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The time-dependent vulnerability of synchronized states is shown for a complex network composed of electronic circuits. We demonstrate that disturbances to the local dynamics of network units can produce different outcomes to synchronization depending on the event timing. We address such time dependence by systematically perturbing the synchronized system at instants of time equally distributed along its trajectory. We find the time instants at which the perturbation desynchronizes the network to be complicatedly mixed with the ones that restore synchronization. Additionally, we characterize perturbation sets obtained for consecutive instants of time by defining a safety index between them. Finally, we demonstrate that the vulnerability is due to state space sensitivities occurring along synchronized trajectories.

We analyze the effect of the Basset history force on the sedimentation of nearly neutrally buoyant particles, exemplified by marine snow, in a three-dimensional turbulent flow. Particles are characterized by Stokes numbers much smaller than unity, and still water settling velocities, measured in units of the Kolmogorov velocity, of order one. The presence of the history force in the Maxey-Riley equation leads to individual trajectories which differ strongly from the dynamics of both inertial particles without this force, and ideal settling tracers. When considering, however, a large ensemble of particles, the statistical properties of all three dynamics become more similar. The main effect of the history force is a rather slow, power-law type convergence to an asymptotic settling velocity of the center of mass, which is found numerically to be the settling velocity in still fluid. The spatial extension of the ensemble grows diffusively after an initial ballistic growth lasting up to ca. one large eddy turnover time. We demonstrate that the settling of the center of mass for such light aggregates is best approximated by the settling dynamics in still fluid found with the history force, on top of which fluctuations appear which follow very closely those of the turbulent velocity field., Comment: 15 pages, 14 figures

We investigate chaotic dynamical systems for which the intensity of trajectories might grow unlimited in time. We show that (i) the intensity grows exponentially in time and is distributed spatially according to a fractal measure with an information dimension smaller than that of the phase space,(ii) such exploding cases can be described by an operator formalism similar to the one applied to chaotic systems with absorption (decaying intensities), but (iii) the invariant quantities characterizing explosion and absorption are typically not directly related to each other, e.g., the decay rate and fractal dimensions of absorbing maps typically differ from the ones computed in the corresponding inverse (exploding) maps. We illustrate our general results through numerical simulation in the cardioid billiard mimicking a lasing optical cavity, and through analytical calculations in the baker map., Comment: 7 pages, 5 figures

A systematic investigation of the effect of the history force on particle advection is carried out for both heavy and light particles. General relations are given to identify parameter regions where the history force is expected to be comparable with the Stokes drag. As an illustrative example, a paradigmatic two-dimensional flow, the von K\'arm\'an flow is taken. For small (but not extremely small) particles all investigated dynamical properties turn out to heavily depend on the presence of memory when compared to the memoryless case: the history force generates a rather non-trivial dynamics that appears to weaken (but not to suppress) inertial effects, it enhances the overall contribution of viscosity. We explore the parameter space spanned by the particle size and the density ratio, and find a weaker tendency for accumulation in attractors and for caustics formation. The Lyapunov exponent of transients becomes larger with memory. Periodic attractors are found to have a very slow, $t^{-1/2}$ type convergence towards the asymptotic form. We find that the concept of snapshot attractors is useful to understand this slow convergence: an ensemble of particles converges exponentially fast towards a snapshot attractor, which undergoes a slow shift for long times.

Chaos is an inherently dynamical phenomenon traditionally studied for trajectories that are either permanently erratic or transiently influenced by permanently erratic ones lying on a set of measure zero. The latter gives rise to the final state sensitivity observed in connection with fractal basin boundaries in conservative scattering systems and driven dissipative systems. Here we focus on the most prevalent case of undriven dissipative systems, whose transient dynamics fall outside the scope of previous studies since no time-dependent solutions can exist for asymptotically long times. We show that such systems can exhibit positive finite-time Lyapunov exponents and fractal-like basin boundaries which nevertheless have codimension one. In sharp contrast with its driven and conservative counterparts, the settling rate to the (fixed-point) attractors grows exponentially in time, meaning that the fraction of trajectories away from the attractors decays super-exponentially. While no invariant chaotic sets exist in such cases, the irregular behavior is governed by transient interactions with transient chaotic saddles, which act as effective, time-varying chaotic sets., Comment: To appear in Physical Review Letters

