bifurcations of subharmonic solutions in reversible systems soit f r r n r r n une application c periodique cut on dit que l equation differentielle x f t x est reversible s il existe un operateur symetrique s sur r n tel que s 2 i de telles equations differentielles sont aussi appelees systemes avec la propriete e par rapport a s on etudie les bifurcations des solutions sous harmoniques de tels systemes reversibles

conditions for local reversibility abstract the paper follows from work by quispel and capel on local reversibility of a dynamical system the approach adopted is to write down conditions for a mapping to behave like an involution in the neighbourhood of a fixed point and then to combine mappings which satisfy these conditions thus producing a composite mapping which is locally reversible in the main this confirms the results obtained by quispel and capel although additional conditions were found in some cases

fick s law and fractality of nonequilibrium stationary states in a reversible multibaker map nonequilibrium stationary states are studied for a multibaker map a simple reversible chaotic dynamical system the probabilistic description is extended by representing a dynamical state in terms of a measure instead of a density function the equation of motion for the cumulative function of this measure is derived and stationary solutions are constructed with the aid of derham type functional equations to select the physical states the time evolution of the distribution under a fixed boundary condition is investigated for an open multibaker chain of scattering type this system corresponds to a diffusive flow experiment through a slab of material for long times any initial distribution approaches the stationary one obeying fick s law at stationarity the intracell distribution is singular in the stable direction and expressed by the takagi function which is continuous but has no finite derivatives the result suggests that singular measures play an important role in the dynamical description of non equilibrium states

chaotic scattering theory of transport and reaction rate coefficients the chaotic scattering theory is here extended to obtain escape rate expressions for the transport coefficients appropriate for a simple classical fluid or for a chemically reacting system this theory allows various transport coefficients such as the coefficients of viscosity thermal conductivity etc to be expressed in terms of the positive lyapunov exponents and kolmogorov sinai entropy of a set of phase space trajectories that take place on an appropriate fractal repeller this work generalizes the previous results of gaspard and nicolis for the coefficient of diffusion of a particle moving in a fixed array of scatterers

hopf bifurcation for equivariant conservative and time reversible systems we study the bifurcation of small periodic solutions at a non semi simple 1 1 resonance in equivariant conservative or equivariant time reversible systems by using an equivariant liapunov schmidt method and restricting to solutions with an appropriate isotropy we reduce the problem to a scalar bifurcation equation the analysis of this equation shows a bifurcation behaviour similar to that found for the hamiltonian hopf bifurcation

coexistence of symmetric periodic points in the standard map the problem of the coexistence of symmetric periodic points of the standard map has been investigated first it is shown that there is another reversibility in addition to the one already known the dominant axis of this reversibility is the remaining coordinate axis using these reversibilities simple recursion formulas that determine the positions of symmetric periodic points have been derived second bifurcation structures of symmetric periodic points from the fixed point at the origin have been investigated it is shown that periodic points bifurcate along both the dominant axes of two reversibilities every time the rotation number of the fixed point passes through a rational number third the problem of the coexistence of symmetric periodic points has been numerically investigated with the aid of the recursion formulas a devil s staircase form of the evolution of the outermost kolmogorov arnold moser kam curve of the fixed point with the change of the perturbation parameter is observed this indicates that bifurcated periodic points do not disappear by inverse bifurcation as long as they are in the invariant region of the fixed point

lyapunov exponent of random walkers on a bond disordered lattice abstract the chaotic properties of a random walker in a quenched random environment are studied analytically following the work of gaspard et al on lorentz gases for systems with closed periodic or open absorbing boundaries the model of interest describes random walkers hopping on a disordered lattice on which the hopping probabilities across bonds are quenched random variables for closed systems an exact expression for the lyapunov exponent is derived which not only depends on the composition but also on the number of clusters of certain type for open systems escape rates and lyapunov exponents are calculated from a mean field approximation the theoretical predictions are compared with the results of extensive computer simulations based on the thermodynamic formalism for large systems the agreement is excellent

error growth in the numerical integration of periodic orbits with application to hamiltonian and reversible systems we analyze in detail the growth with time of the coefficients of the asymptotic expansion of the error in the numerical integration with one step methods of periodic solutions of systems of ordinary differential equations variable stepsizes are allowed we successively consider general hamiltonian and reversible problems for hamiltonian and reversible systems and under fairly general hypotheses on the orbit being integrated numerical methods with relevant geometric properties symplecticness energy conservation reversibility are proved to have better error growth than general methods

a law of order estimation and leading order terms for a family of averaged quantities on a multibaker chain system in this study a family of local quantities defined on each partition and its averaging on a macroscopic small region site are defined on a multibaker chain system on its averaged quantities a law of order estimation in the bulk system is proved making it possible to estimate the order of the quantities with respect to the representative partition scale parameter moreover the form of the leading order terms of the averaged quantities is obtained and the form enables us to have the macroscopic quantity in the continuum limit as 0 and to confirm its partitioning independency these deliverables fully explain the numerical results obtained by ishida consistent with the irreversible thermodynamics

