Much important information about the structural and dynamical properties of complex systems can be extracted from the eigenvalues and eigenvectors of a Markov matrix associated with random walks performed on these systems, and spectral methods have become an indispensable tool in the complex system analysis. In this paper, we study the Markov matrix of a class of scale-free polymer networks. We present an exact analytical expression for all the eigenvalues and determine explicitly their multiplicities. We then use the obtained eigenvalues to derive an explicit formula for the random target access time for random walks on the studied networks. Furthermore, based on the link between the eigenvalues of the Markov matrix and the number of spanning trees, we confirm the validity of the obtained eigenvalues and their corresponding degeneracies.

A bouncing droplet on a vibrated bath can couple to the waves it generates, so that it becomes a propagative walker. Its propulsion at constant velocity means that a balance exists between the permanent input of energy provided by the vibration and the dissipation. Here we seek a simple theoretical description of the resulting non-Hamiltonian dynamics with a walker immersed in a harmonic potential well. We demonstrate that the interaction with the recently emitted waves can be modeled by a Rayleigh-type friction. The Rayleigh oscillator has well defined attractors. The convergence toward them and their stability is investigated through an energetic approach and a linear stability analysis. These theoretical results provide a description of the dynamics in excellent agreement with the experimental data. It is thus a basic framework for further investigations of wave-particle interactions when memory effects are included.

A mechanical system is presented exhibiting a nondeterministic singularity, that is, a point in an otherwise deterministic system where forward time trajectories become nonunique. A Coulomb friction force applies linear and angular forces to a wheel mounted on a turntable. In certain configurations, the friction force is not uniquely determined. When the dynamics evolves past the singularity and the mechanism slips, the future state becomes uncertain up to a set of possible values. For certain parameters, the system repeatedly returns to the singularity, giving recurrent yet unpredictable behavior that constitutes nondeterministic chaotic dynamics. The robustness of the phenomenon is such that we expect it to persist with more sophisticated friction models, manifesting as extreme sensitivity to initial conditions, and complex global dynamics attributable to a local loss of determinism in the limit of discontinuous friction.

This paper presents results on stripe patterns by numerical solution of the Swift-Hohenberg equation. The focus is on the role of initial state and boundary conditions. We choose initial states which generate simple defect configurations and study their evolution. Various classes of defects are identified and their motion and relaxation is studied numerically. We first study the dynamics of a straight front and present a comparison of numerical results with some analytical results. We then study the domain-wall dynamics in configurations containing two and three domains and identify some mechanisms of their relaxation. Rates of domain-wall relaxation depend on several features like incommensuration, dislocations and orientations in neighboring domains, in addition to the curvature of the walls. For a generic class of domain walls the relaxation process has an intrinsic frustration which leads to generation of dislocations. This process also generates stripe curvature thereby making relaxation nonmonotonic. We have also generated some other topological defects and studied their evolution and the effect of boundary conditions on their stability.

Three-dimensional scroll waves of electrical activity are among the mechanisms believed to be responsible for the rapid, unsynchronized contraction of cardiac ventricles, thereby reducing the heart's ability to pump blood. Scroll waves can attach themselves to unexcitable obstacles, and this sometimes highly elongates their life span. Hence, the unpinning and annihilation of these vortices has attracted much attention in recent decades. In this work, we study the influence of a thermal gradient on scroll waves pinned to inert obstacles, in the Belousov-Zhabotinsky reaction. Under a temperature gradient, scroll rings were seen to unpin from these obstacles, thus strikingly reducing their lifetimes. These results were also reproduced by numerical simulations using the Barkley model.

A type of laser-induced surface relief nanostructure—the nanocrown—on thin metallic films was studied both experimentally and theoretically. The nanocrowns, representing a thin corrugated rim of resolidified melt and resembling well-known impact-induced water-crown splashes, were produced by single diffraction-limited nanosecond laser pulses on thin gold films of variable thickness on low-melting copper and high-melting tungsten substrates, providing different transient melting and adhesion conditions for these films. The proposed model of the nanocrown formation, based on a hydrodynamical (thermocapillary Marangoni) surface instability and described by a Kuramoto-Sivashinsky equation, envisions key steps of the nanocrown appearance and gives qualitative predictions of the acquired nanocrown parameters.

We show that transverse electromagnetic waves propagating along an external static electric field in liquid metacrystal (LMC) can provoke spontaneous rearrangement of elongated meta-atoms that changes the direction of the anisotropy axis of the LMC. This kind of instability may reorient the meta-atoms from the equilibrium state parallel to a static field to the state along a high-frequency field and back at the different threshold intensities of electromagnetic waves in such a way that bistability in the system takes place. Reorientation of meta-atoms causes a change in the effective refraction index of LMC that creates, in turn, the conditions for the formation of bright spatial solitons. Such spatial solitons are the self-consistent domains of redirected meta-atoms with trapped photons. We find that the instability thresholds as well as energy flux captured by the spatial soliton can be easily managed by variation of the static electric field applied to the LMC. We study the effects of soliton excitation and collisions via numerical simulations.

