In 1955, Fermi, Pasta, Ulam, and Tsingou reported recurrence over time of energy between modes in a one-dimensional array of nonlinear oscillators. Subsequently, there have been myriad numerical experiments using homogenous FPUT arrays in the form of chains of ideal, nonlinearly coupled oscillators. However, inherent variations (e.g., due to manufacturing tolerance) introduce heterogeneity into the parameters of any physical system. We demonstrate that such tolerances degrade the observance of recurrences, often leading to complete loss in moderately-sized arrays. We numerically simulate heterogeneous FPUT systems to investigate the effects of tolerances on dynamics. Our results illustrate that tolerances in real nonlinear oscillator arrays may limit the applicability of results from numerical experiments on them to physical systems, unless appropriate heterogeneities are taken into account.

Semiconductor lasers with coherent forcing are expected to behave similarly to simple neuron models in response to external perturbations, as long as the physics describing them can be approximated by that of an overdamped pendulum with fluid torque. Beyond the validity range of this approximation, more complex features can be expected. We perform experiments and numerical simulations which show that the system can display resonator and integrator features depending on parameters and that multiple pulses can be emitted in response to larger perturbations.

We interpret the problem of synchronizing multiple coupled laser diodes as a robust stabilization problem. We show that mathematical optimization, specifically constrained nonlinear programming, can be applied to identify stable and robust points of operation with optimal intensities. In contrast to existing methods, the method proposed here does not require multicriteria or Pareto-optimizations for a simultaneous treatment of optimality and robustness. It is based on enforcing a safe distance to manifolds of saddle-node and Hopf bifurcations (or generalizations thereof), where the distance can be chosen to reflect parametric uncertainties of the model or the system operation. While the method involves linearizations, it captures, in the same sense as bifurcation theory, the true nonlinear behavior of the system.

The $C$-dismantling (CD) problem aims at finding the minimum vertex set $\mathcal{D}$ of a graph $\mathcal{G}(\mathcal{V},\mathcal{E})$ after removing which the remaining graph will break into connected components with the size not larger than $C$. In this paper, we introduce a spin-glass model with $C+1$ integer-value states into the CD problem and then study the properties of this spin-glass model with the belief-propagation (BP) equations under the replica-symmetry ansatz. We give the lower bound ${\rho }_c$ of the relative size of $\mathcal{D}$ with finite $C$ on regular random graphs and Erdős-Rényi random graphs. We find ${\rho }_c$ will decrease gradually with growing $C$, and it converges to ${\rho }_{\infty }$ as $C\to \infty$. The CD problem is called the dismantling problem when $C$ is a small finite fraction of $\left|\mathcal{V}\right|$. Therefore, ${\rho }_{\infty }$ is also the lower bound of the dismantling problem when $\left|\mathcal{V}\right|\to \infty$. To reduce the computation complexity of the BP equations, taking the knowledge of the probability of a random selected vertex belonging to a remaining connected component with the size $A$, the original BP equations can be simplified to one with only three states when $C\to \infty$. The simplified BP equations are very similar to the BP equations of the feedback vertex set spin-glass model [H.-J. Zhou, Eur. Phys. J. B 86, 455 (2013)]. Finally, we develop two practical belief-propagation-guide decimation algorithms based on the original BP equations (CD-BPD) and the simplified BP equations (SCD-BPD) to solve the CD problem on a certain graph. Our BPD algorithms and two other state-of-art heuristic algorithms are applied on various random graphs and some real-world networks. Computation results show that the CD-BPD is the best of all tested algorithms in the case of small $C$. But considering the performance and computation consumption, we recommend using SCD-BPD for the network with a small clustering coefficient when $C$ is large.

