homogenization limits and wigner transforms we present a theory for carrying out homogenization limits for quadratic functions called energy densities of solutions of initial value problems ivps with anti self adjoint spatial pseudo differential operators pdos the approach is based on the introduction of phase space wigner matrix measures that are calculated by solving kinetic equations involving the spectral properties of the pdo the weak limits of the energy densities are then obtained by taking moments of the wigner measure the very general theory is illustrated by typical examples like semi classical limits of schrodinger equations with or without a periodic potential the homogenization limit of the acoustic equation in a periodic medium and the classical limit of the dirac equation 1997 john wiley sons inc

intermittent incommensurate chaos abstract near the onset of intermittent chaos from quasiperiodic motion lying on an attracting 2d torus with rotation number 2 1 5 1 2 the power spectrum of the cartesian coordinate of the intersection point on the poincare section is studied the poincare section is distorted from the ellipse near the onset of chaos then a sequence of spectral lines are excited at frequencies i i 2 i 1 2 their intensities are found to obey the power law 4 i or 2 i for i 1 according as the poincare section has a sharp wrinkle or not a similar spectrum is obtained also in the chaotic regime e 0 the mean value of time intervals of quasiperiodic states between two consecutive bursts and the square root of their variance are found to be inversely proportional to e near the onset point g 3 0

fault tolerant cycles embedding in hypercubes with faulty edges let q n be an n dimensional hypercube with f e 5 in this paper we consider the faulty hypercube under the following two additional conditions 1 each vertex is incident to at least two fault free edges and 2 every 4 cycle does not have any pair of non adjacent vertices whose degrees are both two after removing the faulty edges we prove that there exists a fault free cycle of every even length from 4 to 2 n in q n our result improves the result by liu and wang 2014 in terms of the lengths of embedding cycles where under the same conditions a fault free hamiltonian cycle was constructed

bifurcation phenomena in coupled chemical oscillators normal form analysis and numerical simulations abstract a class of diffusively coupled chemical oscillators is mapped into a problem of two interacting hopf bifurcations the normal form analysis predicts a cascade of steady state limit cycle 2 torus 3 torus bifurcations as well as the coexistence of two stable limit cycles numerical simulations of the original system confirm these predictions and in particular show that this system provides an example of bifurcation leading to a stable quasiperiodic regime with three incommensurate frequencies

fault tolerant hamiltonian connectedness of cycle composition networks abstract it is important for a network to tolerate as many faults as possible with the graph representation of an interconnection network a k regular hamiltonian and hamiltonian connected network is super fault tolerant hamiltonian if it remains hamiltonian after removing up to k 2 vertices and or edges and remains hamiltonian connected after removing up to k 3 vertices and or edges super fault tolerant hamiltonian networks have an optimal flavor with regard to the fault tolerant hamiltonicity and fault tolerant hamiltonian connectivity for this reason a cycle composition framework was proposed to construct a k 2 regular super fault tolerant hamiltonian network based on a collection of n k regular super fault tolerant hamiltonian networks containing the same number of vertices for n 3 and k 5 this paper is aimed to emphasize that the cycle composition framework can be still applied even when k 4

network induced chaos in integrate and fire neuronal ensembles it has been shown that a single standard linear integrate and fire if neuron under a general time dependent stimulus cannot possess chaotic dynamics despite the firing reset discontinuity here we address the issue of whether conductance based pulsed coupled network interactions can induce chaos in an if neuronal ensemble using numerical methods we demonstrate that all to all homogeneously pulse coupled if neuronal networks can indeed give rise to chaotic dynamics under an external periodic current drive we also provide a precise characterization of the largest lyapunov exponent for these high dimensional nonsmooth dynamical systems in addition we present a stable and accurate numerical algorithm for evaluating the largest lyapunov exponent which can overcome difficulties encountered by traditional methods for these nonsmooth dynamical systems with degeneracy induced by e g refractoriness of neurons

