We analyze the chaotic emission of an erbium-doped fiber laser with sine-wave pump power modulation for different modulation frequencies and modulation indexes. For each working condition considered, the template which summarizes the corresponding chaotic attractor has been determined by means of topological analysis techniques. The variety of templates obtained is much wider than in previous works: apart from horseshoes, reverse horseshoes, or jellyroll structures with different global torsions, two more kinds of structures have been found. Particularly, one is especially notable as its folding mechanism does not fit the rolling scheme valid for all templates found in former experimental studies. Apart from strengthening the usefulness of templates as convenient objects for characterization of chaotic attractors, the interest of these results lies in the identification of folding mechanisms for chaos generation not found in nature up to now.

This investigation observed large-scale flows in liquid gallium and the oscillation with Rayleigh-Bénard convection. An ultrasonic velocity profiling method was used to visualize the spatiotemporal flow pattern of the liquid gallium in a horizontally long rectangular vessel. Measuring the horizontal component of the flow velocity at several lines, an organized roll-like structure with four cells was observed in the $1\times {10}^4–2\times {10}^5$ range of Rayleigh numbers, and the rolls show clear oscillatory behavior. The long-term fluctuations in temperature observed in point measurements correspond to the oscillations of the organized roll structure. This flow structure can be interpreted as the continuous development of the oscillatory instability of two-dimensional roll convection that is theoretically investigated around the critical Rayleigh number. Both the velocity of the large-scale flows and the frequency of the oscillation increase proportional to the square root of the Rayleigh number. This indicates that the oscillation is closely related to the circulation of large-scale flow.

We investigate numerically the influence of a homogeneous magnetic field on a ferrofluid in the gap between two concentric, independently rotating cylinders. The full Navier-Stokes equations are solved with a combination of a finite difference method and a Galerkin method. Structure, dynamics, symmetry properties, bifurcation, and stability behavior of different vortex structures are investigated for axial and transversal magnetic fields, as well as combinations of them. We show that a transversal magnetic field modulates the Taylor vortex flow and the spiral vortex flow. Thus, a transversal magnetic field induces wavy structures: wavy Taylor vortex flow (wTVF) and wavy spiral vortex flow. In contrast to the classic wTVF, which is a secondarily bifurcating structure, these magnetically generated wavy Taylor vortices are pinned by the magnetic field, i.e., they are stationary and they appear via a primary forward bifurcation out of the basic state of circular Couette flow.

The gravity-driven flow of a liquid film down a vertical flat plate in the presence of an electric field acting in a direction perpendicular to the wall is investigated. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. A strongly nonlinear long-wave evolution equation is developed for flow parameters near the critical instability conditions. The equation retains terms up to second order in the slenderness parameter in order to incorporate the effects of shear-induced growth, short-wave damping due to surface tension, electric stress effects, and dispersive effects. In the additional asymptotic limit of small but finite interfacial perturbations the dynamics are shown to be governed by a Kuramoto–Sivashinsky (KS) equation with Korteweg–de Vries dispersion, also known as the Kawahara or the generalized Kuramoto–Sivashinsky (gKS) equation, which also includes a nonlocal energy growth term that arises from the electrostatics. Extensive numerical experiments are carried out to characterize solutions to this equation. Using perturbation theory and numerical solutions, it is shown that the electric field alters the far-field decay characteristics of bound states of the gKS equation from exponential to algebraic behavior. In addition, it is demonstrated numerically that chaotic solutions of the KS equation that are regularized into traveling-wave pulses when sufficient dispersion is added can in turn become chaotic by applying a sufficiently strong electric field. It is suggested, therefore, that electric fields can be utilized to enhance interfacial turbulence and in turn increase heat or mass transfer in applications. A physical example involving electrified falling film flows of ethelene glycol fluids is furnished and shows that the theory is within reach of experiments.