hyperbolic coxeter groups for triangular potts models the symmetry groups generated by the inversion relations of lattice models of statistical mechanics on triangular lattices are analysed for vertex models and for the standard scalar potts model with two and three site interactions these groups are generated by three inversion relations and are seen to be generically very large ones hyperbolic groups two situations for which the representations of these groups degenerate into smaller ones hopefully compatible with integrability are considered the first reduction for the vertex triangular model corresponds to the situation where the vertex of the triangular model coincides with the left or right hand side of a yang baxter relation in this case the representation of the group is isomorphic up to a semi direct product by a finite group to z z the second reduction for q state ports models occurs for particular values of q the so called tutte beraha numbers for this model algebraic varieties including the known ferromagnetic critical variety happen to be invariant under such large groups of symmetries as a byproduct this analysis provides nice birational representations of hyperbolic coxeter groups

periodic orbits in k symmetric dynamical systems abstract a map l is called k symmetric if itskth iterate lk possesses more symmetry than l for some value of k in k symmetric systems there exists a notion of k symmetric orbits this paper deels with k symmetric periodic orbits we derive a relation between orbits that are k symmetric with respect to reversing k symmetries and symmetric orbits of lk with this relation we set out an efficient method for finding systematically all periodic orbits that are k symmetric with respect to reversing k symmetries this k symmetric fixed set iteration fsi method generalizes a celebrated method due to devogelaere that applies to symmetric periodic orbits in reversible dynamical systems we use the fsi method to study k symmetric periodic orbits of a map of the planer2 possessing a crystallographic reversing k symmetry group the explicit findings illustrate a typically k symmetric phenomenon consisting of a nontrivial relation between the symmetry properties of periodic orbits and their periods

fundamental cycle of a periodic box ball system we investigate a box ball system with periodic boundary conditions since the box ball system is a deterministic dynamical system that takes only a finite number of states it will exhibit periodic motion we determine its fundamental cycle for a given initial state

algebraic decay in hierarchical graphs we study the algebraic decay of the survival probability in open hierarchical graphs we present a model of a persistent random walk on a hierarchical graph and study the spectral properties of the frobenius perron operator using a perturbative scheme we derive the exponent of the classical algebraic decay in terms of two parameters of the model one parameter defines the geometrical relation between the length scales on the graph and the other relates to the probabilities for the random walker to go from one level of the hierarchy to another the scattering resonances of the corresponding hierarchical quantum graphs are also studied the width distribution shows the scaling behavior p 1

invertible cellular automata a review abstract in the light of recent developments in the theory of invertible cellular automata we attempt to give a unified presentation of the subject and discuss its relevance to computer science and mathematical physics

a decomposed equation for local entropy and entropy production in volume preserving coarse grained systems abstract in this study an equation for the local entropy is derived based on the formulation of a master equation and is applied to volume preserving maps the equation consists of the following terms unsteady convection diffusion probability weighted phase space volume expansion rate nonnegative entropy production and residuals the decomposition makes it possible to evaluate entropy production in terms of microscopic dynamics and is expected to be applicable to many coarse grained systems on the phase space when it is applied to two volume preserving multibaker chain systems it is confirmed that the summation of the nonnegative entropy production on each site numerically coincides with the entropy production introduced by gilbert et al t gilbert j r dorfman p gaspard entropy production fractals and relaxation to equilibrium phys rev lett 85 2000 1606 1609 and the phenomenological expression both in nonequilibrium steady and unsteady states the coincidence is brought about by the fact that the residual terms vanish in the thermodynamic limit when they are integrated on each site it follows that the entropy production is dominated by the nonnegative entropy production term and becomes positive in nonequilibrium states

renormalisation in implicit complex maps abstract we study the dynamics of a family of implicit complex maps restricted to an invariant circle the renormalisation analysis for golden mean rotation number is given we find a one parameter family of universality classes and relate this to a line of fixed points of the square of the circle map renormalisation transformation these period 2 points are pairs of fractional linear maps

non algebraic integrability of the chew low reversible dynamical system of the cremona type and the relation with the 7th hilbert problem non resonant case abstract a smooth reversible dynamical system srds and a system of nonlinear functional equations defined by a certain rational quadratic cremona mapping and arising from the static model of the dispersion approach in the theory of strong interactions the chew low equations for p wave n scattering are considered this srds is splitted into 1 and 2 dimensional ones an explicit cremona transformation that completely determines the exact solution of the two dimensional system is found this solution depends on an odd function satisfying a nonlinear autonomous 3 point functional equation non algebraic integrability of srds under consideration is proved using the method of poincare normal forms and the siegel theorem on biholomorphic linearization of a mapping at a non resonant fixed point the proof is based on the classical feldman baker theorem on linear forms of logarithms of algebraic numbers which in turn relies upon solving the 7th hilbert problem by a i gel fond and t schneider and new powerful methods of a baker in the theory of transcendental numbers the general theorem following from the feldman baker theorem on applicability of the siegel theorem to the set of the eigenvalues c n of a mapping at a non resonant fixed point which belong to the algebraic number field a is formulated and proved the main results are presented in theorems 1 3 5 7 8 and remarks 3 7

chaotic hypothesis onsager reciprocity and fluctuation dissipation theorem it is shown that the chaoticity hypothesis recently introduced in statistical mechanics which is analogous to ruelle s principle for turbulence implies the onsager reciprocity and the fluctuation dissipation theorem in various reversible models for coexisting transport phenomena

