It is a challenging problem to establish safe and simple therapeutic methods for various complicated diseases of the nervous system, particularly dynamical diseases such as epilepsy, Alzheimer’s disease, and Parkinson’s disease. From the viewpoint of nonlinear dynamical systems, a dynamical disease can be considered to be caused by a bifurcation induced by a change in the values of one or more regulating parameter. Therefore, the theory of bifurcation control may have potential applications in the diagnosis and therapy of dynamical diseases. In this study, we employ a washout filter-aided dynamic feedback controller to control the onset of Hopf bifurcation in the Hodgkin-Huxley (HH) model. Specifically, by the control scheme, we can move the Hopf bifurcation to a desired point irrespective of whether the corresponding steady state is stable or unstable. In other words, we are able to advance or delay the Hopf bifurcation, so as to prevent it from occurring in a certain range of the externally applied current. Moreover, we can control the criticality of the bifurcation and regulate the oscillation amplitude of the bifurcated limit cycle. In the controller, there are only two terms: the linear term and the nonlinear cubic term. We show that while the former determines the location of the Hopf bifurcation, the latter regulates the criticality of the Hopf bifurcation. According to the conditions of the occurrence of Hopf bifurcation and the bifurcation stability coefficient, we can analytically deduce the linear term and the nonlinear cubic term, respectively. In addition, we also show that mixed-mode oscillations (MMOs), featuring slow action potential generation, which are frequently observed in both experiments and models of chemical and biological systems, appear in the controlled HH model. It is well known that slow firing rates in single neuron models could be achieved only by type-I neurons. However, the controlled HH model is still classified as a type-II neuron, as is the original HH model. We explain that the occurrence of MMOs can be related to the presence of the canard explosion where a small oscillation grows through a sequence of canard cycles to a relaxation oscillation as the control parameter moves through an interval of exponentially small width.

A viscous thread falling from a nozzle onto a surface exhibits the famous rope-coiling effect, in which the thread buckles to form loops. If the surface is replaced by a belt moving with speed $U$, the rotational symmetry of the buckling instability is broken and a wealth of interesting states are observed [see S. Chiu-Webster and J. R. Lister, J. Fluid Mech. 569, 89 (2006)]. We experimentally studied this “fluid-mechanical sewing machine” in a more precise apparatus. As $U$ is reduced, the steady catenary thread bifurcates into a meandering state in which the thread displacements are only transverse to the motion of the belt. We measured the amplitude and frequency $\omega$ of the meandering close to the bifurcation. For smaller $U$, single-frequency meandering bifurcates to a two-frequency “figure-8” state, which contains a significant $2\omega$ component and parallel as well as transverse displacements. This eventually reverts to single-frequency coiling at still smaller $U$. More complex, highly hysteretic states with additional frequencies are observed for larger nozzle heights. We propose to understand this zoology in terms of the generic amplitude equations appropriate for resonant interactions between two oscillatory modes with frequencies $\omega$ and $2\omega$. The form of the amplitude equations captures both the axisymmetry of the $U=0$ coiling state and the symmetry-breaking effects induced by the moving belt.

We study the Olami-Feder-Christensen model with quenched disorder in the coupling parameter $\alpha$. In contrast to an earlier study by Mousseau [Phys. Rev. Lett. 77, 968 (1996)], we do not find a phase diagram with several phase transitions, but continuous crossovers from one type of behavior to another. The crossover behavior is determined by the ratio of three length scales, which are the system size, the penetration depth of the boundary layer, and the correlation length introduced by the disorder.

We derive exact results for the spectral density $S\left(\omega \right)$ of the output of the Preisach model, a standard model for complex, nonlocal hysteresis. We obtain general results for uncorrelated input signals with arbitrary input and Preisach densities. It is shown analytically that uncorrelated input signals are transformed into output exhibiting long-time correlations. For the simplest example of uniform input and Preisach distributions we prove that correlations decay asymptotically with a $t^{-3}$ power law corresponding to a logarithmic low frequency divergence of the second derivative of the spectrum $S\left(\omega \right)$. A simpler expression for symmetric Preisach models is also obtained, which is discussed in detail in Part II, showing that long-time tails or even $1/f$ noise are general features of this class of models.

The Preisach model with symmetric elementary hysteresis loops and uncorrelated input is treated analytically in detail. It is shown that the appearance of long-time tails in the output correlations is a quite general feature of this model. The exponent $\eta$ of the algebraic decay $t^{-\eta }$, which may take any positive value, is determined by the tails of the input and the Preisach density. We identify the system classes leading to identical algebraic tails. These results imply the occurrence of $1/f$ noise for a large class of hysteretic systems.

