Nonlinear Sciences

Vallis proposed a simple model for El-Nino weather phenomenon (referred as Vallis system) by adding an additional parameter p to the Lorenz system. He showed that the chaotic behavior of the Vallis system is related to the El-Nino effect. In the present article we study fractional version of Vallis system in depth. We investigate bifurcations and chaos present in the fractional Vallis system along with the effect of variation of system parameter p. It is observed that the range of values of parameter p for which the Vallis system is chaotic, reduces with the reduction of the fractional order. Further we analyze the incommensurate fractional Vallis system and find the critical value below which the system loses chaos. We also synchronize Vallis system with Bhalekar-Gejji system.

We study the chaotic dynamics of graphene structures, considering both a periodic, defect free, graphene sheet and graphene nanoribbons (GNRs) of various widths. By numerically calculating the maximum Lyapunov exponent, we quantify the chaoticity for a spectrum of energies in both systems. We find that for all cases, the chaotic strength increases with the energy density, and that the onset of chaos in graphene is slow, becoming evident after more than 10 4 natural oscillations of the system. For the GNRs, we also investigate the impact of the width and chirality (armchair or zigzag edges) on their chaotic behavior. Our results suggest that due to the free edges the chaoticity of GNRs is stronger than the periodic graphene sheet, and decreases by increasing width, tending asymptotically to the bulk value. In addition, the chaotic strength of armchair GNRs is higher than a zigzag ribbon of the same width. Further, we show that the composition of 12 C and 13 C carbon isotopes in graphene has a minor impact on its chaotic strength.

We consider a model for chaotic diffusion with amplification on graphs associated with piecewise-linear maps of the interval. We investigate the possibility of having power-law tails in the invariant measure by approximate solution of the Perron-Frobenius equation and discuss the connection with the generalized Lyapunov exponents L(q) . We then consider the case of open maps where trajectories escape and demonstrate that stationary power-law distributions occur when L(q)=r , with r being the escape rate. The proposed system is a toy model for coupled active chaotic cavities or lasing networks and allows to elucidate in a simple mathematical framework the conditions for observing Lévy statistical regimes and chaotic intermittency in such systems.

In this work we study chaotic mixing induced by point micro-rotors in a bounded two dimensional Stokes flow. The dynamics of the pair of rotors, modeled as rotlets, are non Hamiltonian in the bounded domain and produce chaotic advection of fluid tracers in subsets of the domain. A complete parametric investigation of the fluid mixing as a function of the initial locations of the rotlets is performed based on pseudo phase portraits. The mixing of fluid tracers as a function of relative positions of micro-rotors is studied using finite time entropy and locational entropy. The finite time locational entropy is used to identify regions of the fluid that produce good vs poor mixing and this is visualized by the stretching and folding of blobs of tracer particles. Unlike the case of the classic blinking vortex dynamics, the velocity field of the flow modeled using rotlets inside a circular boundary is smooth in time and satisfies the no slip boundary condition. This makes the considered model a more realistic case for studies of mixing in microfluidic devices using magnetic actuated microspheres.

A solvable model of noise effects on globally coupled limit cycle oscillators is proposed. The oscillators are under the influence of independent and additive white Gaussian noise. The averaged motion equation of the system with infinitely coupled oscillators is derived without any approximation through an analysis based on the nonlinear Fokker--Planck equation. Chaotic synchronization associated with the appearance of macroscopic chaotic behavior is shown by investigating the changes in averaged motion with increasing noise intensity.

The mean Poincarré recurrence time as well as the Lyapunov time are measured for the Fermi-Ulam model. We confirm the mean recurrence time is dependent on the size of the window chosen in the phase space to where particles are allowed to recur. The fractal dimension of the region is determined by the slope of the recurrence time against the size of the window and two numerical values were measured: (i) μ = 1 confirming normal diffusion for chaotic regions far from periodic domains and; (ii) μ = 2 leading to anomalous diffusion measured near periodic regions, a signature of local trapping of an ensemble of particles. The Lyapunov time is measured over different domains in the phase space through a direct determination of the Lyapunov exponent, indeed being defined as its inverse.

We numerically investigate the characteristics of chaos evolution during wave packet spreading in two typical one-dimensional nonlinear disordered lattices: the Klein-Gordon system and the discrete nonlinear Schrödinger equation model. Completing previous investigations \cite{SGF13} we verify that chaotic dynamics is slowing down both for the so-called `weak' and `strong chaos' dynamical regimes encountered in these systems, without showing any signs of a crossover to regular dynamics. The value of the finite-time maximum Lyapunov exponent Λ decays in time t as Λ∝ t α Λ , with α Λ being different from the α Λ =−1 value observed in cases of regular motion. In particular, α Λ ≈−0.25 (weak chaos) and α Λ ≈−0.3 (strong chaos) for both models, indicating the dynamical differences of the two regimes and the generality of the underlying chaotic mechanisms. The spatiotemporal evolution of the deviation vector associated with Λ reveals the meandering of chaotic seeds inside the wave packet, which is needed for obtaining the chaotization of the lattice's excited part.

We analyze a pair of delay-coupled FitzHugh-Nagumo oscillators exhibiting in-out intermittency as a part of the generating mechanism of extreme events. We study in detail the characteristics of in-out intermittency and identify the invariant subsets involved --- a saddle fixed point and a saddle periodic orbit --- neither of which are chaotic as in the previously reported cases of in-out intermittency. Based on the analysis of a periodic attractor possessing in-out dynamics, we can characterize the approach to the invariant synchronization manifold and the spiralling out to the saddle periodic orbit with subsequent ejection from the manifold. Due to the striking similarities, this analysis of in-out dynamics explains also in-out intermittency.

When the parameter of a map is chosen, at each iteration step, following a certain rule, is called Parametric Perturbation. If the parameters are drawn from a distribution, then this perturbation is called Random Parametric Perturbation. Studies have already been done on both Periodic and Random perturbations of a continuous map. Here, we have applied this technique on a tent map, which is a piecewise continuous map, and obtained numerical results.

We study the dynamics of coupled systems, ranging from maps supporting chaotic attractors to nonlinear differential equations yielding limit cycles, under different coupling classes, connectivity ranges and initial states. Our focus is the robustness of chimera states in the presence of a few time-varying random links, and we demonstrate that chimera states are often destroyed, yielding either spatiotemporal fixed points or spatiotemporal chaos, in the presence of even a single dynamically changing random connection. We also study the global impact of random links by exploring the Basin Stability of the chimera state, and we find that the basin size of the chimera state rapidly falls to zero under increasing fraction of random links. This indicates the extreme fragility of chimera patterns under minimal spatial randomness in many systems, significantly impacting the potential observability of chimera states in naturally occurring scenarios.