circle map behaviour of a modulated relaxation oscillator abstract the relaxation oscillations of sinusoidally and triangularly driven inductively shunted josephson junctions are studied by reducing the description of the dynamics to one dimensional circle maps the phase diagrams show two critical lines the first one marks the transition to complete phase locking without chaos involved contrary to expectations the second one does not mark the transition to chaos this is found to be due to a point of discontinuity in the corresponding maps the hausdorff dimension of the set of quasiperiodic windings at the first critical lines are computed for the sinusoidally and the triangularly driven oscillator to be 0 875 0 003 and 0 71 0 02 in the case of the second critical lines we give numerical evidence for zero dimension

conservative numerical schemes for the vlasov equation abstract a new scheme for solving the vlasov equation using a phase space grid is proposed the algorithm is based on the conservation of the flux of particles and the distribution function is reconstructed using various techniques that allow control of spurious oscillations or preservation of the positivity several numerical results are presented in two and four dimensional phase space and the scheme is compared with the semilagrangian method this method is almost as accurate as the semilagrangian one and the local reconstruction technique is well suited for parallel computation

propagation of moments and regularity for the 3 dimensional vlasov poisson system we prove that if the initial data has moments inv higher than three then the solution of vlasov poisson has also moments inv higher than three we deduce from this different regularity results on the local density the force field or the solution itself also we give a new uniqueness result and new regularity results for solutions satisfying only the energy andl bounds our proofs are based on a new representation formula and logarithmic estimates for the force field

multi phase computations of the semiclassical limit of the schrodinger equation and related problems whitham vs wigner abstract we present and compare two different techniques to obtain the multi phase solutions for the schrodinger equation in the semiclassical limit the first is whitham s averaging method which gives the modulation equations governing the evolution of multi phase solutions the second is the wigner transform a convenient tool to derive the semiclassical limit equation in the phase space the vlasov equation for the linear schrodinger equation motivated by the linear superposition principle we derive and prove the multi phase ansatz for the wigner function by the stationary phase method and then use the ansatz to close the moment equations of the vlasov equation and obtain the multi phase equations in the physical space we show that the multi phase equations so derived agree with those derived by whitham s averaging method which can be proved using different arguments generic way of obtaining and computing the multi phase equations by the wigner function is given and kinetic schemes are introduced to solve the multi phase equations the numerical schemes based on multiphase closure of the liouville equation are purely eulerian and only operate in the physical space several numerical examples are given to explore the validity of this approach similar studies are conducted for the linearized korteweg de vries equation and the linear wave equation

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

classical representation of the one dimensional anderson model a new approach is applied to the one dimensional anderson model by using a two dimensional hamiltonian map for a weak disorder this approach allows for a simple derivation of correct expressions for the localization length both at the centre and at the edge of the energy band where standard perturbation theory fails approximate analytical expressions for strong disorder are also obtained

map dependence of the fractal dimension deduced from iterations of circle maps every orientation preserving circle mapg with inflection points including the maps proposed to describe the transition to chaos in phase locking systems gives occasion for a canonical fractal dimensiond namely that of the associated set of for whichf g has irrational rotation number we discuss how this dimension depends on the orderr of the inflection points in particular in the smooth case we find numerically thatd r d r 1 r 1 8

a recursively construction scheme for super fault tolerant hamiltonian graphs abstract for the interconnection network topology it is usually represented by a graph when a network is used processors and or links faults may happen thus it is meaningful to consider faulty networks we consider k regular graphs in this paper we define a k regular hamiltonian and hamiltonian connected graph g is super fault tolerant hamiltonian if g remains hamiltonian after removing at most k 2 vertices and or edges and remains hamiltonian connected after removing at most k 3 vertices and or edges a super fault tolerant hamiltonian graph has a certain optimal flavor with respect to the fault tolerant hamiltonicity and fault tolerant hamiltonian connectivity the aim of this paper is to investigate a construction scheme to construct various super fault tolerant hamiltonian graphs along the way the recursive circulant graph is a special case of our construction scheme and the super fault tolerant hamiltonian property of recursive circulant graph is obtained

period doubling bifurcations chaos phase locking and devil s staircase in a bonhoeffer van der pol oscillator abstract the effect of a periodic input current a 1 cos t in the bonhoeffer van der pol oscillator along with a bias a 0 is investigated numerically as the parameter a 1 is varied in the absence of bias by holding the other parameters at constant values typical period doubling bifurcation sequence is found to occur leading to chaotic motion in agreement with the feigenbaum scenario when the bias is switched on at the transition to chaos frequency locking is observed in the system the frequency locked intervals exhibit complete devil s staircase similar to the one observed in belousov zhabotinsky reaction

