Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity

We present eight types of spatial optical solitons which are possible in a model of a planar waveguide that includes a dual-channel trapping structure and competing (cubic-quintic) nonlinearity. The families of trapped beams include “broad” and “narrow” symmetric and antisymmetric solitons, composite states, built as combinations of broad and narrow beams with identical or opposite signs (“unipolar” and “bipolar” states, respectively), and “single-sided” broad and narrow beams trapped, essentially, in a single channel. The stability of the families is investigated via the computation of eigenvalues of small perturbations, and is verified in direct simulations. Three species-narrow symmetric, broad antisymmetric, and unipolar composite states-are unstable to perturbations with real eigenvalues, while the other five families are stable. The unstable states do not decay, but, instead, spontaneously transform themselves into persistent breathers, which, in some cases, demonstrate dynamical symmetry breaking and chaotic internal oscillations. A noteworthy feature is a stability exchange between the broad and narrow antisymmetric states: in the limit when the two channels merge into one, the former species becomes stable, while the latter one loses its stability. Different branches of the stationary states are linked by four bifurcations, which take different forms in the model with the strong and weak coupling between the channels.     

Bursting oscillations as well as the bifurcation mechanism in a non-smooth chaotic geomagnetic field model

Based on the chaotic geomagnetic field model, a non-smooth factor is introduced to explore complex dynamical behaviors of a system with multiple time scales. By regarding the whole excitation term as a parameter, bifurcation sets are derived, which divide the generalized parameter space into several regions corresponding to different kinds of dynamic behaviors. Due to the existence of non-smooth factors, different types of bifurcations are presented in spiking states, such as grazing-sliding bifurcation and across-sliding bifurcation. In addition, the non-smooth fold bifurcation may lead to the appearance of a special quiescent state in the interface as well as a non-smooth homoclinic bifurcation phenomenon. Due to these bifurcation behaviors, a special transition between spiking and quiescent state can also occur.     

Random dynamics of the Morris-Lecar neural model

Determining the response characteristics of neurons to fluctuating noise-like inputs similar to realistic stimuli is essential for understanding neuronal coding. This study addresses this issue by providing a random dynamical system analysis of the Morris-Lecar neural model driven by a white Gaussian noise current. Depending on parameter selections, the deterministic Morris-Lecar model can be considered as a canonical prototype for widely encountered classes of neuronal membranes, referred to as class I and class II membranes. In both the transitions from excitable to oscillating regimes are associated with different bifurcation scenarios. This work examines how random perturbations affect these two bifurcation scenarios. It is first numerically shown that the Morris-Lecar model driven by white Gaussian noise current tends to have a unique stationary distribution in the phase space. Numerical evaluations also reveal quantitative and qualitative changes in this distribution in the vicinity of the bifurcations of the deterministic system. However, these changes notwithstanding, our numerical simulations show that the Lyapunov exponents of the system remain negative in these parameter regions, indicating that no dynamical stochastic bifurcations take place. Moreover, our numerical simulations confirm that, regardless of the asymptotic dynamics of the deterministic system, the random Morris-Lecar model stabilizes at a unique stationary stochastic process. In terms of random dynamical system theory, our analysis shows that additive noise destroys the above-mentioned bifurcation sequences that characterize class I and class II regimes in the Morris-Lecar model. The interpretation of this result in terms of neuronal coding is that, despite the differences in the deterministic dynamics of class I and class II membranes, their responses to noise-like stimuli present a reliable feature.     

On the “expanded local mode” approach applied to the methane molecule

On the base of the “expanded local mode approach” (see, Ref. [12]) a simple expression of the ambiguity parameter sinγ of the CH₄ molecule is estimated and then, using empirical relations between F_i…j force coefficients, simple relations between different spectroscopic parameters of the methane molecule are derived. Comparison with corresponding “experimental” values is made, that shows more than satisfactory correlations between both (predicted and obtained from experimental data) sets of parameters.     

Quantum mechanics and the direction of time

In recent papers the authors have discussed the dynamical properties of large Poincaré systems (LPS), that is, nonintegrable systems with a continuous spectrum (both classical and quantum). An interesting example of LPS is given by the Friedrichs model of field theory. As is well known, perturbation methods analytic in the coupling constant diverge because of resonant denominators. We show that this Poincaré catastrophe can be eliminated by a natural time ordering of the dynamical states. We obtain then a dynamical theory which incorporates a privileged direction of time (and therefore the second law of thermodynamics). However, it is only in very simple situations that this time ordering can be performed in an extended Hilbert space. In general, we need to go to the Liouville space (superspace) and introduce a time ordering of dynamical states according to the number of particles involved in correlations. This leads then to a generalization of quantum mechanics in which the usual Heisenberg's eigenvalue problem is replaced by a complex eigenvalue problem in the Liouville space.     