Numerical simulations reveal emergence of subspinodal length scales in spontaneous dewetting of nonslipping unstable thin liquid films on homogeneous substrates if the liquid viscosity decreases with decrease in film thickness.

We use optical microscopy techniques to directly visualize the structures that emerge in binary mixtures and pure fluids near their respective critical points. We attempt to understand these structures by studying the image formation using both a phase contrast and a dark field filter to our microscope. We found that images of critical fluctuations for both liquid-liquid and liquid-gas critical systems have gray level intensity histograms with Gaussian shape. For all fluids investigated, the temperature-dependent standard deviation of the Gaussian histogram follows a power law with the same exponent. Since the image intensity fluctuations are determined by order parameter fluctuations, this direct imaging method allowed us to estimate the critical exponent of compressibility with very good accuracy.

We study the scaling limit of a planar loop-erased random walk (LERW) on the percolation cluster, with occupation probability $p\ge p_c$. We numerically demonstrate that the scaling limit of planar ${\mathrm{LERW}}_p$ curves, for all $p>p_c$, can be described by Schramm-Loewner evolution (SLE) with a single parameter $\kappa$ that is close to the normal LERW in a Euclidean lattice. However, our results reveal that the LERW on critical incipient percolation clusters is compatible with SLE, but with another diffusivity coefficient $\kappa$. Several geometrical tests are applied to ascertain this. All calculations are consistent with ${\mathrm{SLE}}_{\kappa }$, where $\kappa =1.732\pm 0.016$. This value of the diffusivity coefficient is outside the well-known duality range $2\le \kappa \le 8$. We also investigate how the winding angle of the ${\mathrm{LERW}}_p$ crosses over from Euclidean to fractal geometry by gradually decreasing the value of the parameter $p$ from 1 to $p_c$. For finite systems, two crossover exponents and a scaling relation can be derived. This finding should, to some degree, help us understand and predict the existence of conformal invariance in disordered and fractal landscapes.

Starting from a Langevin description of active particles that move with constant speed in infinite two-dimensional space and its corresponding Fokker-Planck equation, we develop a systematic method that allows us to obtain the coarse-grained probability density of finding a particle at a given location and at a given time in arbitrary short-time regimes. By going beyond the diffusive limit, we derive a generalization of the telegrapher equation. Such generalization preserves the hyperbolic structure of the equation and incorporates memory effects in the diffusive term. While no difference is observed for the mean-square displacement computed from the two-dimensional telegrapher equation and from our generalization, the kurtosis results in a sensible parameter that discriminates between both approximations. We carry out a comparative analysis in Fourier space that sheds light on why the standard telegrapher equation is not an appropriate model to describe the propagation of particles with constant speed in dispersive media.

Within the Rayleigh-Helmholtz model of active Brownian particles, activity is due to a nonlinear velocity-dependent force. In the presence of external trapping potential or constant force, the steady state of the system breaks detailed balance producing a net entropy. Using molecular dynamics simulations, we obtain the probability distributions of entropy production in these steady states. The distribution functions obey fluctuation theorems for entropy production. Using the simulation, we further show that the steady-state response function obeys a modified fluctuation-dissipation relation.

Using a fluid-particle dynamics approach, we numerically study the effects of hydrodynamic interactions on the collective dynamics of active suspensions within a simple model for bacterial motility: each microorganism is modeled as a stroke-averaged dumbbell swimmer with prescribed dipolar force pairs. Using both simulations and qualitative arguments, we show that, when the separation between swimmers is comparable to their size, the swimmers' motions are strongly affected by activity-induced hydrodynamic forces. To further understand these effects, we investigate semidilute suspensions of swimmers in the presence of thermal fluctuations. A direct comparison between simulations with and without hydrodynamic interactions shows these to enhance the dynamic clustering at a relatively small volume fraction; with our chosen model the key ingredient for this clustering behavior is hydrodynamic trapping of one swimmer by another, induced by the active forces. Furthermore, the density dependence of the motility (of both the translational and rotational motions) exhibits distinctly different behaviors with and without hydrodynamic interactions; we argue that this is linked to the clustering tendency. Our study illustrates the fact that hydrodynamic interactions not only affect kinetic pathways in active suspensions, but also cause major changes in their steady state properties.

