Fractal and Wada escape basins in the chaotic particle drift motion in tokamaks with electrostatic fluctuations.

The E×B drift motion of particles in tokamaks provides valuable information on the turbulence-driven anomalous transport. One of the characteristic features of the drift motion dynamics is the presence of chaotic orbits for which the guiding center can experience large-scale drifts. If one or more exits are placed so that they intercept chaotic orbits, the corresponding escape basins structure is complicated and, indeed, exhibits fractal structures. We investigate those structures through a number of numerical diagnostics, tailored to quantify the final-state uncertainty related to the fractal escape basins. We estimate the escape basin boundary dimension through the uncertainty exponent method and quantify final-state uncertainty by the basin entropy and the basin boundary entropy. Finally, we recall the Wada property for the case of three or more escape basins. This property is verified both qualitatively and quantitatively using a grid approach.

Low-dimensional chaos in the single wave model for self-consistent wave-particle Hamiltonian.

We analyze nonlinear aspects of the self-consistent wave-particle interaction using Hamiltonian dynamics in the single wave model, where the wave is modified due to the particle dynamics. This interaction plays an important role in the emergence of plasma instabilities and turbulence. The simplest case, where one particle (N=1) is coupled with one wave (M=1), is completely integrable, and the nonlinear effects reduce to the wave potential pulsating while the particle either remains trapped or circulates forever. On increasing the number of particles ( N=2, M=1), integrability is lost and chaos develops. Our analyses identify the two standard ways for chaos to appear and grow (the homoclinic tangle born from a separatrix, and the resonance overlap near an elliptic fixed point). Moreover, a strong form of chaos occurs when the energy is high enough for the wave amplitude to vanish occasionally.

Transport of blood particles: Chaotic advection even in a healthy scenario.

We study the advection of blood particles in the carotid bifurcation, a site that is prone to plaque development. Previously, it has been shown that chaotic advection can take place in blood flows with diseases. Here, we show that even in a healthy scenario, chaotic advection can take place. To understand how the particle dynamics is affected by the emergence and growth of a plaque, we study the carotid bifurcation in three cases: a healthy bifurcation, a bifurcation with a mild stenosis, and the another with a severe stenosis. The result is non-intuitive: there is less chaos for the mild stenosis case even when compared to the healthy, non-stenosed, bifurcation. This happens because the partial obstruction of the mild stenosis generates a symmetry in the flow that does not exist for the healthy condition. For the severe stenosis, there is more irregular motion and more particle trapping as expected.

Correlated Brownian motion and diffusion of defects in spatially extended chaotic systems.

One of the spatiotemporal patterns exhibited by coupled map lattices with nearest-neighbor coupling is the appearance of chaotic defects, which are spatially localized regions of chaotic dynamics with a particlelike behavior. Chaotic defects display random behavior and diffuse along the lattice with a Gaussian signature. In this note, we investigate some dynamical properties of chaotic defects in a lattice of coupled chaotic quadratic maps. Using a recurrence-based diagnostic, we found that the motion of chaotic defects is well-represented by a stochastic time series with a power-law spectrum 1/fσ with 2.3≤σ≤2.4, i.e., a correlated Brownian motion. The correlation exponent corresponds to a memory effect in the Brownian motion and increases with a system parameter as the diffusion coefficient of chaotic defects.

Synchronous patterns and intermittency in a network induced by the rewiring of connections and coupling.

The connection architecture plays an important role in the synchronization of networks, where the presence of local and nonlocal connection structures are found in many systems, such as the neural ones. Here, we consider a network composed of chaotic bursting oscillators coupled through a Watts-Strogatz-small-world topology. The influence of coupling strength and rewiring of connections is studied when the network topology is varied from regular to small-world to random. In this scenario, we show two distinct nonstationary transitions to phase synchronization: one induced by the increase in coupling strength and another resulting from the change from local connections to nonlocal ones. Besides this, there are regions in the parameter space where the network depicts a coexistence of different bursting frequencies where nonstationary zig-zag fronts are observed. Regarding the analyses, we consider two distinct methodological approaches: one based on the phase association to the bursting activity where the Kuramoto order parameter is used and another based on recurrence quantification analysis where just a time series of the network mean field is required.

