Pattern formation often occurs in spatially extended physical, biological, and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes ...the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue, we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular, revealing for the Swift-Hohenberg equations, a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of a weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.
Two replicas of spatially extended chaotic systems synchronize to a common spatio-temporal chaotic state when coupled above a critical strength. As a prototype of each single spatio-temporal chaotic ...system a lattice of maps interacting via power-law coupling is considered. Furthermore, each unit in the one-dimensional chain is linked to the corresponding one in the replica via a local coupling. The synchronization transition is studied as a nonequilibrium phase transition, and its critical properties are analyzed at varying the spatial interaction range as well as the nonlinearity of the dynamical units composing each system. In particular, continuous and discontinuous local maps are considered. In both cases the transitions are of the second order with critical indices varying with the exponent characterizing the interaction range. For discontinuous maps it is numerically shown that the transition belongs to the anomalous directed percolation (ADP) family of universality classes, previously identified for Levy-flight spreading of epidemic processes. For continuous maps, the critical exponents are different from those characterizing ADP, but apart from the nearest-neighbor case, the identification of the corresponding universality classes remains an open problem. Finally, to test the influence of deterministic correlations for the studied synchronization transitions, the chaotic dynamical evolutions are substituted by suitable stochastic models. In this framework and for the discontinuous case, it is possible to derive an effective Langevin description that corresponds to that proposed for ADP.
We consider pulse-coupled leaky integrate-and-fire neural networks with randomly distributed synaptic couplings. This random dilution induces fluctuations in the evolution of the macroscopic ...variables and deterministic chaos at the microscopic level. Our main aim is to mimic the effect of the dilution as a noise source acting on the dynamics of a globally coupled nonchaotic system. Indeed, the evolution of a diluted neural network can be well approximated as a fully pulse-coupled network, where each neuron is driven by a mean synaptic current plus additive noise. These terms represent the average and the fluctuations of the synaptic currents acting on the single neurons in the diluted system. The main microscopic and macroscopic dynamical features can be retrieved with this stochastic approximation. Furthermore, the microscopic stability of the diluted network can be also reproduced, as demonstrated from the almost coincidence of the measured Lyapunov exponents in the deterministic and stochastic cases for an ample range of system sizes. Our results strongly suggest that the fluctuations in the synaptic currents are responsible for the emergence of chaos in this class of pulse-coupled networks.
In this paper we show that a dynamical description of the protein folding process provides an effective representation of equilibrium properties and it allows for a direct investigation of the ...mechanisms ruling the approach towards the native configuration. The results reported in this paper have been obtained fora two-dimensional toy-model of aminoacid sequences, whosenative configurations were previously determined byMonte Carlo techniques.The somewhat controversial scenario emerging from the comparison among different thermodynamical indicators is definitely better resolved with the help of a truly dynamical description. In particular,we are able to identify the metastable states visited during the folding process by monitoring the temporal evolution of the `long-range' potentialenergy. Moreover, the resulting dynamical scenario is consistent with the picture arising from a reconstruction of the energy landscape in the vicinity of the global minimum. This suggests that the introduction of efficient `static' indicators too should properly account for the complex `orography' of the landscape.
The evolution towards equipartition in the β-FPU chain is studied considering as initial condition the highest frequency mode. Above an analytically derived energy threshold, this zone-boundary mode ...is shown to be modulationally unstable and to give rise to a striking localization process. The spontaneously created excitations have strong similarity with moving exact breathers solutions. But they have a finite lifetime and their dynamics is chaotic. These chaotic breathers are able to collect very efficiently the energy in the chain. Therefore their size grows in time and they can transport a very large quantity of energy. These features can be explained analyzing the dynamics of perturbed exact breathers of the FPU chain. In particular, a close connection between the Lyapunov spectrum of the chaotic breathers and the Floquet spectrum of the exact ones has been found. The emergence of chaotic breathers is convincingly explained by the absorption of high frequency phonons whereas a breather's metastability is for the first time identified. The lifetime of the chaotic breather is related to the time necessary for the system to reach equipartition. The equipartition time turns out to be dependent on the system energy density ε only. Moreover, such time diverges as
ε
−2 in the limit ε → 0 and vanishes as
ε
−1
4
for ε → ∞.
In two quasi-elastic neutron scattering experiments on liquid sodium at 380 K and 900 K at saturated vapour pressure, single particle motion has been studied with high precision. Special emphasis was ...placed on the different single particle dynamics near the melting point and the boiling point of the liquid. The experiments are discussed in the light of a new theory, the mode coupling theory, and reveal two clearly different mode coupling effects in the liquid: Coupling to collective density fluctuations-hindering a particle's diffusion, and coupling to transverse shear modes, which promote single particle dynamics. The interplay of these two basic mechanisms leads to a new understanding of single particle motion in simple liquids, being governed by the temperature dependent competition of two mode coupling effects. In conclusion the non-trivial temperature dependence of the diffusion coefficient in simple liquids can be understood on a purely microscopic basis without phenomenological ingredients.
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