on a periodic soliton cellular automaton we propose a box and ball system with a periodic boundary condition periodic box and ball system pbbs the time evolution rule of the pbbs is represented as a boolean recurrence formula an inverse ultradiscretization of which is shown to be equivalent to the algorithm of the calculus for the 2nth root the relations to the pbbs of the combinatorial r matrix of u q an 1 are also discussed

entropy production in a persistent random walk abstract we consider a one dimensional persistent random walk viewed as a deterministic process with a form of time reversal symmetry particle reservoirs placed at both ends of the system induce a density current which drives the system out of equilibrium the phase space distribution is singular in the stationary state and has a cumulative form expressed in terms of generalized takagi functions the entropy production rate is computed using the coarse graining formalism of gaspard gilbert and dorfman in the continuum limit we show that the value of the entropy production rate is independent of the coarse graining and agrees with the phenomenological entropy production rate of irreversible thermodynamics

dynamical localization of trajectories on a bond disordered lattice the chaotic dynamics of a random walker in a quenched environment is studied via the thermodynamic formalism of ruelle sinai and bowen in which chaotic properties are expressed in terms of a free energy type function of an inverse temperature like parameter localization phenomena in this system are elucidated both analytically and numerically the infinite system limit of the ruelle pressure at 1 and 1 is shown to be controlled by rare configurations of the bond disorder and this is related respectively to the extreme configurations associated to the minimum and maximum lyapunov exponent in finite systems these extreme values and the corresponding configurations are obtained numerically from monte carlo simulations based on the thermodynamic formalism

on the stability of double homoclinic loops we consider 2 degrees of freedom hamiltonian systems with an involutive symmetry and a pair of orbits bi asymptotic homoclinic to a saddle center equilibrium related to pairs of pure real and pure imaginary eigenvalues i we show that the stability of this double homoclinic loop is determined by the reflection coefficient of a one dimensional scattering problem and we also show that the mechanism for losing stability is the creation of an infinite heteroclinic chain connecting a sequence of periodic orbits that accumulates at the double loop

comments on the entropy of nonequilibrium steady states we discuss the entropy of nonequilibrium steady states we analyze the so called spontaneous production of entropy in certain reversible deterministic nonequilibrium system and its link with the collapse of such systems towards an attractor that is of lower dimension than the dimension of phase space this means that in the steady state limit the gibbs entropy diverges to negative infinity we argue that if the gibbs entropy is expanded in a series involving 1 2 body terms the divergence of the gibbs entropy is manifest only in terms involving integrals whose dimension is higher than approximately the kaplan yorke dimension of the steady state attractor all the low order terms are finite and sum in the weak field limit to the local equilibrium entropy of linear irreversible thermodynamics

entropy production from open volume preserving to dissipative systems we generalize gaspard s method for computing the e entropy production rate in hamiltonian systems to dissipative systems with attractors considered earlier by tel vollmer and breymann this approach leads to a natural definition of a coarse grained gibbs entropy which is extensive and which can be expressed in terms of the srb measures and volumes of the coarse graining sets which cover the attractor one can also study the entropy and entropy production as functions of the degree of resolution of the coarse graining process and examine the limit as the coarse graining size approaches zero we show that this definition of the gibbs entropy leads to a positive rate of irreversible entropy production for reversible dissipative systems we apply the method to the case of a two dimensional map based upon a model considered by vollmer tel and breymann that is a deterministic version of a biased random walk we treat both volume preserving and dissipative versions of the basic map and make a comparison between the two cases we discuss the e entropy production rate as a function of the size of the coarse graining cells for these biased random walks and for an open system with flux boundary conditions show regions of exponential growth and decay of the rate of entropy production as the size of the cells decreases this work describes in some detail the relation between the results of gaspard those of of tel vollmer and breymann and those of ruelle on entropy production in various systems described by anosov or anosov like maps

time reversible and equivariant pitchfork bifurcation abstract in this note we describe a new phenomenon in steady state bifurcations of time reversible and equivariant vector fields when the eigenspace associated with a pair of imaginary eigenvalues that passes through zero onto the real line is separable from the nullspace associated with the fixed zeros of the spectrum we prove that a pitchfork bifurcation occurs

on the existence of periodic solutions for a class of symetric hamiltonian systems abstract hamiltonian systems are used to model the motion of discrete mechanical systems this paper establishes the existence of periodic solutions for a class of such systems the method developed to prove existence should be useful for other such problems keywords minimax technique ordinary differential equations and variational problem

perturbed homoclinic solutions in reversible 1 1 resonance vector fields we consider a smooth reversible vector field in r 4 such that the origin is a fixed point the differential at the origin has two double pure imaginary eigenvalues iq for the critical value 0 of the parameter we show by a normal form analysis that the vector field can be approximated by an integrable field in r 4 for which we know all solutions specially interesting ones are the homoclinics to 0 and homoclinics to periodic solutions depending on the sign of a leading nonlinear coefficient we prove in particular the persistence of these homoclinics for the full vector field

time reversible dissipative ergodic maps we construct especially simple families of piecewise linear two dimensional continuous maps these maps generate sets of points resembling continuous dynamical trajectories sampled at discrete times the generated sets of points share many properties with nonequilibrium many body phase space trajectories these characteristic properties include i time reversibility ii multifractal attractor repellor pairs and iii ergodicity without stable fixed points or holes textcopyright 1996 the american physical society