Recurrence quantification analysis for the identification of burst phase synchronisation.

In this work, we apply the spatial recurrence quantification analysis (RQA) to identify chaotic burst phase synchronisation in networks. We consider one neural network with small-world topology and another one composed of small-world subnetworks. The neuron dynamics is described by the Rulkov map, which is a two-dimensional map that has been used to model chaotic bursting neurons. We show that with the use of spatial RQA, it is possible to identify groups of synchronised neurons and determine their size. For the single network, we obtain an analytical expression for the spatial recurrence rate using a Gaussian approximation. In clustered networks, the spatial RQA allows the identification of phase synchronisation among neurons within and between the subnetworks. Our results imply that RQA can serve as a useful tool for studying phase synchronisation even in networks of networks.

Riddling: Chimera's dilemma.

We investigate the basin of attraction properties and its boundaries for chimera states in a circulant network of Hénon maps. It is known that coexisting basins of attraction lead to a hysteretic behaviour in the diagrams of the density of states as a function of a varying parameter. Chimera states, for which coherent and incoherent domains occur simultaneously, emerge as a consequence of the coexistence of basin of attractions for each state. Consequently, the distribution of chimera states can remain invariant by a parameter change, and it can also suffer subtle changes when one of the basins ceases to exist. A similar phenomenon is observed when perturbations are applied in the initial conditions. By means of the uncertainty exponent, we characterise the basin boundaries between the coherent and chimera states, and between the incoherent and chimera states. This way, we show that the density of chimera states can be not only moderately sensitive but also highly sensitive to initial conditions. This chimera's dilemma is a consequence of the fractal and riddled nature of the basin boundaries.

Spatial recurrence analysis: a sensitive and fast detection tool in digital mammography.

Efficient diagnostics of breast cancer requires fast digital mammographic image processing. Many breast lesions, both benign and malignant, are barely visible to the untrained eye and requires accurate and reliable methods of image processing. We propose a new method of digital mammographic image analysis that meets both needs. It uses the concept of spatial recurrence as the basis of a spatial recurrence quantification analysis, which is the spatial extension of the well-known time recurrence analysis. The recurrence-based quantifiers are able to evidence breast lesions in a way as good as the best standard image processing methods available, but with a better control over the spurious fragments in the image.

Chaotic escape of impurities and sticky orbits in toroidal plasmas.

We investigate chaotic impurity transport in toroidal fusion plasmas (tokamaks) from the point of view of passive advection of charged particles due to E×B drift motion. We use realistic tokamak profiles for electric and magnetic fields as well as toroidal rotation effects, and consider also the effects of electrostatic fluctuations due to drift instabilities on particle motion. A time-dependent one degree-of-freedom Hamiltonian system is obtained and numerically investigated through a symplectic map in a Poincaré surface of section. We show that the chaotic transport in the outer plasma region is influenced by fractal structures that are described in topological and metric point of views. Moreover, the existence of a hierarchical structure of islands-around-islands, where the particles experience the stickiness effect, is demonstrated using a recurrence-based approach.

Isochronous island bifurcations driven by resonant magnetic perturbations in tokamaks.

Recent evidence shows that heteroclinic bifurcations in magnetic islands may be caused by the amplitude variation of resonant magnetic perturbations in tokamaks. To investigate the onset of these bifurcations, we consider a large aspect ratio tokamak with an ergodic limiter composed of two pairs of rings that create external primary perturbations with two sets of wave numbers. An individual pair produces hyperbolic and elliptic periodic points, and its associated islands, that are consistent with the Poincaré-Birkhoff fixed-point theorem. However, for two pairs producing external perturbations resonant on the same rational surface, we show that different configurations of isochronous island chains may appear on phase space according to the amplitude of the electric currents in each pair of the ergodic limiter. When one of the electric currents increases, isochronous bifurcations take place and new islands are created with the same winding number as the preceding islands. We present examples of bifurcation sequences displaying (a) direct transitions from the island chain configuration generated by one of the pairs to the configuration produced by the other pair, and (b) transitions with intermediate configurations produced by the limiter pairs coupling. Furthermore, we identify shearless bifurcations inside some isochronous islands, originating nonmonotonic local winding number profiles with associated shearless invariant curves.

