Nonlinear and nonlocal dynamics of spatially extended systems: Stationary states, bifurcations and stability

Spatially extended systems with nonlocal dynamics (e.g. ferromagnetic resonance or current instability) of the type

with uε ⁿ will be studied near the soft-mode instability (wave number k_c ≠ 0) of a stationary and uniform state. An amplitude equation is derived within the framework of a multiple-scale perturbation theory. A particular example of this class of nonlocal dynamics is also treated numerically. As the main result we find that in contrast to the well-known supercritical bifurcation into a stable periodic state, the uniform state can bifurcate supercritically into a stationary state of an amplitude-modulated fast oscillation in space.     

Collapse of solitary waves near the transition from supercritical to subcritical bifurcations

The nonlinear stage of the instability of one-dimensional solitons within a small vicinity of the transition point from supercritical to subcritical bifurcations has been studied both analytically and numerically using the generalized nonlinear Schrödinger equation. It is shown that the pulse amplitude and its width near the collapsing time demonstrate a self-similar behavior with a small asymmetry at the pulse tails due to self-steepening. This theory is applied to solitary interfacial deep-water waves, envelope water waves with a finite depth, and short optical pulses in fibers.     

Energy-flow patterns and bifurcations of two-dimensional cavity solitons

Journal of Experimental and Theoretical Physics - An analysis of two-dimensional spatial optical solitons in a large-aperture class A laser with a saturable absorber is developed. New types of...     

Stationary and oscillatory fronts in a two-component genetic regulatory network model

We investigate a two-component gene network model, originally used to describe the spatiotemporal patterning of the gene products in early Drosophila development. By considering a particular mode of interaction between the two gene products, denoted proteins A and B, we find both stable stationary and time-oscillatory fronts can occur in the reaction-diffusion system. We reduce the system by replacing B with its spatial average (shadow system) and assume an abrupt “on-and-off” switch for the genes. In doing so, explicit formula are obtained for all steady-state solutions and their linear eigenvalues. Using the diffusion of A,D_a, and the basal production rate, r, as bifurcation parameters, we explore ranges in which a monotone, stationary front is stable, and show it can lose stability through a Hopf bifurcation, giving rise to oscillatory fronts. We also discuss the existence and stability of steady-state and time-oscillatory solutions with multiple extrema. An intuitive explanation for the occurrence of stable stationary and oscillatory front solutions is provided based on the behavior of A in the absence of B and the opposite regulation between A and B. Such behavior is also interpreted in terms of the biological parameters in the model, including those governing the connection of the gene network.     

Spontaneous motion of localized structures and localized patterns induced by delayed feedback

We study the properties of 2D dissipative structures in a coherently driven optical resonator subjected to a delayed feedback. It has been predicted that delayed feedback can lead to the spontaneous motion of bright localized structures [M. Tlidi et al., Phys. Rev. Lett. 103, 103904 (2009)]. We study here the phenomenon in detail. In particular, we show that the delayed feedback induces a spontaneous motion of periodic patterns and dark localized structures. We focus our analysis on nascent optical bistability regime where the space time dynamics is described by a variational Swift-Hohenberg equation. In the absence of delayed feedback, dark localized structures and patterns do not move. This behavior occurs when the product of the delay time and the feedback strength exceeds some critical value.     

Delay- and coupling-induced firing patterns in oscillatory neural loops

For a feedforward loop of oscillatory Hodgkin-Huxley neurons interacting via excitatory chemical synapses, we show that a great variety of spatiotemporal periodic firing patterns can be encoded by properly chosen communication delays and synaptic weights, which contributes to the concept of temporal coding by spikes. These patterns can be obtained by a modulation of the multiple coexisting stable in-phase synchronized states or traveling waves propagating along or against the direction of coupling. We derive explicit conditions for the network parameters allowing us to achieve a desired pattern. Interestingly, whereas the delays directly affect the time differences between spikes of interacting neurons, the synaptic weights control the phase differences. Our results show that already such a simple neural circuit may unfold an impressive spike coding capability.     

Instabilities,bifurcations and catastrophes

Two independently developed bifurcational theories for the instability of a slowly evolving system, such as a stellar mass, are correlated. Applied to the fracture of a mechanically stressed perfect crystal, they predict a sharp cusp on the failure stress locus.     

Stability of periodic waves generated by subcritical oscillatory instability under the action of feedback control

Periodic solutions of a subcritical cubic complex Ginzburg-Landau equation are considered in the limit of large dispersion and nonlinear frequency shift. Results obtained formerly by Schöpf and Kramer are revisited and extended to the case of a defocusing nonlinearity. It is shown that a global feedback control can extend existence and stability regions of the stationary solutions in both focusing and defocusing cases.     