the bakerian lecture 1987 quantum chaology bounded or driven classical systems often exhibit chaos exponential instability that persists but their quantum counterparts do not nevertheless there are new regimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic and this motivates the following definition definition quantum chaology is the study of semiclassical but nonclassical behaviour characteristic of systems whose classical motion exhibits chaos this is illustrated by the statistics of energy levels on scales comparable with the mean level spacing of order h n for n freedoms these fall into universality classes for classically chaotic systems the statistics are those of random matrices real symmetric or complex hermitian depending on the presence or absence of time reversal symmetry for classically regular ones the statistics are poisson on larger scales of order h i e classically small but semiclassically large universality breaks down these phenomena are being explained by representing spectra in terms of classical closed orbits universal spectral behaviour has its origin in very long orbits non universal behaviour depends only on short ones

tupling in three dimensional reversible mappings feigenbaum s parameter scaling exponents s are calculated for l 3m l 5m l 7m l 9m sequences of periodic orbits in a class of three dimensional non measure preserving reversible mappings some preliminary results on orbit scaling exponents for l 3m sequences are also included scaling exponents found so far are the same as those found for two dimensional reversible mappings

asymptotic behaviour of fundamental cycle of periodic box ball systems we investigate asymptotic behaviour of fundamental cycle of periodic box ball systems pbbss when the system size n goes to infinity according to integrable nature of the pbbs the trajectory is confined to qualitatively smaller number of states than that of the total states we prove that although the maximum fundamental cycle is of order of exp n almost all fundamental cycle is less than exp logn 2

transversal homoclinic points of a class of conservative diffeomorphisms abstract we develop a global graph transformation to obtain estimates for certain invariant manifolds of a class of area preserving diffeomorphisms with symmetries the existence of homoclinic or heteroclinic points and the analyticity of the angle between the invariant manifolds at them is proved for the henon and the standard maps a lower bound of this angle is given for a large set of values of the parameters

an iterated implicit complex map abstract iterated complex maps defined by a root of a biquadratic equation g z z 0 are studied numerical studies suggest conjugacy functions with differentiable natural boundaries and maps having island chains occupying much of the riemann sphere an analogy with hamiltonian systems is discussed

exact results for an approximate renormalisation scheme and some predictions for the breakup of invariant ori abstract an approximate renormalisation scheme is derived for the breakup of invariant tori of arbitrary winding ratio in hamiltonian systems of one and a half degrees of freedom similar to that of escande and doveil it is a free semi group with two generators this scheme is solved exactly for its orbits stable manifolds unstable manifolds and critical set various results are found including a cantor set of universal fractal diagrams the robustness of noble tori and a scaling law for areas near critical circles

lock on to tape like behaviour in the torsional buckling of anisotropic rods abstract through nonlinear normal form analysis of the equilibrium equations developed in a neighbourhood of the straight rod solution we show the existence of a phase transition from mildly anisotropic to full tape like buckling in non symmetric rods subject to terminal loads provided the material has a sufficiently low poisson s ratio for a solid elliptical rod with a poisson s ratio of 0 2 the critical point is found to occur at a cross sectional aspect ratio of just over 3 for larger aspect ratios the rod locks on to a one twist per wave buckling mode with no internal twist in the mildly anisotropic regime there turn out to be two physically distinct pairs of symmetric so called primary localised buckling modes differing in the phase of the internal twist these four modes homoclinic orbits remain of the full circle of localised buckling modes existing in the symmetric case after breaking of the circular symmetry of the rod s cross section in the strongly anisotropic regime only the energetically favourable pair of flat buckling modes survives

capillary gravity waves on the free surface of an inviscid fluid of infinite depth existence of solitary waves permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated an application of the hodograph transform converts the free boundary value problem into a boundary value problem for the cauchy riemann equations in the lower halfplane with nonlinear differential boundary conditions this can be converted to an integro differential equation with symbol k2 4 k 4 1 where is a bifurcation parameter a normal form analysis is presented which shows that the boundary value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for small this normal form system has been studied thoroughly by several authors iooss kirchgassner 8 iooss peroueme 10 dias iooss 5 it admits a pair of solitary wave solutions which are reversible in the sense ofkirchgassner 11 by applying a method introduced in 11 it is shown that this pair of reversible solitary waves persists for the boundary value problem and that the decay at infinity of these solitary waves is at least like 1 x

fourth order resonant collisions of multipliers in reversible maps period 4 orbits and invariant curves resonant collision of multipliers at i of a symmetric fixed point for a 2 parameter family of 4 dimensional reversible maps is considered bifurcation of period 4 orbits from the fixed point and their linear stability characteristics are briefly reviewed in one of the three possible types of bifurcation see text a small angle secondary collision of the floquet multipliers of the bifurcating periodic orbit takes place leading to the bifurcation of invariant curves from the orbit the invariant curves are calculated in a perturbation scheme in the leading order of perturbation the secondary bifurcation is found to be of superthreshold type an interesting pattern in the vicinity of the resonant collision involving families of invariant curves and 2 tori emerges results of numerical iterations corroborating the picture conjectured on the basis of perturbation calculations are presented corresponding results on resonant collisions at 1 are briefly stated