We study coupled dynamical systems wherein the influence of one system on the other is cumulative: coupling signals are integrated over a time interval $\tau$. A major consequence of integrative coupling is that amplitude death occurs over a wider range and in a single region in parameter space. For coupled limit cycle oscillators (the Landau-Stuart model) we obtain an analytic estimate for the boundary of this region while for coupled chaotic Lorenz oscillators numerical results are presented. For given $\tau$ we find that there is a critical coupling strength at which the frequency of oscillations changes discontinuously.

We consider a two-dimensional system in which a dilute stream of particles collides with an oblique planar wall. Both collisions between particles and collisions between particles and the wall are inelastic. We perform numerical simulations in two dimensions and show that the mean force experienced by the wall can be a nonmonotonic function of the angle between the wall and the particle stream. We show that this occurs because particles that rebound from the wall can collide with incoming particles and be scattered. This kind of particle-particle collision can reduce the force experienced by the wall. We refer to this effect as shielding. Furthermore, we show that the force experienced by the wall may be an increasing, decreasing or nonmonotonic function of the restitution coefficient in particle-particle collisions. We derive an exact solution for the mean force on the wall if the system is dilute, and the theoretical prediction is found to be in good agreement with our numerical results. The theory allows us to explicitly quantify the effects of shielding, and thus to explain a number of interesting features. The theory generally provides a useful upper bound for the mean force.

We investigate the collapse of granular rodpiles as a function of particle (length/diameter) and pile (height/radius) aspect ratio. We find that, for all particle aspect ratios below 24, there exists a critical height $H_l$ below which the pile never collapses, maintaining its initial shape as a solid, and a second height $H_u$ above which the pile always collapses. Intermediate heights between $H_l$ and $H_u$ collapse with a probability that increases linearly with increasing height. The linear increase in probability is independent of particle length, width, or aspect ratio. When piles collapse, the runoff scales as a piecewise power law with pile height, with $r_f\sim {\tilde{H}}^{1.2\pm 0.1}$ for pile heights below ${\tilde{H}}_c\approx 0.74$ and $r_f\approx H^{0.6\pm 0.1}$ for taller piles.

We report results of calorimetric and optical investigations of binary mixtures of rodlike and bent-shaped molecules. We find that the observed critical heat anomaly associated with the smectic-$A_2$ to biaxial smectic-$A_{2b}$ transition is well described with a Fisher-renormalized form of the usual scaling expression. The effect of renormalization is large in this system in part because of the moderately steep slope of the phase boundary ($d{T}_c/dX\sim 100\text{K}$, where $X$ is the mole fraction of the bent-core molecules) and in part because of the proximity to the tricritical point. The magnitude of heat anomaly at the smectic-$A_2$–smectic-$A_{2b}$ transition showed a drastic decrease as $X$ becomes smaller. Moreover, the nematic–smectic-$A_2$ transitions investigated turned out to be always first order and the transition enthalpy showed only weak dependence on the concentration $X$. The results imply that the energy fluctuations around the smectic-$A_2$–smectic-$A_{2b}$ transition are very sensitive to the underlying ordering of the smectic-$A_2$ background.

We obtain a Fokker-Planck equation describing experimental data on the collective motion of locusts. The noise is of internal origin and due to the discrete character and finite number of constituents of the swarm. The stationary probability distribution shows a rich phenomenology including nonmonotonic behavior of several order and disorder transition indicators in noise intensity. This complex behavior arises naturally as a result of the randomness in the system. Its counterintuitive character challenges standard interpretations of noise induced transitions and calls for an extension of this theory in order to capture the behavior of certain classes of biologically motivated models. Our results suggest that the collective switches of the group’s direction of motion might be due to a random ergodic effect and, as such, they are inherent to group formation.

