We study the influence of noise on the spatially localized, temporally regular states (stationary, one frequency, two frequencies) in the regime of anomalous dispersion for the cubic-quintic complex Ginzburg-Landau equation as a function of the bifurcation parameter. We find that noise of a fairly small strength η is sufficient to reach a chaotic state with exploding dissipative solitons. That means that noise can induce explosions over a fairly large range of values of the bifurcation parameter μ. Three different routes to chaos with exploding dissipative solitons are found as a function of μ. As diagnostic tools we use the separation to characterize chaotic behavior and the energy to detect spatially localized explosive behavior as a function of time.
We investigate the route to exploding dissipative solitons in the complex cubic-quintic Ginzburg-Landau equation, as the bifurcation parameter, the distance from linear onset, is increased. We find for a large class of initial conditions the sequence: stationary localized solutions, oscillatory localized solutions with one frequency, oscillatory localized solutions with two frequencies, and exploding localized solutions. The transition between localized solutions with one and with two frequencies, respectively, is analyzed in detail. It is found to correspond to a forward Hopf bifurcation for these localized solutions as the bifurcation parameter is increased. In addition, we make use of power spectra to characterize all time-dependent states. On the basis of all information available, we conclude that the sequence oscillatory localized solutions with one frequency, oscillatory localized solutions with two frequencies, and exploding dissipative solitons can be interpreted as the analog of the Ruelle-Takens-Newhouse route to chaos for spatially localized solutions.
We investigate a two-dimensional extended system showing chaotic and localized structures. We demonstrate the robust and stable existence of two types of exploding dissipative solitons. We show that the center of mass of asymmetric dissipative solitons undergoes a random walk despite the deterministic character of the underlying model. Since dissipative solitons are stable in two-dimensional systems we conjecture that our predictions can be tested in systems as diverse as nonlinear optics, parametric excitation of granular media and clay suspensions, and sheared electroconvection.
Generalized equations of motion for the Weber-Clebsch potentials that reproduce Navier-Stokes dynamics are derived. These depend on a new parameter, with the dimension of time, and reduce to the Ohkitani and Constantin equations in the singular special case where the new parameter vanishes. Let us recall that Ohkitani and Constantin found that the diffusive Lagrangian map became noninvertible under time evolution and required resetting for its calculation. They proposed that high frequency of resetting was a diagnostic for vortex reconnection. Direct numerical simulations are performed. The Navier-Stokes dynamics is well reproduced at small enough Reynolds number without resetting. Computation at higher Reynolds numbers is achieved by performing resettings. The interval between successive resettings is found to abruptly increase when the new parameter is varied from 0 to a value much smaller than the resetting interval.
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