Estimating localized oscillatory tissue motion for assessment of the underlying mechanical modulus

The technique of harmonic motion imaging (HMI) uses the localized stimulus of the oscillatory ultrasonic radiation force as produced by two overlapping beams of distinct frequencies, and estimates the resulting harmonic displacement in the tissue in order to assess its underlying mechanical properties. In this paper, we studied the relationship between measured displacement and stiffness in gels and tissues in vitro. Two focused ultrasound transducers with a 100 mm focal length were used at frequencies of 3.7500 MHz and either 3.7502 (or 3.7508 MHz), respectively, in order to produce an oscillatory motion at 200 Hz in the gel or tissue. A 1.1 MHz diagnostic transducer (Imasonics, Inc.) was also focused at 100 mm and acquired 5 ms RF signals (pulse repetition frequency (PRF)=3.5 kHz) at 100 MHz sampling frequency during radiation force application. First, three 50x50 mm(2) acrylamide gels were prepared at concentrations of 4%, 8% and 16%. The resulting displacement was estimated using crosscorrelation techniques between successively acquired RF signals with a 2 mm window and 80% window overlap at 1260 W/cm(2). A normal 1-D indentation instrument (TeMPeST) applied oscillatory loads at 0.1-200 Hz with a 5 mm-diameter flat indenter. Then, 12 displacement measurements in 6 porcine muscle specimens (two measurements/case, as above) were made in vitro, before and after ablation which was performed for 10 s at 1260 W/cm(2). In all gel cases, the harmonic displacement was found to linearly increase with intensity and exponentially decrease with gel concentration. The TeMPeST measurements showed that the elastic moduli for the 4%, 8% and 16% gels equaled 3.93+/-0.06, 17.1+/-0.2 and 75+/-2 kPa, respectively, demonstrating that the HMI displacement estimate depends directly on the gel stiffness. Finally, in the tissues samples, the mean displacement amplitude showed a twofold decrease between non-ablated and ablated tissue, demonstrating a correspondence between the HMI response and an increase in stiffness measured with the TeMPeST instrument.     

Improved functional-weight approach to oscillatory patterns in excitable networks

Studies of sustained oscillations on complex networks with excitable node dynamics received much interest in recent years. Although an individual unit is non-oscillatory, they may organize to form various collective oscillatory patterns through networked connections. An excitable network usually possesses a number of oscillatory modes dominated by different Winfree loops and numerous spatiotemporal patterns organized by different propagation path distributions. The traditional approach of the so-called dominant phase-advanced drive method has been well applied to the study of stationary oscillation patterns on a network. In this paper, we develop the functional-weight approach that has been successfully used in studies of sustained oscillations in gene-regulated networks by an extension to the high-dimensional node dynamics. This approach can be well applied to the study of sustained oscillations in coupled excitable units. We tested this scheme for different networks, such as homogeneous random networks, small-world networks, and scale-free networks and found it can accurately dig out the oscillation source and the propagation path. The present approach is believed to have the potential in studies competitive non-stationary dynamics.     

Self-organized current-density patterns and bifurcations in n-GaAs with a circular symmetry of contacts

Self-organized nucleation of current-density filaments has been investigated in an n-GaAs epitaxial layer with circular contact symmetry. The first filament arises from a spatially uniform state, further filaments are generated by filament division processes. Filament growth, filament division and rearrangement of filaments reveal hysteretic behavior accompanied with subcritical bifurcations.     

Stable squares and other oscillatory turing patterns in a reaction-diffusion model

We study the Brusselator reaction-diffusion model under conditions where the Hopf mode is supercritical and the Turing band is subcritical. Oscillating Turing patterns arise in the system when bulk oscillations lose their stability to spatial perturbations. Spatially uniform external periodic forcing can generate oscillating Turing patterns when both the Turing and Hopf modes are subcritical in the autonomous system. Most of the symmetric patterns show period doubling in both space and time. Patterns observed include squares, rhombi, stripes, and hexagons.     

Experimental study of traveling waves and target patterns in oscillatory reacting media

Recent experimental work on is presented traveling waves and target patterns developing in oscillatory Belousov-Zhabotinsky reagents. It only deals with trigger waves and relaxation oscillations. Measurements have been made on (1) the dispersion relation of waves, (2) the statistical properties of centers and targets, (3) the wavefront and speed near the core of the pattern. Several unanticipated results are reported.