T-points: A codimension two heteroclinic bifurcation

The local bifurcation structure of a heteroclinic bifurcation which has been observed in the Lorenz equations is analyzed. The existence of a particular heteroclinic loop at one point in a two-dimensional parameter space (a T point) implies the existence of a line of heteroclinic loops and a logarithmic spiral of homoclinic orbits, as well as countably many other topologically more complicatedT points in a small neighborhood in parameter space.     

Bifurcation analysis of the sixteen-vertex model

We shall construct a hierarchy of subclasses of the 16-vertex model having qualitatively different symmetry properties. We determine the bifurcation points in the parameter space of the model where new symmetry elements are added to the invariance group of the partition function. In this paper we restrict ourselves to the study of site-dependent transformations converting a homogeneous 16-vertex model into a different homogeneous model. Apart from a trivial transformation, resulting in a change of sign of all vertex weights, such site-dependent transformations exist only for those points in parameter space where particular relations are satisfied. The solution of these relations gives rise to three 6-parameter families of models, two of which are equivalent to the general 8-vertex model, and two families of 4-parameter models. The primary bifurcation models depending on six parameters contain three different types of secondary bifurcation models, depending on 4 parameters, one of which is equivalent to Baxter's symmetric 8-vertex model.     

Jerk系统耦合的分岔和混沌行为

通过分析耦合的Jerk系统的平衡点及其稳定性,给出了参数空间中不同的分岔集,进而将参数空间划分为对应于各种动力学行为的不同区域.探讨了耦合系统随不同参数变化的动力学演化过程,重点分析了系统耦合强度变化对其动力学行为的影响.揭示了多种运动模式共存及倍周期分岔等各种非线性现象的产生机理.     

非线性切换系统的动力学行为分析

建立了周期切换下的非线性电路模型,基于子系统平衡点及其稳定性分析,分别给出了其相应的fold分岔和Hopf分岔条件,讨论了子系统在不同平衡态下由周期切换导致的各种复杂行为,指出切换系统的周期解随参数的变化存在着倍周期分岔和鞍结分岔两种失稳情形,并相应地导致不同的混沌振荡,进而结合系统轨迹及其相应的分岔分析,揭示了各种振荡模式的动力学机理. 关键词： 周期切换 倍周期分岔 鞍结分岔 混沌     

Bursting phenomena as well as the bifurcation mechanism in a coupled BVP oscillator with periodic excitation

We explore the complicated bursting oscillations as well as the mechanism in a high-dimensional dynamical system.By introducing a periodically changed electrical power source in a coupled BVP oscillator, a fifth-order vector field with two scales in frequency domain is established when an order gap exists between the natural frequency and the exciting frequency.Upon the analysis of the generalized autonomous system, bifurcation sets are derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two typical cases are focused on as examples,in which different types of bursting oscillations such as sub Hopf/sub Hopf burster, sub Hopf/fold-cycle burster, and doublefold/fold burster can be observed. By employing the transformed phase portraits, the bifurcation mechanism of the bursting oscillations is presented, which reveals that different bifurcations occurring at the transition between the quiescent states(QSs) and the repetitive spiking states(SPs) may result in different forms of bursting oscillations. Furthermore, because of the inertia of the movement, delay may exist between the locations of the bifurcation points on the trajectory and the bifurcation points obtained theoretically.     

Bifurcation structures in two-dimensional maps: The endoskeletons of shrimps

Bifurcation structures for nonlinear dynamical systems in a space of two parameters often display geometric shapes resembling shrimps. For one-dimensional maps with two parameters and multiple extrema, the underlying structure of the shrimps can be elucidated by computing the locus of superstable cycles which form a “skeleton” that supports the shrimps. Here we use continuation methods to identify and compute structures in two-dimensional maps that play the same role as the skeleton in one-dimensional maps. This facilitates determining the complex geometries for situations in which there is multistability, and for which the regions of parameter space supporting stable orbits get vanishingly small.     

At the other side of a saddle-node

We describe phenomena occurring just before a saddle-node bifurcation for one-parameter families of interval maps. In particular, as a parameter approaches the bifurcation value, attracting periodic orbits of periodsk, k+1,k+2,k+3,... can appear. We make a detailed study of a family of cusp-shaped maps, where this phenomenon occurs in a pure form.     

Some flesh on the skeleton: The bifurcation structure of bimodal maps

One of the main tools in the numerical study of two-parameter families of one-dimensional maps is the drawing of curves in parameter space corresponding to the existence of superstable periodic orbits. We use kneading theory to describe the structure of these sets of curves for the case of maps with at most two turning points. Then we explain how the bifurcation structure hangs on this “skeleton”.     

