Nonlinear Sciences

A recent model of Ariel et al. [1] for explaining the observation of Lévy walks in swarming bacteria suggests that self-propelled, elongated particles in a periodic array of regular vortices perform a super-diffusion that is consistent with Lévy walks. The equations of motion, which are reversible in time but not volume preserving, demonstrate a new route to L'evy walking in chaotic systems. Here, the dynamics of the model is studied both analytically and numerically. It is shown that the apparent super-diffusion is due to "sticking" of trajectories to elliptic islands, regions of quasi-periodic orbits reminiscent of those seen in conservative systems. However, for certain parameter values, these islands coexist with asymptotically stable periodic trajectories, causing dissipative behavior on very long time scales.

In dynamical systems limit cycles arise as a result of a Hopf bifurcation, after a control parameter has crossed its critical value. In this study we present a constructive method to produce dissipative dynamics which lead to stable periodic orbits as time grows, with predesigned transient dynamics. Depending on the construction method a) the limiting orbit can be a regular circle, an ellipse or a more complex closed orbit and b) the approach to the limiting orbit can follow an exponential law or a power law. This technique allows to design nonlinear models of dynamical systems with desired (exponential or power law) relaxation properties.

We use the weight δ I, deduced from the estimation of Lyapunov vectors, in order to characterise regions in the kinetic (x, v) space with particles that most contribute to chaoticity. For the paradigmatic model, the cosine Hamiltonian mean field model, we show that this diagnostic highlights the vicinity of the separatrix, even when the latter hardly exists.

We explore the behaviour of chaotic oscillators in hierarchical networks coupled to an external chaotic system whose intrinsic dynamics is dissimilar to the other oscillators in the network. Specifically, each oscillator couples to the mean-field of the oscillators below it in the hierarchy, and couples diffusively to the oscillator above it in the hierarchy. We find that coupling to one dissimilar external system manages to suppress the chaotic dynamics of all the oscillators in the network at sufficiently high coupling strength. This holds true irrespective of whether the connection to the external system is direct or indirect through oscillators at another level in the hierarchy. Investigating the synchronization properties show that the oscillators have the same steady state at a particular level of hierarchy, whereas the steady state varies across different hierarchical levels. We quantify the efficacy of control by estimating the fraction of random initial states that go to fixed points, a measure analogous to basin stability. These quantitative results indicate the easy controllability of hierarchical networks of chaotic oscillators by an external chaotic system, thereby suggesting a potent method that may help design control strategies.

In this Letter we propose a method to control a set of arbitrary nodes in a directed network such that they follow a synchronous trajectory which is, in general, not shared by the other units of the network. The problem is inspired to those natural or artificial networks whose proper operating conditions are associated to the presence of clusters of synchronous nodes. Our proposed method is based on the introduction of distributed controllers that modify the topology of the connections in order to generate outer symmetries in the nodes to be controlled. An optimization problem for the selection of the controllers, which includes as a special case the minimization of the number of the links added or removed, is also formulated and an algorithm for its solution is introduced.

We investigate how the diffusion exponent is affected by controlling small domains in the phase space.The main Kolomogorov-Arnold-Moser - KAM island of the Standard Map is considered to validate the investigation. The bifurcation scenario where the periodic island emits smaller resonance regions is considered and we show how closing paths escape from the island shore by controlling points and hence making the diffusion exponent smaller. We notice the bigger controlled area the smaller the diffusion exponent. We show that controlling around the hyperbolic points associated to the bifurcation is better than a random control to reduce the diffusion exponent. The recurrence plot shows us channels of escape and a control applied there reduces the diffusion exponent.

In this work we present a method to control chimera states through linear augmentation. Using an ensemble of globally coupled oscillators, we demonstrate that control over the spatial location of the incoherent or coherent regions of a chimera state in a network can be achieved. This is characterized by exploring their basins of attraction. We also verify the stability of different states present in the coupled systems before and after linear augmentation by calculating the transverse Lyapunov exponent and Master stability function. We observe that the control through this technique is independent of the coupling mechanisms and also on the initial conditions chosen for the creation of chimera states.

We present an efficient technique for control of synchrony in a globally coupled ensemble by pulsatile action. We assume that we can observe the collective oscillation and can stimulate all elements of the ensemble simultaneously. We pay special attention to the minimization of intervention into the system. The key idea is to stimulate only at the most sensitive phase. To find this phase we implement an adaptive feedback control. Estimating the instantaneous phase of the collective mode on the fly, we achieve efficient suppression using a few pulses per oscillatory cycle. We discuss the possible relevance of the results for neuroscience, namely for the development of advanced algorithms for deep brain stimulation, a medical technique used to treat Parkinson's disease.

We demonstrate the control of vortical motion of neutral classical particles in driven superlattices. Our superlattice consists of a superposition of individual lattices whose potential depths are modulated periodically in time but with different phases. This driving scheme breaks the spatial reflection symmetries and allows an ensemble of particles to rotate with an average angular velocity. An analysis of the underlying dynamical attractors provides an efficient method to control the angular velocities of the particles by changing the driving amplitude. As a result, spatially periodic patterns of particles showing different vortical motion can be created. Possible experimental realizations include holographic optical lattice based setups for colloids or cold atoms.

Resonances in particle transmission through a 1D finite lattice are studied in the presence of a finite number of impurities. Although this is a one-dimensional system that is classically integrable and has no chaos, studying the statistical properties of the spectrum such as the level spacing distribution and the spectral rigidity shows quantum chaos signatures. Using a dimensionless parameter that reflects the degree of state localization, we demonstrate how the transition from regularity to chaos is affected by state localization. The resonance positions are calculated using both the Wigner-Smithtime-delay and a Siegert state method, which are in good agreement. Our results give evidence for the existence of quantum chaos in one dimension which is a counter-example to the Bohigas-Giannoni-Schmit conjecture.