Chaotic saddles and interior crises in a dissipative nontwist system.

We consider a dissipative version of the standard nontwist map. Nontwist systems present a robust transport barrier, called the shearless curve, that becomes the shearless attractor when dissipation is introduced. This attractor can be regular or chaotic depending on the control parameters. Chaotic attractors can undergo sudden and qualitative changes as a parameter is varied. These changes are called crises, and at an interior crisis the attractor suddenly expands. Chaotic saddles are nonattracting chaotic sets that play a fundamental role in the dynamics of nonlinear systems; they are responsible for chaotic transients, fractal basin boundaries, and chaotic scattering, and they mediate interior crises. In this work we discuss the creation of chaotic saddles in a dissipative nontwist system and the interior crises they generate. We show how the presence of two saddles increases the transient times and we analyze the phenomenon of crisis induced intermittency.

Suppression of bursting synchronization in clustered scale-free (rich-club) neuronal networks.

Functional brain networks are composed of cortical areas that are anatomically and functionally connected. One of the cortical networks for which more information is available in the literature is the cat cerebral cortex. Statistical analyses of the latter suggest that its structure can be described as a clustered network, in which each cluster is a scale-free network possessing highly connected hubs. Those hubs are, on their hand, connected together in a strong fashion ("rich-club" network). We have built a clustered scale-free network inspired in the cat cortex structure so as to study their dynamical properties. In this article, we focus on the synchronization of bursting activity of the cortical areas and how it can be suppressed by means of neuron deactivation through suitably applied light pulses. We show that it is possible to effectively suppress bursting synchronization by acting on a single, yet suitably chosen neuron, as long as it is highly connected, thanks to the "rich-club" structure of the network.

Two-state on-off intermittency and the onset of turbulence in a spatiotemporally chaotic system.

We show the existence of two-state on-off intermittent behavior in spatially extended dynamical systems, using as an example the damped and forced drift wave equation. The two states are stationary solutions corresponding to different wave energies. In the language of (Fourier-mode) phase space these states are embedded in two invariant manifolds that become transversely unstable in the regime where two-state on-off intermittency sets in. The distribution of laminar duration sizes is compatible with the similar phenomenon occurring in time only in the presence of noise. In an extended system the noisy effect is provided by the spatial modes excited by the perturbation. We show that this intermittency is a precursor of the onset of strong turbulence in the system.

Transport properties in nontwist area-preserving maps.

Nontwist systems, common in the dynamical descriptions of fluids and plasmas, possess a shearless curve with a concomitant transport barrier that eliminates or reduces chaotic transport, even after its breakdown. In order to investigate the transport properties of nontwist systems, we analyze the barrier escape time and barrier transmissivity for the standard nontwist map, a paradigm of such systems. We interpret the sensitive dependence of these quantities upon map parameters by investigating chaotic orbit stickiness and the associated role played by the dominant crossing of stable and unstable manifolds.

Fractal structures in the parameter space of nontwist area-preserving maps.

Fractal structures are very common in the phase space of nonlinear dynamical systems, both dissipative and conservative, and can be related to the final state uncertainty with respect to small perturbations on initial conditions. Fractal structures may also appear in the parameter space, since parameter values are always known up to some uncertainty. This problem, however, has received less attention, and only for dissipative systems. In this work we investigate fractal structures in the parameter space of two conservative dynamical systems: the standard nontwist map and the quartic nontwist map. For both maps there is a shearless invariant curve in the phase space that acts as a transport barrier separating chaotic orbits. Depending on the values of the system parameters this barrier can break up. In the corresponding parameter space the set of parameter values leading to barrier breakup is separated from the set not leading to breakup by a curve whose properties are investigated in this work, using tools as the uncertainty exponent and basin entropies. We conclude that this frontier in parameter space is a complicated curve exhibiting both smooth and fractal properties, that are characterized using the uncertainty dimension and basin and basin boundary entropies.