time correlations in mode hopping of coupled oscillators we study the dynamics in a system of coupled oscillators when arnold tongues overlap by varying the initial conditions the deterministic system can be attracted to different limit cycles adding noise the mode hopping between different states become a dominating part of the dynamics we simplify the system through a poincare section and derive a 1d model to describe the dynamics we explain that for some parameter values of the external oscillator the time distribution of occupancy in a state is exponential and thus memoryless in the general case on the other hand it is a sum of exponential distributions characteristic of a system with time correlations

fractal dimensions and homeomorphic conjugacies we investigate the behavior of the spectrum of singularities associated with the invariant measure of some dynamical systems under nonsmooth coordinate changes when the homeomorphic conjugacy is not lipschitz continuous we discuss how its singularities can affect the whole set of generalized fractal dimensions we give applications to homeomorphisms that conjugate critical circle maps with irrational golden mean winding numbers we present numerical studies corroborating the theoretical predictions

devil s staircase in an optomechanical cavity we study self excited oscillations seos in an on fiber optomechanical cavity while the phase of seos randomly diffuses in time when the laser power injected into the cavity is kept constant phase locking may occur when the laser power is periodically modulated in time we investigate the dependence of phase locking on the amplitude and frequency of the laser power modulation we find that phase locking can be induced with a relatively low modulation amplitude provided that the ratio between the modulation frequency and the frequency of seos is tuned close to a rational number of relatively low hierarchy in the farey tree to account for the experimental results a one dimensional map which allows evaluating the time evolution of the phase of seos is theoretically derived by calculating the winding number of the one dimensional map the regions of phase locking can be mapped in the plane of modulation amplitude and modulation frequency comparison between the theoretical predictions and the experimental findings yields a partial agreement

a numerical study of the likelihood of phase locking abstract we report the results of numerical computation of the size of the set of s for which the hemeomorphism of the circle f k x x k 2 sin 2 x mod 1 is phased locked for values of k ranging from 0 1 to 1 the results suggest that for moderate values of k phase locking is unlikely except for small periods but that for k 1 almost all s give phase locking with large periods accounting for a surprisingly large portion of the total

universality and self similarity in the bifurcations of circle maps abstract the bifurcation structure in a two parameter family of circle maps is considered these maps have a topological degree that may be different from one a generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described continuity arguments are used to establish the existence of periodic orbits by plotting the locus of parameter values associated with superstable cycles self similar bifurcations are found these bifurcations are a generalization of the familiar period doubling cascade in maps with one extrema to two parameter maps with two extrema finally a scheme for the global organization of bifurcation in these maps is proposed

paths and cycles identifying vertices in twisted cubes the hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement the twisted cube is an important variation of the hypercube and preserves many of its desirable properties karpovsky et al introduced the concept of identifying codes to model fault detection in multiprocessor systems and honkala et al developed an identifying code by using cycles to identify the faulty processors in the hypercube in this paper we study the vertex identification problem on the twisted cube we first propose an interesting construction scheme to build paths and cycles and furthermore apply a minimum number of paths and cycles to identify the faulty processors of the twisted cube

renormalization in a circle map with two inflection points abstract an extended version of the sine circle map becomes non invertible by developing simultaneously two critical points in part of parameter space universality in the corresponding transition from quasiperiodicity to chaos manifests itself in the organisation of doubly superstable orbits four operators are needed to generate all such orbits the existence of non trivial equivalence relations for operation sequences makes the present theory different from other multi operator renormalisation approaches possible implications of the theory in higher dimensional systems are discussed

complete devil s staircase fractal dimension and universality of mode locking structure in the circle map it is shown numerically that the stability intervals for limit cycles of the circle map form a complete devil s staircase at the onset of chaos the complementary set to the stability intervals is a cantor set of fractal dimension d 0 87 this exponent is found to be universal for a large class of functions