Braess' paradox is an observation in traffic networks in which changes to the structure of the network—such as the addition of a road—which are intended to enhance flow can often instead have the paradoxical effect of reducing flow through the network. Versions of Braess' paradox have subsequently been observed in many other network types. Homeostasis is a seemingly unrelated biological concept in which interacting regulatory mechanisms work in concert to regulate a particular system output such that it is relatively insensitive to changes in input. A classic example is the broad range of ambient temperatures (input) over which mammalian body temperature (output) is close to constant. In this work, we propose a connection between these concepts using the mathematical formulation of infinitesimal homeostasis and argue that perturbations of homeostatic systems in disease may often lead to observations of paradoxical behavior via the universal unfoldings of infinitesimal homeostasis. We illustrate the concept with an example system drawn from the study of the pathophysiology and treatment of asthma; as a network flow model, this exhibits a form of paradoxical behavior akin to Braess' paradox which we argue arises from perturbation of homeostasis due to disease.

The spreading of a cap-shaped spherical droplet of non-Newtonian power-law liquids on a completely wettable spherical substrate is theoretically studied. Both convex spherical substrates and concave spherical cavities with smooth or rough surfaces are considered. The droplet on a rough substrate is modeled by either the Wenzel or the Cassie-Baxter model. The two sources of driving force of spreading by the surface tension and the line tension are considered. Also, the two channels of energy dissipation by the viscous dissipation within the bulk and the frictional dissipation at the contact line are considered. A combined theory of spreading on a spherical substrate is constructed by including those four factors. The spreading process is divided into four stages, each of which is governed by one of two driving forces and one of two dissipations. It is found that the dynamic contact angle $\theta$ has a characteristic time ($t$) dependence at each stage. It does not necessarily follow the standard power law $\theta \sim t^{-\alpha }$. Instead, the relaxation can be a power law with the exponent $\alpha$ different from that on a flat substrate, or it can be exponential or it can finish within a finite time. Therefore, various spreading scenarios on a spherical substrate and in a spherical cavity are predicted.

We study the stability of granular systems constituted by particles with inhomogeneous distribution of inner mass, which is a factor that has not been previously analyzed. In particular, we report experimental results concerning the stability of two-dimensional piles of hollow plexiglass cylinders containing an inserted smaller metal rod. These particles partially fill a rough-edged drum whose rotation produces successive avalanches. The distribution of the maximum angle of stability, the angle of repose, and the avalanche size are analyzed for systems in which the inserts are fixed or free to move. We reveal two interesting effects related to the nonuniform inner mass distribution. On the one hand, the disorder in the array of the center of mass, enhanced in the system of particles with a fixed insert, decreases the stability of the pile. On the other hand, both mobile or fixed inserts lead to larger erosion of the free surface once the avalanche is triggered.

We theoretically and experimentally investigate low-Reynolds-number propulsion of geometrically achiral planar objects that possess a dipole moment and that are driven by a rotating magnetic field. Symmetry considerations (involving parity $\widehat{P}$ and charge conjugation $\widehat{C}$) establish correspondence between propulsive states depending on orientation of the dipolar moment. Although basic symmetry arguments do not forbid individual symmetric objects to efficiently propel due to spontaneous symmetry breaking, they suggest that the average ensemble velocity vanishes. Some additional arguments show, however, that highly symmetrical ($\widehat{P}$-even) objects exhibit no net propulsion while individual less symmetrical ($\widehat{C}\widehat{P}$-even) propellers do propel. Particular magnetization orientation, rendering the shape $\widehat{C}\widehat{P}$-odd, yields unidirectional motion typically associated with chiral structures, such as helices. If instead of a structure with a permanent dipole we consider a polarizable object, some of the arguments have to be modified. For instance, we demonstrate a truly achiral ($\widehat{P}$- and $\widehat{C}\widehat{P}$-even) planar shape with an induced electric dipole that can propel by electrorotation. We thereby show that chirality is not essential for propulsion due to rotation-translation coupling at low Reynolds number.