Curvature-sensing and curvature-remodeling proteins, such as Amphiphysin, Epsin, and Exo70, are known to reshape cell membranes, and this remodeling event is essential for key biophysical processes such as tubulation, exocytosis, and endocytosis. Curvature-inducing proteins can act as curvature sensors; they aggregate to membrane regions matching their intrinsic curvature; as well as induce curvature in cell membranes to stabilize emergent high curvature, nonspherical, structures such as tubules, discs, and caveolae. A definitive understanding of the interplay between protein recruitment and migration, the evolution of membrane curvature, and membrane morphological transitions is emerging but remains incomplete. Here, within a continuum framework and using the machinery of Monte Carlo simulations, we introduce and compare three free-energy methods to delineate the free-energy landscape of curvature-inducing proteins on bilayer membranes. We demonstrate the utility of the Widom test particle (or field) insertion methodology in computing the excess chemical potentials associated with curvature-inducing proteins on the membrane—in particular, we use this method to track the onset of morphological transitions in the membrane at elevated protein densities. We validate this approach by comparing the results from the Widom method with those of thermodynamic integration and Bennett acceptance ratio methods. Furthermore, the predictions from the Widom method have been tested against analytical calculations of the excess chemical potential at infinite dilution. Our results are useful in precisely quantifying the free-energy landscape, and also in determining the phase boundaries associated with curvature-induction, curvature-sensing, and morphological transitions. This approach can be extended to studies exploring the role of thermal fluctuations and other external (control) variables, such as membrane excess area, in shaping curvature-mediated interactions on bilayer membranes.

We report about the main dynamical features of a model of leaky integrate-and-fire excitatory neurons with short-term plasticity defined on random massive networks. We investigate the dynamics by use of a heterogeneous mean-field formulation of the model that is able to reproduce dynamical phases characterized by the presence of quasisynchronous events. This formulation allows one to solve also the inverse problem of reconstructing the in-degree distribution for different network topologies from the knowledge of the global activity field. We study the robustness of this inversion procedure by providing numerical evidence that the in-degree distribution can be recovered also in the presence of noise and disorder in the external currents. Finally, we discuss the validity of the heterogeneous mean-field approach for sparse networks with a sufficiently large average in-degree.

We study the stability of the synchronized state in time-varying complex networks using the concept of basin stability, which is a nonlocal and nonlinear measure of stability that can be easily applied to high-dimensional systems [P. J. Menck, J. Heitzig, N. Marwan, and J. Kurths, Nature Phys. 9, 89 (2013)]. The time-varying character is included by stochastically rewiring each link with the average frequency $f$. We find that the time taken to reach synchronization is lowered and the stability range of the synchronized state increases considerably in dynamic networks. Further we uncover that small-world networks are much more sensitive to link changes than random ones, with the time-varying character of the network having a significant effect at much lower rewiring frequencies. At very high rewiring frequencies, random networks perform better than small-world networks and the synchronized state is stable over a much wider window of coupling strengths. Lastly we show that the stability range of the synchronized state may be quite different for small and large perturbations, and so the linear stability analysis and the basin stability criterion provide complementary indicators of stability.

We present a method to find the best temporal partition at any time scale and rank the relevance of partitions found at different time scales. This method is based on random walkers coevolving with the network and as such constitutes a generalization of partition stability to the case of temporal networks. We show that, when applied to a toy model and real data sets, temporal stability uncovers structures that are persistent over meaningful time scales as well as important isolated events, making it an effective tool to study both abrupt changes and gradual evolution of a network mesoscopic structures.

We study the stability of orbits in large Boolean networks. We treat the case in which the network has a given complex topology, and we do not assume a specific form for the update rules, which may be correlated with local topological properties of the network. While recent past work has addressed the separate effects of complex network topology and certain classes of update rules on stability, only crude results exist about how these effects interact. We present a widely applicable solution to this problem. Numerical simulations confirm our theory and show that local correlations between topology and update rules can have profound effects on the qualitative behavior of these systems.

We study a model of globally coupled phase oscillators that contains two groups of oscillators with positive (synchronizing) and negative (desynchronizing) incoming connections for the first and second groups, respectively. This model was previously studied by Hong and Strogatz (the Hong-Strogatz model) in the case of a large number of oscillators. We consider a generalized Hong-Strogatz model with a constant phase shift in coupling. Our approach is based on the study of invariant manifolds and bifurcation analysis of the system. In the case of zero phase shift, various invariant manifolds are analytically described and a new dynamical mode is found. In the case of a nonzero phase shift we obtained a set of bifurcation diagrams for various systems with three or four oscillators. It is shown that in these cases system dynamics can be complex enough and include multistability and chaotic oscillations.