a lower bound on the number of hamiltonian cycles through a prescribed edge in a crossed cube the crossed cube proposed by efe is one of the most notable variations of the hypercube but some properties of the former are superior to those of the latter for example the diameter of the crossed cube is almost the half that of the hypercube in this paper we consider the problem of embedding a hamiltonian cycle passing through a prescribed edge in crossed cubes a concept termed the cycle pattern is used to construct a linear algorithm for embedding a structural hamiltonian cycle so the cycle passes through an arbitrary given edge in the crossed cube further we give the necessary and sufficient conditions for determining what kind of permutation generates a hamiltonian cycle pattern of the crossed cube as a result we obtain a lower bound for the number of hamiltonian cycles through a given edge in an n dimensional crossed cube our work extends some recently obtained results

on some super fault tolerant hamiltonian graphs a k regular hamiltonian and hamiltonian connected graph g is super fault tolerant hamiltonian if g remains hamiltonian after removing at most k 2 nodes and or edges and remains hamiltonian connected after removing at most k 3 nodes and or edges a super fault tolerant hamiltonian graph has a certain optimal flavor with respect to the fault tolerant hamiltonicity and hamiltonian connectivity in this paper we investigate a construction scheme to construct super fault tolerant hamiltonian graphs in particularly twisted cubes crossed cubes and mobius cubes are all special cases of this construction scheme therefore they are all super fault tolerant hamiltonian graphs

bifurcations of circle maps arnol d tongues bistability and rotation intervals we study the bifurcations of two parameter families of circle maps that are similar tofb w x x w b 2 sin 2 x mod1 the bifurcation diagram is constructed in terms of setstr wheretr is the set of parameter values b w for whichfb w has an orbit with rotation numberr we show that the known structure whenb 1 wherefb w is no longer injective and can have an interval of rotation numbers specifically the tongues overlap in a uniform monotonic manner and forr irrational tr opens into a tongue our other theorems give information about the dynamics offb w e g bistability or aperiodicity for b w in various subsets of parameter space

existence of global weak solutions to one component vlasov poisson and fokker planck poisson systems in one space dimension with measures as initial data we consider cauchy problems for the 1 d one component vlasov poisson and fokker planck poisson equations with the initial electron density being in the natural space of arbitrary non negative finite measures in particular the initial density can be a dirac measure concentrated on a curve which we refer to as electron sheet initial data these problems resemble both structurally and functional analytically cauchy problems for the 2 d euler and navier stokes equations in vorticity formulation with vortex sheet initial data here we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved we give a natural definition of the product establish its weak stability and existence of weak solutions follows our concept of weak solutions through the newly defined product is justified since solutions to the fokker planck poisson equation the analogue of navier stokes equation are shown to converge to weak solutions of the vlasov poisson equation as the fokker planck term vanishes the main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations this is needed because the problem studied here is beyond the range of applicability of the velocity averaging compactness methods of diperna lions 1994 john wiley sons inc

the scaling of arnol d tongues for differentiable homeomorphisms of the circle we show both for diffeomorphisms of the circle and for differentiable homeomorphisms that are not diffeomorphisms that the widths of the arnol d tongues in a one parameter family scale asq 3 whenq is the denominator of the rotation number

concentrations in the one dimensional vlasov poisson equations i temporal development and non unique weak solutions in the single component case abstract weak solutions of the one component vlasov poisson equation in a single space dimension are proposed and studied here as a simpler analogue problem for the behavior of weak solutions of the two dimensional incompressible euler equations with non negative vorticity the physical structural and functional analytic analogies between these two problems are developed in detail here with this background explicit solutions for electron sheet initial data the analogue of vortex sheet initial data are presented which display the phenomena of singularity formation in finite time as well as the explicit temporal development of charge concentrations other rigorous explicit examples with charge concentration are developed where there are non unique weak solutions with the same initial data in one of these non unique weak solutions an electron sheet completely collapses to a point charge in finite time the detailed limiting behavior of regularizations such as the diffusive fokker planck equation are developed through a very efficient numerical method which yields extremely high resolution for these simpler analogue problems a striking consequences of the numerical results reported here is the fact that there is not a selection principle for a unique weak solution in some situations where there are several weak solutions with charge concentration for the same initial data in particular two explicit weak solutions with the same initial data are constructed here where it is demonstrated that the zero smoothing limit of time reversible particle methods converges to one of these solutions while the zero diffusion limit of the fokker planck equation converges to the other weak solution