We analyse the effect of the Basset history force on the sedimentation or rising of inertial particles in a two-dimensional convection flow. When memory effects are neglected, the system exhibits rich dynamics, including periodic, quasi-periodic and chaotic attractors. Here we show that when the full advection dynamics is considered, including the history force, both the nature and the number of attractors change, and a fractalization of their basins of attraction appears. In particular, we show that the history force significantly weakens the horizontal diffusion and changes the speed of sedimentation or rising. The influence of the history force is dependent on the size of the advected particles, being stronger for larger particles., Comment: 13 pages, 16 figures

Motivated by applications in optics and acoustics we develop a dynamical-system approach to describe absorption in chaotic systems. We introduce an operator formalism from which we obtain (i) a general formula for the escape rate $\kappa$ in terms of the natural conditionally-invariant measure of the system; (ii) an increased multifractality when compared to the spectrum of dimensions $D_q$ obtained without taking absorption and return times into account; and (iii) a generalization of the Kantz-Grassberger formula that expresses $D_1$ in terms of $\kappa$, the positive Lyapunov exponent, the average return time, and a new quantity, the reflection rate. Simulations in the cardioid billiard confirm these results.

The dynamics of a spring-block train placed on a moving conveyor belt is investigated both by simple experiments and computer simulations. The first block is connected by spring to an external static point, and due to the dragging effect of the belt the blocks undergo complex stick-slip dynamics. A qualitative agreement with the experimental results can only be achieved by taking into account the spatial inhomogeneity of the friction force on the belt's surface, modeled as noise. As a function of the velocity of the conveyor belt and the noise strength, the system exhibits complex, self-organized critical, sometimes chaotic dynamics and phase transition-like behavior. Noise induced chaos and intermittency is also observed. Simulations suggest that the maximum complexity of the dynamical states is achieved for a relatively small number of blocks, around five., Comment: 13 pages, 22 figures, updated figure 13

There are numerous physical situations in which a hole or leak is introduced in an otherwise closed chaotic system. The leak can have a natural origin, it can mimic measurement devices, and it can also be used to reveal dynamical properties of the closed system. In this paper we provide an unified treatment of leaking systems and we review applications to different physical problems, both in the classical and quantum pictures. Our treatment is based on the transient chaos theory of open systems, which is essential because real leaks have finite size and therefore estimations based on the closed system differ essentially from observations. The field of applications reviewed is very broad, ranging from planetary astronomy and hydrodynamical flows, to plasma physics and quantum fidelity. The theory is expanded and adapted to the case of partial leaks (partial absorption/transmission) with applications to room acoustics and optical microcavities in mind. Simulations in the lima .con family of billiards illustrate the main text. Regarding billiard dynamics, we emphasize that a correct discrete time representation can only be given in terms of the so- called true-time maps, while traditional Poincar \'e maps lead to erroneous results. We generalize Perron-Frobenius-type operators so that they describe true-time maps with partial leaks., Comment: Review article, 54 pages, final published version

Inertial particles suspended in many natural and industrial flows undergo coagulation upon collisions and fragmentation if their size becomes too large or if they experience large shear. Here we study this coagulation-fragmentation process in time-periodic incompressible flows. We find that this process approaches an asymptotic, dynamical steady state where the average number of particles of each size is roughly constant. We compare the steady-state size distributions corresponding to two fragmentation mechanisms and for different flows and find that the steady state is mostly independent of the coagulation process. While collision rates determine the transient behavior, fragmentation determines the steady state. For example, for fragmentation due to shear, flows that have very different local particle concentrations can result in similar particle size distributions if the temporal or spatial variation of shear forces is similar., Comment: 8 pages, 7 figures

Inertial particles advected in chaotic flows often accumulate in strange attractors. While moving in these fractal sets they usually approach each other and collide. Here we consider inertial particles aggregating upon collision. The new particles formed in this process are larger and follow the equation of motion with a new parameter. These particles can in turn fragment when they reach a certain size or shear forces become sufficiently large. The resulting system consists of a large set of coexisting dynamical systems with a varying number of particles. We find that the combination of aggregation and fragmentation leads to an asymptotic steady state. The asymptotic particle size distribution depends on the mechanism of fragmentation. The size distributions resulting from this model are consistent with those found in rain drop statistics and in stirring tank experiments., Comment: 4 pages, 4 figures