on the initial value problem of a periodic box ball system we show that the initial value problem of a periodic box ball system can be solved in an elementary way using simple combinatorial methods

trace maps as 3d reversible dynamical systems with an invariant one link between the theory of quasicrystals and the theory of nonlinear dynamics is provided by the study of so called trace maps a subclass of them are mappings on a one parameter family of 2d surfaces that foliate 3 and also 3 they are derived from transfer matrix approaches to properties of 1d quasicrystals in this article we consider various dynamical properties of trace maps we first discuss the fibonacci trace map and give new results concerning boundedness of orbits on certain subfamilies of its invariant 2d surfaces we highlight a particular surface where the motion is integrable and semiconjugate to an anosov system i e the mapping acts as a pseudo anosov map we identify properties of symmetry and reversibility time reversal symmetry in the fibonacci trace map dynamics and discuss the consequences for the structure of periodic orbits we show that a conservative period boubling sequence can be identified when moving through the one parameter family of 2d surfaces by using generator trace maps in terms of which all trace maps obtained from invertible two letter substitution rules can be expressed we show that many features of the fibonacci trace map hold in general the role of the fricke character hat i x y z x 2 y 2 z 2 2xyz 1 its symmetry group and reversibility for the nielsen trace maps are described algebraically finally we outline possible higher dimensional generalizations

revaluation of the first order upwind difference scheme to solve coarse grained master equations this study addresses the initial boundary value problem of coarse grained probability measure on the state space in which a differentiable vector field v is given and as a consequence the differenced continuity equation using the first order upwind difference scheme uds based on the finite volume method appears as the physical substance on the coarse grained dynamics at first the uds is theoretically shown to be equivalent to a class of coarse grained master equations cgme brought by a principle that we cannot distinguish state points in the same partition with each other the principle is based on the formulation of non equilibrium statistical mechanics to resolve the macroscopic irreversibility moreover the entropy production evaluated by the uds is also shown to be in accord with the average volume contraction rate in the steady state this is essential for the non equilibrium statistical dynamics and was numerically confirmed under the principle of coarse graining the uds is very superior to the conventional monte carlo method in computer time and storage and is very useful to solve the cgme

dynamical chaos in the lorentz lattice gas this paper provides an introduction to the applications of dynamical systems theory to nonequilibrium statistical mechanics in particular to a study of nonequilibrium phenomena in lorentz lattice gases with stochastic collision rules using simple arguments based upon discussions in the mathematical literature we show that such lattice gases belong to the category of dynamical systems with positive lyapunov exponents this is accomplished by showing how such systems can be expressed in terms of continuous phase space variables expressions for the lyapunov exponent of a one dimensional lorentz lattice gas with periodic boundaries are derived other quantities of interest for the theory of irreversible processes are discussed

reversible systems with first integrals abstract a reversible system of ordinary differential equations ode possessing some first integrals is considered a simple description of the first integral space is given it is shown that under the auxiliary assumption that the integrals are symmetric i e they do not change sign under the action of the reversing involution the presence of the integrals allows one to bring the problem of dynamical stability for a symmetric equilibrium to the similar problem but for the reduced system the reduced system is merely the restriction of the initial system to the intersection of the surfaces of level of the integrals passing through the equilibrium

dynamical system models of transport chaos characteristics the macroscopic limit and irreversibility the escape rate formalism and the thermostating algorithm describe relaxation towards a decaying state with absorbing boundaries and a steady state of periodic systems respectively it has been shown that the key features of the transport properties of both approaches if modeled by low dimensional dynamical systems can conveniently be described in the framework of multibaker maps in the present paper we discuss in detail the steps required to reach a meaningful macroscopic limit the limit involves a sequence of coarser and coarser descriptions projections until one reaches the level of irreversible macroscopic advection diffusion equations the influence of boundary conditions is studied in detail only a few of the chaos characteristics possess a meaningful macroscopic limit but none of these is sufficient to determine the entropy production in a general non equilibrium state 2003 elsevier b v all rights reserved

existence and stability of n pulses on optical fibers with phase sensitive amplifiers abstract the propagation of pulses in optical communication systems in which attenuation is compensated by phase sensitive amplifiers is investigated a central issue is whether optical fibers are capable of carrying several pieces of information at the same time in this paper multiple pulses are shown to exist for a fourth order nonlinear diffusion model due to kutz and co workers 1994 moreover criteria are derived for determining which of these pulses are stable the pulses arise in a reversible orbit flip a homoclinic bifurcation investigated here for the first time numerical simulations are used to study multiple pulses far away from the actual bifurcation point they confirm that properties of the multiple pulses including their stability are surprisingly well predicted by the analysis carried out near the bifurcation

coexistence of periodic points in reversible dynamical systems on a surface coexistence of periodic points is investigated in area preserving reversible dynamical systems on a plane which are not necessarily close to an integrable one it is shown that on the symmetry axis a higher cycle implies a lower cycle and any cycle implies an infinite number of higher cycles the standard map is investigated as an example it is shown that k cycles for any k 2 densely fill the phase plane of the map as the parameter goes to infinity in addition the number of these cycles are estimated