exact treatment of mode locking for a piecewise linear map a piecewise linear map with one discontinuity is studied by analytic means in the two dimensional parameter space when the slope of the map is less than unity periodic orbits are present and we give the precise symbolic dynamic classification of these the localization of the periodic domains in parameter space is given by closed expressions the winding number forms a devil s terrace a two dimensional function whose cross sections are complete devils s staircases in such a cross section the complementary set to the periodic intervals is a cantor set with dimensiond 0

a particle method for the semiclassical limit of the schrodinger equation and the vlasov poisson equations when using standard deterministic particle methods point values of the computed solutions have to be recovered from their singular particle approximations by using some smoothing procedure the choice of the smoothing procedure is rather flexible moreover there is always a parameter associated with the smoothing procedure if this parameter is too large the numerical solution loses its accuracy if it is too small oscillation appears no explicit formula has been given on how to choose this parameter in this paper we develop a particle method for the semiclassical limit of the schrodinger equation and the vlasov poisson equations in which we use the property of conservation of charge which was studied in d wei kinetic and related models 3 2010 pp 729 754 to construct the density this method avoids the recovery step of the particle methods thus it is simpler and more accurate in particular it gives more accurate field quantities consequently we apply this method to the vlasov poi

the dynamics of phase locking and points of resonance in a forced magnetic oscillator abstract we report data on an experimental system a forced symmetric oscillator containing a saturable inductor with magnetic hysteresis it displays a hopf bifurcation to quasiperiodicity entrainment horns and chaos we study in detail the bifurcations and hysteresis occurring near points of resonance particularly strong resonance and show how the observed behavior can be understood using arnold s theory much of the behavior relating to the entrainment horns is explored period doubling and symmetry breaking bifurcations homoclinic bifurcations and crises and other bifurcations taking place at the horn boundaries important features of the behavior related to symmetry properties of the oscillator are studied and explained through the concept of a half cycle map the system is shown to exhibit a hopf bifurcation from a phase locked state to periodic islands similar to those found in hamiltonian systems an initialization technique is used to observe the manifolds of saddle orbits and other hidden structure an unusual differential equation model is developed which is irreversible and generates a noninvertible poincare map of the plane noninvertibility of this planar map has important effects on the behavior observed the poincare map may also be approximated through experimental measurements resulting in a planar map with parameter dependence this model gives good correspondence with the system in a region of the parameter space

spectra of scaling indices for fractal measures theory and experiment abstract we propose a new formalism to characterize fractal measures such as strange attractors of dynamical systems which uses a function f here is the scaling index of the measure about a point on the fractal and f is the dimension of the set of points on the fractal with the same value of the spectrum describes the global scaling structure of the measure the spectrum includes as special points the hausdorff dimension of the fractal and the scaling indices of the most rarefied and the most concentrated regions of the measure we apply this formalism to two well known transitions to chaos the route via period doubling and the route via quasiperiodicity in both cases we find a universal spectrum which includes previously discovered scaling numbers both spectra have been measured experimentally on a forced rayleigh benard system

a semiclassical transport model for two dimensional thin quantum barriers we present a two dimensional time dependent semiclassical transport model for mixed state scattering with thin quantum films the stationary schrodinger equation is solved in the quantum barrier to obtain the scattering coefficients used to supply the interface condition that connects two classical domains the solution in the classical regions is solved using a particle method and interface condition combined with the hamiltonian preserving scheme the overall cost is roughly the same as solving a classical barrier we construct a numerical method based on this semiclassical approach and validate the model using two numerical examples

transition to topological chaos for circle maps abstract many biperiodic flows can be modelled by maps of a circle to itself for such maps the transition from zero to positive topological entropy can be achieved in several ways we describe all the possible routes for smooth circle maps and discuss the relevance of our results to the transition to chaos for two frequency systems