In order to simulate observational and experimental situations, we consider a leak in the phase space of a chaotic dynamical system. We obtain an expression for the escape rate of the survival probability applying the theory of transient chaos. This expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble. With a subtle choice of the initial ensemble, we obtain an equivalence to the classical problem of Poincare recurrences in closed systems, which is treated in the same framework. Finally, we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle in hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential and asymptotic power-law decays of the survival probability., Comment: Corrected version, as published. 12 pages, 9 figures

We obtain a description of the Poincar\'e recurrences of chaotic systems in terms of the ergodic theory of transient chaos. It is based on the equivalence between the recurrence time distribution and an escape time distribution obtained by leaking the system and taking a special initial ensemble. This ensemble is atypical in terms of the natural measure of the leaked system, the conditionally invariant measure. Accordingly, for general initial ensembles, the average recurrence and escape times are different. However, we show that the decay rate of these distributions is always the same. Our results remain valid for Hamiltonian systems with mixed phase space and validate a split of the chaotic saddle in hyperbolic and non-hyperbolic components., Comment: 4 pages and 4 figures, final published version

Aerosols under chaotic advection often approach a strange attractor. They move chaotically on this fractal set but, in the presence of gravity, they have a net vertical motion downwards. In practical situations, observational data may be available only at a given level, for example at the ground level. We uncover two fractal signatures of chaotic advection of aerosols under the action of gravity. Each one enables the computation of the fractal dimension $D_{0}$ of the strange attractor governing the advection dynamics from data obtained solely at a given level. We illustrate our theoretical findings with a numerical experiment and discuss their possible relevance to meteorology., Comment: Accepted for publication in Phys. Rev. E (Rapid Communications)

Many examples of chemical and biological processes take place in large-scale environmental flows. Such flows generate filamental patterns which are often fractal due to the presence of chaos in the underlying advection dynamics. In such processes, hydrodynamical stirring strongly couples into the reactivity of the advected species and might thus make the traditional treatment of the problem through partial differential equations difficult. Here we present a simple approach for the activity in in-homogeneously stirred flows. We show that the fractal patterns serving as skeletons and catalysts lead to a rate equation with a universal form that is independent of the flow, of the particle properties, and of the details of the active process. One aspect of the universality of our appraoch is that it also applies to reactions among particles of finite size (so-called inertial particles)., Comment: 10 pages

In their paper Cohen and Rondoni severely question the physical relevance of studies of transport by means of multibaker maps and other classes of dynamics considering the motion of independen particles - like e.g. the Lorentz gas. We argue that this is to a large extent due to inappropriate interpretation of the models. In particular, the arguments concerning the lack of local thermodynamic equilibrium are inconsistent for the models previously worked out by ourselves., Comment: 3 pages, revtex4 -- with a 4th item addressed

We study the evolution of a localized perturbation in a chemical system with multiple homogeneous steady states, in the presence of stirring by a fluid flow. Two distinct regimes are found as the rate of stirring is varied relative to the rate of the chemical reaction. When the stirring is fast localized perturbations decay towards a spatially homogeneous state. When the stirring is slow (or fast reaction) localized perturbations propagate by advection in form of a filament with a roughly constant width and exponentially increasing length. The width of the filament depends on the stirring rate and reaction rate but is independent of the initial perturbation. We investigate this problem numerically in both closed and open flow systems and explain the results using a one-dimensional "mean-strain" model for the transverse profile of the filament that captures the interplay between the propagation of the reaction-diffusion front and the stretching due to chaotic advection., Comment: to appear in Chaos, special issue on Chaotic Flow

In summer 1997 we were sitting with Bob Dorfman and a few other friends interested in chaotic systems and transport theory on a terrace close to Oktogon in Budapest. While taking our (decaf) coffee after a very nice Italian meal, we discussed about logarithmic divergences in the entropy production of systems with absorbing boundary conditions and their consequences for the escape-rate formalism. It was guessed at that time that the problem could be resolved by a careful discussion of the physical content of the absorbing boundary conditions. To our knowledge a thorough analysis of this long-standing question is still missing. We dedicate it hereby to Bob on occasion of his 65th birthday., Comment: 16 pages; RevTex 4 with graphicx package; eps-figures