the finite dimensional reversible kam theory abstract the main results of the theory of quasi periodic motions in finite dimensional reversible dynamical systems are surveyed lower dimensional invariant tori and excitation of elliptic normal modes are discussed in detail the case where the reversing involution is perturbed is considered several fundamental differences between the reversible and hamiltonian theories are listed

reversible long term integration with variable stepsizes the numerical integration of reversible dynamical systems is considered a backward analysis for variable stepsize one step methods is developed and it is shown that the numerical solution of a symmetric one step method implemented with a reversible stepsize strategy is formally equal to the exact solution of a perturbed differential equation which again is reversible this explains geometrical properties of the numerical flow such as the nearby preservation of invariants in a second part the efficiency of symmetric implicit runge kutta methods linear error growth when applied to integrable systems is compared with explicit nonsymmetric integrators quadratic error growth

generating functions for dynamical systems with symmetries integrals and differential invariants abstract we give a survey and some new examples of generating functions for systems with symplectic structure systems with a first integral systems that preserve volume and systems with symmetries and or time reversing symmetries both odes and maps are treated and we discuss how generating functions may be used in the structure preserving numerical integration of odes with the above properties

a visual shape descriptor using sectors and shape context of contour lines this paper describes a visual shape descriptor based on the sectors and shape context of contour lines to represent the image local features used for image matching the proposed descriptor consists of two component feature vectors first the local region is separated into sectors and their gradient magnitude and orientation values are extracted a feature vector is then constructed from these values second local shape features are obtained using the shape context of contour lines another feature vector is then constructed from these contour lines the proposed approach calculates the local shape feature without needing to consider the edges this can overcome the difficulty associated with textured images and images with ill defined edges the combination of two component feature vectors makes the proposed descriptor more robust to image scale changes illumination variations and noise the proposed visual shape descriptor outperformed other descriptors in terms of the matching accuracy 14 525 better than sift 21 better than pca sift 11 86 better than gloh and 25 66 better than the shape context

invariant circles for the piecewise linear standard map we investigate invariant circles for a one parameter family of piecewise linear twist homeomorphisms of the annulus we show that invariant circles of all types and rotation numbers occur and we classify them into families we compute parameter ranges in which there are no invariant circles

passive scalars three dimensional volume preserving maps and chaos the dynamics of a medium sized particle passive scalar suspended in a general time periodic incompressible fluid flow can be described by three dimensional volume preserving maps in this paper these maps are studied in limiting cases in which some of the variables change very little in each iteration and others change quite a lot the former are called slow variables or actions the latter fast variables or angles the maps are classified by their number of actions for maps with only one action we find strong evidence for the existence of invariant surfaces that survive the nonlinear perturbation in a kam like way on the other hand for the two action case the motion is confined to invariant lines that break for arbitrary small size of the nonlinearity instead we find that adiabatic invariant surfaces emerge and typically intersect the resonance sheet of the fast motion at these intersections surfaces are locally broken and transitions from one to another can occur we call this process which is analogous to arnold diffusion singularity induced diffusion it is characteristic of two action maps in one action maps this diffusion is blocked by kam like surfaces

period doubling in two parameter families abstract we study two aspects of the cross over from dissipative to area preserving behaviour for period doubling cascades first we consider the smoothness of the limiting bifurcation curve at the crossover point we make a conjecture and give supporting numerical evidence second we study the spectrum of the linearised doubling operator acting on arbitrary mappings and show that it can be expressed in terms of the spectrum of its restriction to area preserving mappings

poisson structure of the reversible 1 1 resonance abstract vector fields that are reversible but non hamiltonian are considered in the neighborhood of a 1 1 resonance it is found that the normalized vector field has a hidden poisson structure on the orbit space that is reminiscent of the equations governing the rigid body in mechanics the theory is used to give a geometric characterization of the phase space in the neighborhood of a reversible 1 1 resonance certain similarities and also essential differences with the hamiltonian 1 1 resonance are found

diffusion effusion and chaotic scattering an exactly solvable liouvillian dynamics we study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area preserving mapping of an infinitely extended strip onto itself this exactly solvable mapping sustains chaotic behaviors and diffusion processes the relationship between the diffusion coefficient the lyapunov exponent and the entropy per unit time is derived the long lived classical resonances of the liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation in this sense there is a quasi isomorphism between the resonance spectrum of the liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation furthermore we show that a fractal repeller is associated to each non equilibrium state in the isolated and finite multibaker chain the nonequilibrium states are all unstable with respect to the equilibrium validating a weak form of the second principle of thermodynamics for the present dynamical system consequences of nonequilibrium fractals on classical measurements are discussed we then describe the open multibaker chain as a scattering system fractal properties of chaotic scattering are here shown to be related to diffusion in the chain

time reversibility and particle sedimentation this paper studies an ordinary differential equation ode model called the stokeslet model and describes sedimentation of small clusters of particles in a highly viscous fluid this model has a trivial solution in which the n particles arrange themselves at the vertices of a regular n sided polygon when n 3 hocking j fluid mech 20 1964 pp 129 139 and caflisch et al phys fluids 31 1988 pp 3175 3179 prove the existence of periodic motion in the frame moving with the center of gravity in the cluster in which the particles form an isosceles triangle the study of periodic and quasiperiodic solutions of the stokeslet model is continued with emphasis on the spatial and time reversal symmetry of the model time reversibility is due to infinite viscosity and spatial dihedral symmetry is due to the assumption of identical particles and the symmetry of the trivial solution for three particles the existence of a second family of periodic solutions and a family of quasiperiodic solu

