frequently observed in the collective behavior of a large number of coupled limit cycle oscillators. Our time delayed feedback model offers a simple paradigm for obtaining and investigating these temporal states in a single oscillator.We construct a detailed bifurcation diagram of the oscillator as a function of the time delay parameter and the driving strengths of the feedback terms. We find some new states in the presence of the quadratic nonlinear feedback term with interesting characteristics like birhythmicity, phase reversals, radial trapping, phase jumps and spiraling patterns in the amplitude space. Our results may find useful applications in physical, chemical or biological systems.

Vortex line and magnetic line representations are introduced for description of flows in ideal hydrodynamics and MHD, respectively. For incompressible fluids it is shown that the equations of motion for vorticity Ω and magnetic field with the help of this transformation follow from the variational principle. By means of this representation it is possible to integrate the system of hydrodynamic type with the Hamiltonian H=∫|Ω|dr . It is also demonstrated that these representations allow to remove from the noncanonical Poisson brackets, defined on the space of divergence-free vector fields, degeneracy connected with the vorticity frozenness for the Euler equation and with magnetic field frozenness for ideal MHD. For MHD a new Weber type transformation is found. It is shown how this transformation can be obtained from the two-fluid model when electrons and ions can be considered as two independent fluids. The Weber type transformation for ideal MHD gives the whole Lagrangian vector invariant. When this invariant is absent this transformation coincides with the Clebsch representation analog introduced by Zakharov and Kuznetov.

This is an introductory course on fully developed turbulence. It discusses: in Lecture 1: the Navier Stokes equations, existence of solutions, statistical description, energy balance and cascade picture; in Lecture 2: the Kolmogorov theory of three-dimensional turbulence versus intermittency, the Kraichnan-Batchelor theory of two-dimensional turbulence; in Lecture 3: the Richardson dispersion law and the breakdown of the Lagrangian flow; in Lecture 4: direct and inverse cascades and intermittency in the Kraichnan model of passive advection.