We report the identification of global phase synchronization (GPS) in a linear array of unidirectionally coupled Mackey-Glass time-delay systems exhibiting highly non-phase-coherent chaotic attractors with complex topological structure. In particular, we show that the dynamical organization of all the coupled time-delay systems in the array to form GPS is achieved by sequential synchronization as a function of the coupling strength. Further, the asynchronous ones in the array with respect to the main sequentially synchronized cluster organize themselves to form clusters before they achieve synchronization with the main cluster. We have confirmed these results by estimating instantaneous phases including phase difference, average phase, average frequency, frequency ratio, and their differences from suitably transformed phase coherent attractors after using a nonlinear transformation of the original non-phase-coherent attractors. The results are further corroborated using two other independent approaches based on recurrence analysis and the concept of localized sets from the original non-phase-coherent attractors directly without explicitly introducing the measure of phase.

We study a triangular network of three populations of coupled phase oscillators with identical frequencies. The populations interact nonlocally, in the sense that all oscillators are coupled to one another, but more weakly to those in neighboring populations than to those in their own population. This triangular network is the simplest discretization of a continuous ring of oscillators. Yet it displays an unexpectedly different behavior: in contrast to the lone stable chimera observed in continuous rings of oscillators, we find that this system exhibits two coexisting stable chimeras. Both chimeras are, as usual, born through a saddle-node bifurcation. As the coupling becomes increasingly local in nature they lose stability through a Hopf bifurcation, giving rise to breathing chimeras, which in turn get destroyed through a homoclinic bifurcation. Remarkably, one of the chimeras reemerges by a reversal of this scenario as we further increase the locality of the coupling, until it is annihilated through another saddle-node bifurcation.

With an underlying common theme of competing length scales, we study the many-body Schrödinger equation in a quasiperiodic potential and discuss its connection with the Kolmogorov-Arnold-Moser (KAM) problem of classical mechanics. We propose a possible visualization of such connection in experimentally accessible many-body observables. Those observables are useful probes for the three characteristic phases of the problem: the metallic, Anderson and band insulator phases. In addition, they exhibit fingerprints of nonlinear phenomena such as bifurcations and devil’s staircases. Our numerical treatment is complemented with a perturbative analysis which provides insight on the underlying physics. The perturbation theory approach is particularly useful in illuminating the distinction between the Anderson insulator and the band insulator phases in terms of paired sets of dimerized states.

The mutual interaction of two oscillating gas bubbles in different concentrations of sulfuric acid is numerically investigated. A nonlinear oscillation for spherical symmetric bubbles with equilibrium radii smaller than $10\mu \text{m}$ at a frequency of 37 kHz in a strong driving acoustical field $P_a=1.8\text{bar}$ is assumed. The calculations are based on the investigation of the secondary Bjerknes force with regard to adiabatic model for the bubble interior which appears as repulsion or attraction interaction force. In this work the influence of the various concentrations of sulfuric acid in uncoupled and coupled distances between bubbles has been investigated. It is found that the sign and value of the secondary Bjerknes force depend on the sulfuric acid viscosity and its amount would be decreased by liquid viscosity enhancement. The results show that big change in the parameters of produced bubbles occurs in the sulfuric acid with concentrations from 65% to 85%.

Three-dimensional flows are addressed in porous media with resurgences. These media are characterized by a double structure, i.e., a continuous porous medium and capillaries with impermeable walls which relate distant points of the continuous medium. Spatially periodic harmonic functions with singularities at given points are constructed by means of the Berdichevskij functions; these functions extend to three dimensions the classical Weierstrass functions. The solution of the same medium with two vertices is obtained in two and three dimensions and thoroughly compared and discussed.

Shear-induced deformation of liquid capsule enclosed by thin shell causes the development of in-plane tensions and bending moments due to the shell thickness or to a preferred three-dimensional unstressed configuration. This paper considers the effect of bending stiffness due to a preferred three-dimensional structure on the deformation and motion of the liquid capsule. To perform the numerical simulations, an improved formulation for computing the forces generated on the capsule surface during deformation is proposed. This formulation takes full account of large deformation kinematics and the development of in-plane tensions and bending moments. The deformation and orientation dynamics of capsules with different reference shapes are studied under various shear rates, viscosity ratios, and bending modulus. The numerical results show that the bending stiffness not only restricts the deformation but also affects the motion mode of the capsules. In addition, raising bending stiffness amplifies the shape deformation oscillations in tank-treading mode but reduces the oscillations in tumbling mode.