entropy production in open volume preserving systems we describe a mechanism leading to positive entropy production in volume preserving systems under nonequilibrium conditions we consider volume preserving systems sustaining a diffusion process like the multibaker map or the lorentz gas a continuous flux of particles is imposed across the system resulting in a steady gradient of concentration in the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient the invariant measure becomes singular for instance in the multibaker map the limit invariant measure has a cumulative function given in terms of the nondifferentiable takagi function because of this singularity of the invariant measure the entropy must be defined as a coarse grained entropy instead of the fined grained gibbs entropy which would require the existence of a regular measure with a density the coarse grained entropy production is then shown to be asymptotically positive and moreover given by the entropy production expected from irreversible thermodynamics

gender classification from unaligned facial images using support subspaces rough face alignments result in suboptimal performance of face identification in this study we present an approach for identifying the gender based on facial images without proper face alignments instead of just using only the detected face patch for identification a set of patches is randomly cropped around the face detection region each patch set is represented by a linear subspace and compared with other linear subspaces by measuring their canonical correlations a similarity matrix comprised of the canonical correlations is then incorporated into an indefinite kernel support vector machine svm formulation the number of support vectors which we call support subspaces can be decided automatically hence we can avoid the dimension selection problem observed in our previous work our experimental results demonstrate that the proposed approach outperforms state of the art methods

mating quadratic maps with the modular group in 1994 s bullett and c penrose introduced the one complex parameter family of 2 2 holomorphic correspondences mathcal f a begin aligned left frac aw 1 w 1 right 2 left frac aw 1 w 1 right left frac az 1 z 1 right left frac az 1 z 1 right 2 3 end aligned and proved that for every value of a in 4 7 subset mathbb r the correspondence mathcal f a is a mating between a quadratic polynomial q c z z 2 c c in mathbb r and the modular group vargamma psl 2 mathbb z they conjectured that this is the case for every member of the family mathcal f a which has a in the connectedness locus we show here that matings between the modular group and rational maps in the parabolic quadratic family per 1 1 provide a better model we prove that every member of the family mathcal f a which has a in the connectedness locus is such a mating

capillary gravity solitary waves with damped oscillations abstract capillary gravity solitary waves with damped oscillations are studied analytically the analysis follows the work of iooss and kirchgassner who proved that these waves exist for all values of the froude number smaller than one the water wave problem is reduced to a system of ordinary differential equations by using the center manifold theorem the normal form of this reduced system can be obtained and a good approximation to these waves for small amplitude is constructed the limit as the water depth becomes infinite is considered as a special case a comparison with existing numerical results is made for small amplitude waves

recent results for the thermostatted lorentz gas abstract we review the behaviour of a driven thermostatted lorentz gas the two variable poincare section confirms that the attractor fills the whole phase space and that the associated stationary measure is ergodic and multifractal for fields below 2 2 such a property of the attractor ends with either a crisis or the emergence of elliptical regions followed eventually by stable orbits we present accurate periodic orbit expansion calculations for the diffusion coefficient pressure and lyapunov exponent the periodic orbit approach suggests the definition of a partition function and gives a simple explanation for a positive conductivity and diffusion coefficient

conserved quantities of generalized periodic box ball systems constructed from the ndkp equation we investigate periodic box ball systems pbbss with several kinds of balls and box capacity greater than or equal to one conserved quantities of the pbbss are constructed from those of the nonautonomous discrete kp ndkp equation using the lax representation of the ndkp equation

recognition by linear combinations of models an approach to visual object recognition in which a 3d object is represented by the linear combination of 2d images of the object is proposed it is shown that for objects with sharp edges as well as with smooth bounding contours the set of possible images of a given object is embedded in a linear space spanned by a small number of views for objects with sharp edges the linear combination representation is exact for objects with smooth boundaries it is an approximation that often holds over a wide range of viewing angles rigid transformations with or without scaling can be distinguished from more general linear transformations of the object by testing certain constraints placed on the coefficients of the linear combinations three alternative methods of determining the transformation that matches a model to a given image are proposed

euler poisson equations and existence of quasi periodic motions of heavy tops abstract we study the existence of quasi periodic motions of an asymmetric heavy top by using two different approaches the standard moser weinstein theorem and an extended liapounov center theorem we take advantage of the multiple reversibility of the euler poisson equations and are able to show that the extended liapounov center theorem yields more normal modes than does the moser weinstein theorem

on the symmetry lines of the standard mapping abstract the taylor and chirikov standard mapping is used as a model to represent the dynamics of non integrable systems by means of the poincare section of the trajectories intersecting a fixed plane of the phase space this study is based on a systematic use of the decomposition of the standard mapping in a product of two involutions as was birkhoff s in the restricted three body problem an infinite family of involutions is generated by these two involutions forming a group including the iterates of the mapping and their inverses the symmetry lines of the family of involutions determine the position of the periodic points and explain the onset of new chains of periodic points produced by the winding of the symmetry lines around the stable periodic points