We describe an approach for exploring microscopic properties of granular media that couples x-ray microtomography and distinct-element-method (DEM) simulations through image analysis. We illustrate it via the study of the intriguing phenomenon of instant arching in an hourglass (in our case a cylinder filled with a polydisperse mixture of glass beads that has a small circular shutter in the bottom). X-ray tomography provides three-dimensional snapshots of the microscopic conditions of the system both prior to opening the shutter, and thereafter, once jamming is completed. The process time in between is bridged using DEM simulation, which settles to positions in remarkably good agreement with the x-ray images. Specifically designed image analysis procedures accurately extract the geometrical information, i.e., the positions and sizes of the beads, from the raw x-ray tomographs, and compress the data representation from initially 5 gigabytes to a few tens of kilobytes per tomograph. The scope of the approach is explored through a sensitivity analysis to input data perturbations in both bead sizes and positions. We establish that accuracy of size—much more than position—estimates is critical, thus explaining the difficulty in considering a mixture of beads of different sizes. We further point to limits in the replication ability of granular flows away from equilibrium; i.e., the difficulty of numerically reproducing chaotic motion.

To understand the self-organization of magnetic nanocrystals in an applied field, we perform Monte Carlo simulations of Stockmayer fluids confined between two parallel walls. The system is examined in the gas-liquid coexistence region of its phase diagram and the field is applied perpendicular to the walls. Gibbs ensemble simulations are carried out to determine the phase coexistence curves of the confined Stockmayer fluid. In canonical simulations, different types of organizations appear dependent on particle density: columns, walls, and elongated and spherical holes. The morphology and size of structures are in good agreement with results obtained by free energy minimization and experiments. The influence of a distribution of particle sizes on the particle organization is investigated.

To design conformally invisible cloaks for arbitrarily shaped objects, we use the nonuniform rational $B$-spline (NURBS) to represent the geometrical modeling of the arbitrary object. Based on the method of optical transformation, analytical formulas of the permittivity and permeability tensors are proposed for arbitrarily shaped invisible cloaks. Such formulas can be easily implemented in the design of arbitrary cloaks. Full-wave simulations are given for heart-shaped invisible cloaks and perfectly electrical conducting (PEC) objects, in which we observe that the power-flow lines of incoming electromagnetic waves will be bent smoothly in the cloaks and will return to their original propagation directions after propagating around the object. We also show that the scattered field from the PEC object coated with the invisible cloak is much smaller than that from the PEC core. The application of NURBS in the design of arbitrary cloaks shows transformation optics to be a very general tool to interface with commercial softwares like 3D STUDIOMAX and MAYA.

We investigate a reversible polymerization process in which individual polymers aggregate and fragment at a rate proportional to their molecular weight. We find a nonequilibrium phase transition despite the fact that the dynamics are perfectly reversible. When the strength of the fragmentation process exceeds a critical threshold, the system reaches a thermodynamic steady state where the total number of polymers is proportional to the system size. The polymer length distribution has a sharp exponential tail in this case. When the strength of the fragmentation process falls below the critical threshold, the steady state becomes nonthermodynamic as the total number of polymers grows sublinearly with the system size. Moreover, the length distribution has an algebraic tail and the characteristic exponent varies continuously with the fragmentation rate.

A discrete model exhibiting conserved dynamics with nonconserved noise involving particles of different nature, termed as linear and nonlinear, is proposed here. The morphology of the surface has been studied with different abundances of these particles. The saturated surface, slowly evolved from a lower contribution of nonlinear particles to a higher contribution of nonlinear particles, splits into four distinct scaling regimes with three crossover lengths. Each regime is characterized by different scaling property. It is shown that when the contribution of the nonlinear particles crosses a critical value, the surface morphology shows a linear-nonlinear “phase transition.” The roughness exponent in a nonlinear regime is well compared with that of the continuum nonlinear equation in a molecular beam epitaxy (MBE) class as well as a MBE motivated discrete model.

The three-dimensional structure of a protein can be treated as a complex network composed of amino acids, and the network properties can help us to understand the relationship between structure and function. Since the amino acid network of a protein is formed in the process of protein folding, it is difficult for general network models to explain its evolving mechanism. Based on the perspective of protein folding, we propose an evolving model for amino acid networks. In our model, the evolution starts from the amino acid sequence of a native protein and it is guided by two generic assumptions: i.e., the neighbor preferential rule and the energy preferential rule. We find that the neighbor preferential rule predominates the general network properties and the energy preferential rule predominates the specific biological structure characteristics. Applied to native proteins, our model mimics the features of amino acid networks well.

The wake behind conducting dust grains in a supersonic plasma flow with a directed photon flux is studied by the particle-in-cell method. The electron emission leads to a positive charge on the dust. The resulting plasma wake differs significantly from the case without photoelectrons. This wake is studied for different photon fluxes and different angles between the incoming unidirectional photons and the plasma flow velocity. The simulations are carried out in two spatial dimensions, treating ions and electrons as individual particles.