The linear stability of magnetohydrodynamic Taylor-Couette flows in axially unbounded cylinders is considered for magnetic Prandtl number unity. Magnetic background fields varying from purely axial to purely azimuthal are imposed, with a general helical field parametrized by $\beta =B_{\varphi }/B_z$. We map out the transition from the standard magnetorotational instability (MRI) for $\beta =0$ to the nonaxisymmetric azimuthal magnetorotational instability for $\beta \to \infty$. For finite $\beta$, positive and negative wave numbers $m$, corresponding to right and left spirals, are no longer degenerate. For the nonaxisymmetric modes, the most unstable mode spirals in the opposite direction to the background field. The standard $\left(\beta =0\right)$ MRI is axisymmetric for weak fields (including the instability with the lowest Reynolds number) but is nonaxisymmetric for stronger fields. If the azimuthal field is due in part to an axial current flowing through the fluid itself (and not just along the central axis), then it is also unstable to the nonaxisymmetric Tayler instability which is most effective without rotation. For purely toroidal fields the solutions for $m=\pm 1$ are identical so that in this case no preferred helicity results. For large $\beta$ the wave number $m=-1$ is preferred, whereas for $\beta \lesssim 1$ the mode with $m=-2$ is most unstable. The most unstable modes always spiral in the same direction as the background field. For background fields with positive and not too large $\beta$ the kinetic helicity of the fluctuations proves to be negative for all the magnetic instabilities considered.

The generalized traveling wave method (GTWM) is developed for the nonlinear Schrödinger equation (NLSE) with general perturbations in order to obtain the equations of motion for an arbitrary number of collective coordinates. Regardless of the particular ansatz that is used, it is shown that this alternative approach is equivalent to the Lagrangian formalism, but has the advantage that only the Hamiltonian of the unperturbed system is required, instead of the Lagrangian for the perturbed system. As an explicit example, we take 4 collective coordinates, namely the position, velocity, amplitude and phase of the soliton, and show that the GTWM yields the same equations of motion as the perturbation theory based on the Inverse Scattering Transform and as the time variation of the norm, first moment of the norm, momentum, and energy for the perturbed NLSE.

We introduce the heterogeneous voter model (HVM), in which each agent has its own intrinsic rate to change state, reflective of the heterogeneity of real people, and the partisan voter model (PVM), in which each agent has an innate and fixed preference for one of two possible opinion states. For the HVM, the time until consensus is reached is much longer than in the classic voter model. For the PVM in the mean-field limit, a population evolves to a preference-based state, where each agent tends to be aligned with its internal preference. For finite populations, discrete fluctuations ultimately lead to consensus being reached in a time that scales exponentially with population size.

The classical setting of stochastic resonance is extended to account for the presence of an arbitrary number of simultaneously stable steady states. General expressions for the linear response are derived for systems involving one variable. The existence of an optimal value of noise strength and of an optimal number of stable states for which the response is maximized is established.

The phase behavior of confined nematogens is studied using the Lebwohl-Lasher model. For three-dimensional systems the model is known to exhibit a discontinuous nematic-isotropic phase transition, whereas the corresponding two-dimensional systems apparently show a continuous Berezinskii-Kosterlitz-Thouless-like transition. In this paper, we study the phase transitions of the Lebwohl-Lasher model when confined between planar slits of different widths in order to establish the behavior of intermediate situations between the pure planar model and the three-dimensional system, and compare with previous estimates for the critical thickness, i.e., the slit width at which the transition switches from continuous to discontinuous.