Parabolic resonance: a route to hamiltonian spatiotemporal chaos

Excitation of Peregrine-Type Waveforms from Vanishing Initial Conditions in the Presence of Periodic Forcing

We show by direct numerical simulations that spatiotemporally localised waveforms, strongly reminiscent of the Peregrine rogue wave, can be excited by vanishing initial conditions for the periodically driven nonlinear Schrödinger equation. The emergence of the Peregrine-type waveforms can be potentially justified, in terms of the existence and modulational instability of spatially homogeneous solutions of the model and the continuous dependence of the localised initial data for small time intervals. We also comment on the persistence of the above dynamics, under the presence of small damping effects, and justify that this behaviour should be considered as far from approximations of the corresponding integrable limit.

Chaoticons described by nonlocal nonlinear Schrödinger equation

It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrödinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-like properties (with invariant statistic width and interaction of quasi-elastic collisions).

Dynamical playground of a higher-order cubic Ginzburg-Landau equation: From orbital connections and limit cycles to invariant tori and the onset of chaos

The dynamical behavior of a higher-order cubic Ginzburg-Landau equation is found to include a wide range of scenarios due to the interplay of higher-order physically relevant terms. We find that the competition between the third-order dispersion and stimulated Raman scattering effects gives rise to rich dynamics: this extends from Poincaré-Bendixson-type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections or to space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. One of our main results is that third-order dispersion has a dominant role in the development of such complex dynamics, since it can be chiefly responsible (even in the absence of other higher-order effects) for the existence of periodic, quasiperiodic, and chaotic spatiotemporal structures. Suitable low-dimensional phase-space diagnostics are devised and used to illustrate the different possibilities and identify their respective parametric intervals over multiple parameters of the model.

Control of extreme events in the bubbling onset of wave turbulence

We show the existence of an intermittent transition from temporal chaos to turbulence in a spatially extended dynamical system, namely, the forced and damped one-dimensional nonlinear Schrödinger equation. For some values of the forcing parameter, the system dynamics intermittently switches between ordered states and turbulent states, which may be seen as extreme events in some contexts. In a Fourier phase space, the intermittency takes place due to the loss of transversal stability of unstable periodic orbits embedded in a low-dimensional subspace. We mapped these transversely unstable regions and perturbed the system in order to significantly reduce the occurrence of extreme events of turbulence.

Targeted activation in deterministic and stochastic systems

Metastable escape is ubiquitous in many physical systems and is becoming a concern in engineering design as these designs (e.g., swarms of vehicles, coupled building energetics, nanoengineering, etc.) become more inspired by dynamics of biological, molecular and other natural systems. In light of this, we study a chain of coupled bistable oscillators which has two global conformations and we investigate how specialized or targeted disturbance is funneled in an inverse energy cascade and ultimately influences the transition process between the conformations. We derive a multiphase averaged approximation to these dynamics which illustrates the influence of actions in modal coordinates on the coarse behavior of this process. An activation condition that predicts how the disturbance influences the rate of transition is then derived. The prediction tools are derived for deterministic dynamics and we also present analogous behavior in the stochastic setting and show a divergence from Kramers activation behavior under targeted activation conditions.