The physical factors controlling the power-law behavior of impact energy in a composite granular chain remain elusive. Based on event-driven simulations and the on-off intermittency of wave reflections, we obtain the probability distribution functions of the waiting time $\tau$ and the energy leakage $\mathrm{\Delta }E$. They exhibit lognormal distributions, which together with the relationship between $\mathrm{\Delta }E$ and $\tau$ allow one to explain directly the power-law behavior of the confined energy. This work may be extended to higher dimensions and help us understand the complex dynamics in granular materials.

The dynamics of a discrete polymer in time-dependent external potentials is studied with the master equation approach. We consider both stochastic and deterministic switching mechanisms for the potential states and give the essential equations for computing the stationary-state properties of molecules with internal structure in time-dependent periodic potentials on a lattice. As an example, we consider standard and modified Rubinstein-Duke polymers and calculate their mean drift and effective diffusion coefficient in the two-state nonsymmetric flashing potential and symmetric traveling potential. Rich nonlinear behavior of these observables is found. By varying the polymer length, we find current inversions caused by the rebound effect that is only present for molecules with internal structure. These results depend strongly on the polymer type. We also notice increased transport coherence for longer polymers.

We compute the effect of hydrodynamic interaction and stretching on the fluctuation properties of a polymer, with its end points held fixed. Computing the preaveraged hydrodynamic tensor exactly for this geometry, we study both flexible and semiflexible polymer chains, such as Zimm, freely jointed chain, and wormlike chain (WLC) models. We compare the spectra of relaxation-time scales for the effective normal modes of these models. The spectra differ across models with respect to the degree of stretch, but their power-law scaling with low mode numbers turns out to be the same. The characteristics of the transverse modes of WLC agree very well with the experimental data on DNA. The crossover scaling function for $⟨1∕r⟩$, the inverse of the distance along the polymer contour, yields a modified formula for the size of a “Pincus blob,” appropriate for the fixed-end boundary condition.

We study the structural characteristics of complex networks using the representative eigenvectors of the adjacent matrix. The probability distribution function of the components of the representative eigenvectors are proposed to describe the localization on networks where the Euclidean distance is invalid. Several quantities are used to describe the localization properties of the representative states, such as the participation ratio, the structural entropy, and the probability distribution function of the nearest neighbor level spacings for spectra of complex networks. Whole-cell networks in the real world and the Watts-Strogatz small-world and Barabasi-Albert scale-free networks are considered. The networks have nontrivial localization properties due to the nontrivial topological structures. It is found that the ascending-order-ranked series of the occurrence probabilities at the nodes behave generally multifractally. This characteristic can be used as a structural measure of complex networks.

The impossibility to use the MCMC (Markov chain Monte Carlo) methods for long noisy chaotic time series (TS) (due to high computational complexity) is a serious limitation for reconstruction of dynamical systems (DSs). In particular, it does not allow one to use the universal Bayesian approach for reconstruction of a DS in the most interesting case of the unknown evolution operator of the system. We propose a technique that makes it possible to use the MCMC methods for Bayesian reconstruction of a DS from noisy chaotic TS of arbitrary long duration.

We present a complexity measure for any finite time series. This measure has invariance under any monotonic transformation of the time series, has a degree of robustness against noise, and has the adaptability of satisfying almost all the widely accepted but conflicting criteria for complexity measurements. Surprisingly, the measure is developed from Kolmogorov complexity, which is traditionally believed to represent only randomness and to satisfy one criterion to the exclusion of the others. For familiar iterative systems, our treatment may imply a heuristic approach to transforming symbolic dynamics into permutation dynamics and vice versa.

In the first part of this article we study the hysteretic bistable response of Duffing oscillators and show ways to control the switching between stable branches of this nonlinear response. The control mechanism is either applied through a pulse that can be in phase or out of phase with the periodic driving force or through a frequency-modulated driving force. In the second part we show how memory effects in dissipation qualitatively and quantitatively alter the dynamics of Duffing oscillators. We show how memory functions corresponding to different dissipative regimes (diffusion, subdiffusion, and superdiffusion) affect the oscillator. In particular, we obtain universal power laws for the absoption when the driving frequency $\omega \to 0$. For subdiffusive memories the power law exponents $\nu <2$, for diffusive memories $\nu =2$, and for superdiffusive memories $\nu >2$.

We consider the logistic map wherein the nonlinearity parameter is periodically modulated. For low periods, there is multistability, namely two or more distinct dynamical attractors coexist. The case of period 2 is treated in detail, and it is shown how an extension of the kneading theory for one-dimensional maps can be applied in order to analyze the origin of bistability, and to demarcate the principal regions of bistability in the phase space. When the period of the modulation is increased—and here we choose periods which are the Fibonacci numbers—the measure of multistable regions decreases. The limit of quasiperiodic driving is approached in two different ways, by increasing the period and keeping the drive dichotomous, or by increasing the period and varying the modulation sinusoidally. In the former case, we find that multistability persists in small regions of the phase space, while in the latter, there is no evidence of multistability but strange nonchaotic attractors are created.