thermodynamic formalism and localization in lorentz gases and hopping models the thermodynamic formalism expresses chaotic properties of dynamical systems in terms of the ruelle pressure the inverse temperature like variable allows one to scan the structure of the probability distributin in the dynamic phase space this formalism is applied here to a lorentz lattice gas where a particle moving on a lattice of sizeld collides with fixed scatterers placed at random locations here we give rigorous arguments that the ruelle pressure in the limit of infinite systems has two branches joining with a slope discontinuity at 1 the low and high branches correspond to localization of trajectories on respectively the most chaotic highest density region and the most deterministic lowest density region i e is completely controlled by rare fluctuations in the distribution of scatterers on the lattice and it dose not carry and information on the global structure of the static disorder as approaches unity from either side a localization delocalization transition leads to a state where trajectories are extended and carry information on transprot properties at finitel the narrow region around 1 where the trajectories are extended scales as inl 2 where depends on the sign of 1 ifd 1 and as l inl 1 ifd 1 this result appears to be general for diffusive systems with static disorder such as random walks in random environments or for the continuous lorentz gas other models of random walks on disordered lattices showing the same phenomenon are discussed

multi view face identification and pose estimation using b spline interpolation the available face views in the training set are mostly limited in this paper we present a view interpolation method using nonlinear b spline on face manifolds two models the inner outer ellipse model and the moment of inertia model are developed to estimate the pose orientation we use the limited view pose face images to form the pose eigen space then based on these nonlinear manifolds we form a b spline for each individual identification is to compute the shortest euclidean distance from a given test view to the nearest point within one of these b splines once the test view is classified as a familiar individual in the training set not only can the individual be identified but also the pose angle can be estimated experimental results show that b spline interpolation can achieve a recognition rate of 95

on the numerical integration of ordinary differential equations by symmetric composition methods differential equations of the form dot x x a b are considered where the vector fields a and b can be integrated exactly enabling numerical integration of x by composition of the flows of a and b various symmetric compositions are investigated for order complexity and reversibility free lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order a new more accurate way of applying the methods thus obtained to compositions of an arbitrary first order integrator is described and tested the determining equations are explored and new methods up to 100 times more accurate at constant work than those previously known are given

period doubling bifurcations and universality in conservative systems we have studied numerically a sequence of period doubling bifurcations of periodic orbits of henon s conservative two dimensional mapping in agreement with other recent studies we also find evidence that such sequences possess universality properties similar to the ones observed for dissipative systems the corresponding universal constants however have significantly different values in each case we suggest that a possible explanation for this discrepancy may be the fact that conservative systems owing to their measure preserving property do not become asymptotically one dimensional as dissipative systems do

the existence of transverse homoclinic points in the sitnikov problem abstract using melnikov s method we are able to prove the existence of transverse homoclinic orbits and therefore the existence of a horseshoe in a special restricted three body problem this analysis is an alternative to the one described by moser stable and random motions in dynamical systems princeton univ press princeton nj 1973 based on sitnikov s original work dokl akad nauk ussr 133 no 2 1960 303 306 where the task is accomplished using a more direct construction of the horseshoe

adaptive nonlinear manifolds and their applications to pattern recognition dimensionality reduction has long been associated with retinotopic mapping for understanding cortical maps multisensory information is processed fused and mapped to an essentially 2 d cortex in an information preserving manner data processing and projection techniques inspired by this biological mechanism are playing an increasingly important role in pattern recognition computational intelligence data mining information retrieval and image recognition dimensionality reduction involves reduction of features or volume of data and has become an essential step of information processing in many fields the topic of manifold learning has recently attracted a great deal of attention and a number of advanced techniques for extracting nonlinear manifolds and reducing data dimensions have been proposed from statistics geometry theory and adaptive neural networks this paper provides an overview of this challenging and emerging topic and discusses various recent methods such as self organizing map som kernel pca principal manifold isomap local linear embedding and laplacian eigenmap many of them can be considered in a learning manifold framework the paper further elaborates on the biologically inspired som model and its metric preserving variant visom under the framework of adaptive manifold and their applications in dimensionality reduction with face recognition are investigated the experiments demonstrate that adaptive visom based methods produce markedly improved performance over the others due to their metric scaling and preserving properties along the nonlinear manifold

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

spatially periodic steady state solutions of a reversible system at strong and subharmonic resonances abstract we study steady state spatially periodic solutions of a system of reaction diffusion equations introduced by shyldkrot and ross by both numerical integration of the time dependent system and application of theorems of arnol d and sevryuk to the steady state reversible system the results illustrate rich behavior of reversible systems recently discovered by vanderbauwhede

on universality for area preserving maps of the plane abstract we present detailed evidence that one parameter families of area preserving maps exhibit cascades of period doubling with universal geometric scaling in the parameter we relate this behaviour to a fixed point equation of the form 1 and det d 1 r 2 r 2 in particular we argue that the scaling transformation r 2 r 2 is conjugate to the transformation 0 x y x y with 2 and in fact 2 we present some numerical evidence that 8 721 1 4 018 1 16 36 where is the asymptotic ratio of the differences of the parameter values corresponding to the successive periods 2k described above