A consistent description of shear flow and the accompanied viscous heating as well the associated entropy balance is given in the framework of a deterministic dynamical system. A laminar shear flow is modeled by a Hamiltonian multibaker map which drives velocity and temperature fields. In an appropriate macroscopic limit one recovers the Navier-Stokes and heat conduction equations along with the associated entropy balance. This indicates that results of nonequilibrium thermodynamics can be described by means of an abstract, sufficiently chaotic and mixing dynamics. A thermostating algorithm can also be incorporated into this framework., Comment: 11 pages; RevTex with multicol+graphicx packages; eps-figures

A consistent description of a shear flow, the accompanied viscous heating, and the associated entropy balance is given in the framework of a deterministic dynamical system, where a multibaker dynamics drives two fields: the velocity and the temperature distributions. In an appropriate macroscopic limit their transport equations go over into the Navier-Stokes and the heat conduction equation of viscous flows. The inclusion of an artificial heat sink can stabilize steady states with constant temperatures. It mimics a thermostating algorithm used in non-equilibrium molecular-dynamics simulations., Comment: LaTeX 2e (epl.cls + sty-files for Europhys Lett included); 7 pages + 1 eps-figure

A generalized multibaker map with periodic boundary conditions is shown to model boundary-driven transport, when the driving is applied by a ``perturbation'' of the dynamics localised in a macroscopically small region. In this case there are sustained density gradients in the steady state. A non-uniform stationary temperature profile can be maintained by incorporating a heat source into the dynamics, which deviates from the one of a bulk system only in a (macroscopically small) localized region such that a heat (or entropy) flux can enter an attached thermostat only in that region. For these settings the relation between the average phase-space contraction, the entropy flux to the thermostat and irreversible entropy production is clarified for stationary and non-stationary states. In addition, thermoelectric cross-effects are described by a multibaker chain consisting of two parts with different transport properties, modelling a junction between two metals., Comment: 15 pages (revtex), 6 figures (eps)

A consistent description of simultaneous heat and particle transport, including cross effects, and the associated entropy balance is given in the framework of a deterministic dynamical system. This is achieved by a multibaker map where, besides the phase-space density of the multibaker, a second field with appropriate source terms is included in order to mimic a spatial temperature distribution and its time evolution. Conditions are given to ensure consistency in an appropriately defined continuum limit with the thermodynamic entropy balance. They leave as the only free parameter of the model the entropy flux let directly into a surroundings. If it vanishes in the bulk, the transport properties of the model are described by the thermodynamic transport equations. Another choice leads to a uniform temperature distribution. It represents transport problems treated by means of a thermostatting algorithm, similar to the one considered in non-equilibrium molecular dynamics., Comment: 18 pages, 3 postscript figures

We give a thermodynamically consistent description of simultaneous heat and particle transport, as well as of the associated cross-effects, in the framework of a chaotic dynamical system, a generalized multibaker map. Besides the density, a second field with appropriate source terms is included in order to mimic, after coarse graining, a spatial temperature distribution and its time evolution. A new expression is derived for the irreversible entropy production in a steady state, as the average of the growth rate of the relative density, a unique combination of the two fields., Comment: 4 pages, 2 postscript figures

The structure of the concentration field of a decaying substance produced by chemical sources and advected by a smooth incompressible two-dimensional flow is investigated. We focus our attention on the non-uniformities of the H\"older exponent of the resulting filamental chemical field. They appear most evidently in the case of open flows where irregularities of the field exhibit strong spatial intermittency as they are restricted to a fractal manifold. Non-uniformities of the H\"older exponent of the chemical field in closed flows appears as a consequence of the non-uniform stretching of the fluid elements. We study how this affects the scaling exponents of the structure functions, displaying anomalous scaling, and relate the scaling exponents to the distribution of finite-time Lyapunov exponents of the advection dynamics. Theoretical predictions are compared with numerical experiments., Comment: 10 pages, REVTeX, 7 figures included in the text. Related material at http://www.imedea.uib.es/Nonlinear/