The second- and third-order zonal oscillations of metallic droplets at high temperatures beyond 2000 K were experimentally achieved by electrostatic levitation. To quantitatively describe the suspension stability of different metallic droplets, a stability factor model was proposed as a function of the surface tension and density. The influences of droplet size and temperature on the oscillation pattern, oscillation frequency, and oscillation amplitude were analyzed. As a supplement to experiment, a feasible mathematical model of droplet deformation and oscillation, coupling the effects of electrostatic and flow fields, was established to study the underlying mechanism of droplet dynamics at high temperatures. The simulation not only reproduced the experimental observations, but also predicted the evolution characteristics of higher-order oscillations. The inherent relationships between the oscillation frequency and the droplet size, density, and surface tension of liquid metals were systematically investigated by simulation. Moreover, the suspension stabilities of nine typical metallic droplets were derived and compared with the simulation results.

The interaction of a high-power laser pulse with a thin foil can generate energetic, broadband terahertz radiation. Here, we report an experimental investigation on the influence of incident laser polarization and wavelength on the terahertz emission and maximum proton energy from the target rear surface. For similar incident laser intensities, the characteristics of the particle beams and the terahertz radiation show a wavelength dependence. The results fit well with the established scaling laws for the terahertz yield and the maximum proton energy as a function of the incident laser irradiance ($I{\lambda }^2$).

Evolutionary game theory has been widely used to study the evolution of cooperation in social dilemmas where imitation-led strategy updates are typically assumed. However, results of recent behavioral experiments are not compatible with the predictions based on the imitation dynamics, casting doubts on the assumption of the imitation-led updates and calling for alternative mechanisms of strategy updates. An aspiration-led update is often considered as an alternative to the imitation-led one. While details of update rules can have significant impacts on the evolutionary outcomes and many variations in imitation-led updates are thus studied, however, few variations exist in aspiration-led updates. We introduce an aspiration-led update mechanism (“Satisfied-Defect, Unsatisfied-Cooperate”) that is psychologically intuitive and can yield a behavior richer than the conventional aspiration-led update does in Prisoner's Dilemma games. Using analytical and numerical methods, we study and link the stochastic dynamics of it in well-mixed finite populations and the deterministic dynamics of infinite populations.

The transverse-field Ising model on the Sierpiński fractal, which is characterized by the fractal dimension ${log}_2^{}3\approx 1.585$, is studied by a tensor-network method known as the higher-order tensor renormalization group. We analyze the ground-state energy and the spontaneous magnetization in the thermodynamic limit. The system exhibits the second-order phase transition at the critical transverse field $h_{\mathrm{c}}^{}=1.865$. The critical exponents $\beta \approx 0.198$ and $\delta \approx 8.7$ are obtained. Complementary to the tensor-network method, we make use of the real-space renormalization group and improved mean-field approximations for comparison.

We consider the product of a large number of matrices chosen randomly (with some correlation) as $A$ or $B$. We introduce an exact functional equation, whose solution gives the limiting distribution and the Lyapunov exponent for the product of large number of matrices. Thus we suggest new semianalytical method to solve the random product problem of correlated matrices. The method can be easily generalized to high order matrices, as we deal with a larger systems of functional equations. We have provided a known numerical algorithm for the solution of the matrix product problem. We have also applied our method for a solution of discrete time random evolution. Our functional equation method provides exact transition point.

We investigate percolation on a randomly directed lattice, an intermediate between standard percolation and directed percolation, focusing on the isotropic case in which bonds on opposite directions occur with the same probability. We derive exact results for the percolation threshold on planar lattices, and we present a conjecture for the value of the percolation-threshold in any lattice. We also identify presumably universal critical exponents, including a fractal dimension, associated with the strongly connected components both for planar and cubic lattices. These critical exponents are different from those associated either with standard percolation or with directed percolation.

We present a detailed study of instabilities arising in optoelectronic oscillators with a single narrow-band time-delayed feedback loop. Such optoelectronic oscillators produce periodic solutions and may be useful as high-purity signal generators, but their nonlinear dynamics sets limits on available signal amplitudes. Starting from an integrodifferential model, we utilize approximate analytic solutions to find the stability boundaries of the periodic solutions as well as regions of multistability. Our analytical predictions are confirmed by numerical simulations and experiments.

