Featured Researches

Chaotic Dynamics

Bicritical Behavior of Period Doublings in Unidirectionally-Coupled Maps

We study the scaling behavior of period doublings in two unidirectionally-coupled one-dimensional maps near a bicritical point where two critical lines of period-doubling transition to chaos in both subsystems meet. Note that the bicritical point corresponds to a border of chaos in both subsystems. For this bicritical case, the second response subsystem exhibits a new type of non-Feigenbaum critical behavior, while the first drive subsystem is in the Feigenbaum critical state. Using two different methods, we make the renormalization group analysis of the bicritical behavior and find the corresponding fixed point of the renormalization transformation with two relevant eigenvalues. The scaling factors obtained by the renormalization group analysis agree well with those obtained by a direct numerical method.

Read more
Chaotic Dynamics

Bifurcation analysis of the plane sheet pinch

A numerical bifurcation analysis of the electrically driven plane sheet pinch is presented. The electrical conductivity varies across the sheet such as to allow instability of the quiescent basic state at some critical Hartmann number. The most unstable perturbation is the two-dimensional tearing mode. Restricting the whole problem to two spatial dimensions, this mode is followed up to a time-asymptotic steady state, which proves to be sensitive to three-dimensional perturbations even close to the point where the primary instability sets in. A comprehensive three-dimensional stability analysis of the two-dimensional steady tearing-mode state is performed by varying parameters of the sheet pinch. The instability with respect to three-dimensional perturbations is suppressed by a sufficiently strong magnetic field in the invariant direction of the equilibrium. For a special choice of the system parameters, the unstably perturbed state is followed up in its nonlinear evolution and is found to approach a three-dimensional steady state.

Read more
Chaotic Dynamics

Billiard Sequences and the Property of Splittability of Integrable Hamilton Systems

The paper establishes the property of splittability of billiard boundary sequences in n dimensional cube into subsequences of fractional parts. This reveals a new property of integrable and weak perturbated Hamilton systems: under a simple assumption, the boundary motion of elliptic orbits on stable KAM tori, if considering in cartesian coordinates, can be splitted into a countable set of discrete rotations. The rate of the split process, expressed in terms of some exceptional sets density, in dependence of number-theoretical characteristics of the orbits frequencies, is also examined.

Read more
Chaotic Dynamics

Birth of resonances in the spin-orbit problem of Celestial Mechanics

The behaviour of resonances in the spin-orbit coupling in Celestial Mechanics is investigated. We introduce a Hamiltonian nearly-integrable model describing an approximation of the spin-orbit interaction. A parametric representation of periodic orbits is presented. We provide explicit formulae to compute the Taylor series expansion in the perturbing parameter of the function describing this parametrization. Then we compute approximately the radius of convergence providing an indication of the stability of the periodic orbit. This quantity is used to describe the different probabilities of capture into resonance. In particular, we notice that for low values of the orbital eccentricity the only significative resonance is the synchronous one. Higher order resonances (including 1:2, 3:2, 2:1) appear only as the orbital eccentricity is increased.

Read more
Chaotic Dynamics

Boosting Sonoluminescence

Single bubble sonoluminescence has been experimentally produced through a novel approach of optimized sound excitation. A driving consisting of a first and second harmonic with selected amplitudes and relative phase results in an increase of light emission compared to sinusoidal driving. We achieved a raise of the maximum photo current of up to 300% with the two-mode sound signal. Numerical simulations of multimode excitation of a single bubble are compared to this result.

Read more
Chaotic Dynamics

Border Collision Bifurcations in Two Dimensional Piecewise Smooth Maps

Recent investigations on the bifurcations in switching circuits have shown that many atypical bifurcations can occur in piecewise smooth maps which can not be classified among the generic cases like saddle-node, pitchfork or Hopf bifurcations occurring in smooth maps. In this paper we first present experimental results to establish the need for the development of a theoretical framework and classification of the bifurcations resulting from border collision. We then present a systematic analysis of such bifurcations by deriving a normal form --- the piecewise linear approximation in the neighborhood of the border. We show that there can be eleven qualitatively different types of border collision bifurcations depending on the parameters of the normal form, and these are classified under six cases. We present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur. This theoretical framework will help in explaining bifurcations in all systems which can be represented by two dimensional piecewise smooth maps.

Read more
Chaotic Dynamics

Breakdown of Modulational Approximations in Nonlinear Wave Interaction

In this work we investigate the validity limits of the modulational approximation as a method to describe the nonlinear interaction of conservative wave fields. We focus on a nonlinear Klein-Gordon equation and suggest that the breakdown of the approximation is accompanied by a transition to regimes of spatiotemporal chaos.

Read more
Chaotic Dynamics

Bulk Properties of Anharmonic Chains in Strong Thermal Gradients: Non-Equilibrium ϕ 4 Theory

We study nonequilibrium properties of a one-dimensional lattice Hamiltonian with quartic interactions in strong thermal gradients. Nonequilibrium temperature profiles, T(x), are found to develop significant curvature and boundary jumps. From the determination of the bulk thermal conductivity, we develop a quantitative description of T(x) including the jumps.

Read more
Chaotic Dynamics

Can Strange Nonchaotic Dynamics be induced through Stochastic Driving?

Upon addition of noise, chaotic motion in low-dimensional dynamical systems can sometimes be transformed into nonchaotic dynamics: namely, the largest Lyapunov exponent can be made nonpositive. We study this phenomenon in model systems with a view to understanding the circumstances when such behaviour is possible. This technique for inducing ``order'' through stochastic driving works by modifying the invariant measure on the attractor: by appropriately increasing measure on those portions of the attractor where the dynamics is contracting, the overall dynamics can be made nonchaotic, however {\it not} a strange nonchaotic attractor. Alternately, by decreasing measure on contracting regions, the largest Lyapunov exponent can be enhanced. A number of different chaos control and anticontrol techniques are known to function on this paradigm.

Read more
Chaotic Dynamics

Cascades in helical turbulence

The existence of a second quadratic inviscid invariant, the helicity, in a turbulent flow leads to coexisting cascades of energy and helicity. An equivalent of the four-fifth law for the longitudinal third order structure function, which is derived from energy conservation, is easily derived from helicity conservation cite{Procaccia,russian}. The ratio of dissipation of helicity to dissipation of energy is proportional to the wave-number leading to a different Kolmogorov scale for helicity than for energy. The Kolmogorov scale for helicity is always larger than the Kolmogorov scale for energy so in the high Reynolds number limit the flow will always be helicity free in the small scales, much in the same way as the flow will be isotropic and homogeneous in the small scales. A consequence is that a pure helicity cascade is not possible. The idea is illustrated in a shell model of turbulence.

Read more

Ready to get started?

Join us today