Periodic-orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system.

Chaotic dynamical systems with two or more attractors lying on invariant subspaces may, provided certain mathematical conditions are fulfilled, exhibit intermingled basins of attraction: Each basin is riddled with holes belonging to basins of the other attractors. In order to investigate the occurrence of such phenomenon in dynamical systems of ecological interest (two-species competition with extinction) we have characterized quantitatively the intermingled basins using periodic-orbit theory and scaling laws. The latter results agree with a theoretical prediction from a stochastic model, and also with an exact result for the scaling exponent we derived for the specific class of models investigated. We discuss the consequences of the scaling laws in terms of the predictability of a final state (extinction of either species) in an ecological experiment.

Short-time memories in a network with randomly distributed connections.

Coupled map lattices are able to store short-term memories when an external periodic input is applied. We consider short-term memory formation in networks with both regular (nearest-neighbor) and randomly chosen connections. The regimes under which single or multiple memorized patterns are stored are studied in terms of the coupling and nonlinear parameters of the network.

Dynamical characterization of transport barriers in nontwist Hamiltonian systems.

The turnstile provides us a useful tool to describe the flux in twist Hamiltonian systems. Thus, its determination allows us to find the areas where the trajectories flux through barriers. We show that the mechanism of the turnstile can increase the flux in nontwist Hamiltonian systems. A model which captures the essence of these systems is the standard nontwist map, introduced by del Castillo-Negrete and Morrison. For selected parameters of this map, we show that chaotic trajectories entering in resonances zones can be explained by turnstiles formed by a set of homoclinic points. We argue that for nontwist systems, if the heteroclinic points are sufficiently close, they can connect twin-islands chains. This provides us a scenario where the trajectories can cross the resonance zones and increase the flux. For these cases the escape basin boundaries are nontrivial, which demands the use of an appropriate characterization. We applied the uncertainty exponent and the entropies of the escape basin boundary in order to quantify the degree of unpredictability of the asymptotic trajectories.

Inference of topology and the nature of synapses, and the flow of information in neuronal networks.

The characterization of neuronal connectivity is one of the most important matters in neuroscience. In this work, we show that a recently proposed informational quantity, the causal mutual information, employed with an appropriate methodology, can be used not only to correctly infer the direction of the underlying physical synapses, but also to identify their excitatory or inhibitory nature, considering easy to handle and measure bivariate time series. The success of our approach relies on a surprising property found in neuronal networks by which nonadjacent neurons do "understand" each other (positive mutual information), however, this exchange of information is not capable of causing effect (zero transfer entropy). Remarkably, inhibitory connections, responsible for enhancing synchronization, transfer more information than excitatory connections, known to enhance entropy in the network. We also demonstrate that our methodology can be used to correctly infer directionality of synapses even in the presence of dynamic and observational Gaussian noise, and is also successful in providing the effective directionality of intermodular connectivity, when only mean fields can be measured.

Synchronous behaviour in network model based on human cortico-cortical connections.

OBJECTIVE: We consider a network topology according to the cortico-cortical connection network of the human brain, where each cortical area is composed of a random network of adaptive exponential integrate-and-fire neurons. APPROACH: Depending on the parameters, this neuron model can exhibit spike or burst patterns. As a diagnostic tool to identify spike and burst patterns we utilise the coefficient of variation of the neuronal inter-spike interval. MAIN RESULTS: In our neuronal network, we verify the existence of spike and burst synchronisation in different cortical areas. SIGNIFICANCE: Our simulations show that the network arrangement, i.e. its rich-club organisation, plays an important role in the transition of the areas from desynchronous to synchronous behaviours.