This paper reports a robust regime in the dynamics of three coupled modes when one mode is pumped, the second is dissipated, and the third is not subject to any external action. In this regime, the first and the second modes vanish, while the third mode becomes large. This is applied to the problem of plasma confinement by magnetic field, when the third mode represents zonal (poloidal) flow. This paper demonstrates that in order to generate strong zonal (poloidal) flow—a transport barrier—in magnetically confined fusion plasma, it is beneficial to have significant decrement in addition to increment of other modes.

Reentrant spiral waves (also called rotors) have been observed in the heart muscle during cardiac arrhythmias and are nowadays targeted by ablation therapy in order to cure certain heart rhythm disorders. Phase singularity (PS) is considered to represent the organizing center of the spiral wave and the spatiotemporal behavior of the spiral wave can be extensively quantified by tracking PS. Recent clinical studies suggested that ablating the tissue at PS locations may cure atrial fibrillation. However, for experimental data, typically only one state variable, i.e., the voltage, is recorded. Accordingly, the calculation of phase and the identification of PSs must be carried out by using one variable. Here, a Jacobian-determinant method using one variable in reconstructed state space is proposed and the advantages of the method are demonstrated.

Solid-solid phase transitions are commonly encountered at the atomic scale in alloys and in superatomic mesoscopic systems of colloidal particles. Here we investigate a solid-solid phase transition occurring at the macroscopic scale between lattices of liquid jets with different symmetries generated by convection in a horizontal layer of a binary liquid mixture. In the absence of a shear stress, upwelling and downwelling jets arrange into two staggered square lattices with a spacing of approximately 3 mm. Applying a shear stress triggers a phase separation of the square patterns into two centered-rectangular lattices drifting into opposite directions, each lattice being made either by upwelling or downwelling jets. This structural phase transition is reversible. The macroscopic nature of the system allows us to investigate the kinetics of the transition by direct visualization with shadowgraphy. The mechanism of the transition depends on the path followed. It occurs through a nucleation and growth mechanism when the shear stress is imposed, and through a martensitic transformation of the lattice when the stress is removed.

We investigate the phonon spectra of two-dimensional liquid dusty plasmas on a one-dimensional periodic substrate using numerical simulations. The propagation of the waves across the potential wells of the substrate is inhibited due to the confinement of the dust particles by the substrate minima. If the substrate wells are narrow or deep, one-dimensional chains of particles are formed in each minimum, and the longitudinal motion of an individual chain dominates the propagation of waves along the potential wells of the substrate. Increasing the width or decreasing the depth of the substrate minima allows the particles to buckle into a zigzag structure, and the resulting spectra develop two branches, one for sloshing motion and one for breathing motion. The repulsion between neighboring dust particles produces backward propagation of the sloshing wave for small wave numbers.

Using numerical simulations, we examine the structure and diffusion of a two-dimensional dusty plasma (2DDP) in the presence of a one-dimensional periodic substrate (1DPS) as a function of increasing substrate strength. Both the pair correlation function perpendicular to the substrate modulation and the mean-squared displacement (MSD) of dust particles are calculated. It is found that both the structure and dynamics of 2DDP exhibit strong anisotropic effects, due to the applied 1DPS. As the substrate strength increases from zero, the structure order of dusty plasma along each potential well of 1DPS increases first probably due to the competition between the interparticle interactions and the particle-substrate interactions, and then decreases gradually, which may be due to the reduced dimensionality and the enhanced fluctuations. The obtained MSD along potential wells of 1DPS clearly shows three processes of diffusion in our studied 2DDP. Between the initial ballistic and finally diffusive motion, there is the intermediate subdiffusion discovered here, which may result from the substrate-induced distortion of the caging dynamics.