uncertain destination dynamics certain dynamical systems exhibit a sensitivity to initial conditions in which the asymptotic state is selected from an infinite number of possible states the phase space of such systems is foliated with attractors each of which is associated with a particular set of initial conditions the associated uncertain destination dynamics can be analyzed by an appropriate reduction of the full system to a subsystem that explicitly yields the dynamics s1063 651x 99 09909 2 pacs number s 05 45 2a recent studies of high dimensional dynamical systems have revealed many new modes of complex evolution such as synchronization 1 on off intermittency 2 and fractal riddled and intermingled basins 3 many of these studies have raised fundamental questions concerning the relationship between model systems and the real systems they attempt to describe 4 in this paper we report on another gross complexity in the evolution of high dimensional systems for many dynamical systems governed by evolution equations of the form

complete devil s staircase in an ising model with competing interactions an ising model on a regular cayley tree with competing ferro and antiferromagnetic nearest neighbour interactions can be formulated as a discrete two dimensional mapping the authors use this mapping to obtain sequences of modulated phases at low temperatures associated with complete devil s staircases the fractal dimensionality of the staircases increase with temperature at high temperatures the incommensurate phases may occupy regions of finite measure of the phase diagram

winding numbers for the supercritical sine circle map abstract the dependence of the winding number on the initial value and on the nonlinearity for the supercritical sine circle map is discussed analytical investigations and numerical results show that some scaling behavior for the winding number can be extracted

fractal bifurcation sets renormalization strange sets and their universal invariants the transition structure of the most common routes to chaos are organized by fractal bifurcation sets examples include the quasiperiodic transitions to chaos and the period doubling structure found in arnol d tongues in this paper i discuss the universality of such fractal bifurcation sets and their relation to strange invariant sets of renormalization transformations an important result is that fractal bifurcation sets from within the same universality chaos are lipeomorphic this implies that they have the same fractal structure and in particular the same hausdorff dimension and scaling spectra some other invariants are introduced

reconstructing bifurcation diagrams only from time waveforms abstract a simple algorithm is proposed for reconstructing bifurcation diagrams only from time waveforms of a dynamical system without knowing an explicit form of the dynamics and its dependence on the parameters reconstruction means that the obtained bifurcation diagram is qualitatively similar to that of the original system the algorithm consists of two steps first a nonlinear predictor is sought using a particular class of functions with many parameters second the karhunen loeve transform is used to find only significant parameters contributing the bifurcations the algorithm is tested against two families of dynamical system the henon family and the coupled logistic delayed logistic family

concentrations in the one dimensional vlasov poisson equations ii screening and the necessity for measure valued solutions in the two component case abstract weak and measure valued solutions for the two component vlasov poisson equations in a single space dimension are proposed and studied here as a simpler analogue problem for the limiting behavior of approximations for the two dimensional euler equations with general vorticity of two signs from numerical experiments and mathematical theory it is known that much more complex behavior can occur in limiting processes for vortex sheets with general vorticity of two signs as compared with non negative vorticity here such behavior is confirmed rigorously for the simpler analogue problem through explicit examples involving singular charge concentration for the two component vlasov poisson equations the concepts of measure valued and weak solution are introduced explicit examples with charge concentration establish that the limit of weak solutions in a dynamic process is necessarily a measure valued solution in some cases rather than the anticipated weak solution i e no concentration cancellation occurs the limiting behavior of computational regularizations involving high resolution particle methods is presented here both for the instances with measure valued solutions and also for new examples with non unique weak solutions the authors demonstrate that different computational regularizations can exhibit completely different limiting behavior in situations with measure valued and or non unique weak solutions

symbolic dynamics of order preserving orbits abstract the maps we consider are roughly those that can be obtained by truncating non invertible maps to weakly monotonic maps they have a flat piece the binary sequences that correspond to order preserving orbits are shown to satisfy a minimax principle which was already known for order preserving orbits with rational rotation number the converse is also proven all minimax orbits are order preserving with respect to some rotation number for certain families of such circle maps one can solve exactly for the parameter values for which the map has a specified rotation number rho for rho rational we obtain the endpoints of the resonance intervals recursively these parameter values can be organized in a natural way as the nodes of a farey tree we give some applications of the ideas discussed