The multi-agent Parrondo’s model based on the network evolution

A multi-agent Parrondo’s model is proposed in the paper. The model includes link A based on the rewiring mechanism (the network evolution) + game B (dependent on the spatial neighbors). Moreover, to produce the paradoxical effect and analyze the “agitating” effect of the network evolution, the dynamic processes of the network evolution + game B are studied. The simulation results and the theoretical analysis both show that the network evolution can make game B which is losing produce the winning paradoxical effect. Furthermore, we obtain the parameter space where the strong or weak Parrondo’s paradox occurs. Each size of the region of the parameter space is larger than the one in the available multi-agent Parrondo’s model of game A + game B. This result shows that the “agitating” effect of rewiring based on the network evolution is better than that of the zero-sum game between individuals.     

On some properties of nearly conservative dynamics of Ikeda map and its relation with the conservative case

The behavior of the well-known Ikeda map with very weak dissipation (so-called nearly conservative case) is investigated. The changes in the bifurcation structure of the parameter plane while decreasing the dissipation are revealed. It is shown that when the dissipation is very weak the system demonstrates an “intermediate” type of dynamics combining the peculiarities of conservative and dissipative dynamics. The correspondence between the trajectories in the phase space in the conservative case and the transformations of the set of initial conditions in the nearly conservative case has been obtained. The dramatic increase of the number of coexisting low-period attractors and the extraordinary growth of the transient time while the dissipation decreases have been revealed. The method of plotting a bifurcation tree for the set of initial conditions has been used to classify the existing attractors by their structure. Also it was shown that most of the coexisting attractors are destroyed by rather small external noise, and the transient time in noisy driven systems increases still more. The new method of two-parameter analysis for conservative systems was proposed.     

Measurement of relevant elastic and damping material properties in sandwich thick plates

An easy-to-implement method to measure relevant elastic and damping properties of the constituents of a sandwich structure, possibly with a heterogeneous core, is proposed. The method makes use of a one-point dynamical measurement on a thick-plate. The hysteretic model for each (possibly orthotropic) constituent is written generically as “E(1+jη)” for all mechanical parameters. The estimation method of the parameters relies on a mixed experimental/numerical procedure. The frequencies and dampings of the natural modes of the plate are obtained from experimental impulse responses by means of a high-resolution modal analysis technique. This allows for considerably more experimental data to be used. Numerical modes (frequencies, dampings, and modal shapes) are computed by means of an extended Rayleigh-Ritz procedure under the “light damping” hypothesis, for given values of the mechanical parameters. Minimising the differences between the modal characteristics yields an estimation of the values of the mechanical parameters describing the hysteretic behaviour. A sensitivity analysis assesses the reliability of the method for each parameter. Validations of the method are proposed by (a) applying it to virtual plates on which a finite-element model replaces the experimental modal analysis, (b) some comparisons with results obtained by static mechanical measurements, and (c) by comparing the results on different plates made of the same sandwich material.     

A scaling theory of bifurcations in the symmetric weak-noise escape problem

We consider two-dimensional overdamped double-well systems perturbed by white noise. In the weak-noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two wells. However, as the parameters of a symmetric double-well system are varied, a unique MPEP may bifurcate into two equally likely MPEPs. At the bifurcation point in parameter space, the activation kinetics of the system become non-Arrhenius. We quantify the non-Arrhenius behavior of a system at the bifurcation point, by using the Maslov-WKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix. The approximation is a formal asymptotic solution of the Smoluchowski equation. Our construction relies on a new scaling theory, which yields critical exponents describing weak-noise behavior at the bifurcation point, near the saddle.     

Routes to bursting in a periodically driven oscillator

The dynamical behaviors of a periodic excited oscillator with multiple time scales in the form that order gap exists between the frequency of the excitation and the natural frequency, are investigated in this Letter. By regarding the whole excitation term as a parameter, bifurcation sets are derived, which divide the generalized parameter space into several regions corresponding to different kinds of dynamics. Different types of bursting phenomena, such as fold/Hopf bursting, fold/Hopf/homoclinic bursting and Hopf/homoclinic bursting, are presented, the mechanism of which is obtained based on the bifurcations of the generalized autonomous system as well as the introduction of the so-called transformed phase portraits. Furthermore, the evolution of the bursting is discussed in details, in which one may find that when the two limit cycles caused by the Hopf bifurcations of the two related equilibrium points interact with each other, homoclinic bifurcation may occur, leading to the merge of the two cycles to form a large amplitude cycle. The homoclinic bifurcation may cause the two asymmetric bursters to merge into a symmetric enlarged burster, in which the large amplitude of the spiking state agrees well with the amplitude of the cycle caused by the homoclinic bifurcation.