feedback mediated control of spiral waves abstract a theoretical approach is developed which allows one to describe the drift of a spiral wave mediated by a feedback signal taken as an integral of a system variable over a certain domain of an excitable medium the underlying reaction diffusion equations are reduced to an autonomous system of ordinary differential equations for the coordinates of the core center of the spiral wave a new method to determine the drift velocity field of the spiral core is proposed based on a superposition principle this approach is applied to analyze the drift velocity field for a feedback signal generated from a one point detector and from integration domains of different size and shape including global feedback it is shown that a variation of the shape of the integration domain can induce bifurcations that change the attractor structure of the drift velocity field these theoretical predictions are in good quantitative agreement with numerical simulations performed for the oregonator model and the complex ginzburg landau equation the results are also confirmed by experiments with the light sensitive belousov zhabotinsky reaction

adaptive coupling and enhanced synchronization in coupled phase oscillators we study the dynamics of an adaptive coupled array of phase oscillators the adaptive law is designed in such a way that the coupling grows stronger for the pairs which have larger phase incoherence the proposed scheme enhances the synchronization and achieves a more reasonable coupling dynamics for the network of oscillators with different intrinsic frequencies the synchronization speed and the steady state phase difference can be adjusted by the parameters of the adaptive law besides global coupling nearest neighbor ring coupling is also considered to demonstrate the generality of the method

classification of steady solutions of the full kinematic model abstract the geometric type of kinematic model governs any moving plane curve which propagates in the normal direction in the medium there have appeared some results in the literature via this model however these results are all based upon some simplification on the model or prior phenomenological assumptions on the solutions in this paper we use really full kinematic model to classify steady solutions i e curvature is independent of time with positive or negative curvature various type of solutions have been obtained such as rotating and translating waves especially for the rotating spiral waves spiral waves arise in many biological chemical and physiological systems the kinematical model can be used to describe the motion of the spiral arms approximated as curves in the plane in fact using our results theorem 3 8 we can answer the following questions is there any steadily rotating wave for a given weakly excitable medium if yes what kind of information we can know about these rotating waves e g the tip s curvature the tip s tangential velocity and the rotating frequency comparing our results with previous ones in the literature there are some differences between them the only evolving plane curves with positive or negative curvature must have monotonous curvatures via simplified model but full model admits solutions with any given oscillating number of the curvatures

stable particle like solutions to the nonlinear wave equations of three dimensional excitable media so called excitable media support nonlinear waves that propagate as pulses e g the action potential in nerve fibers and heart muscle and the oxidation pulse in certain chemical reactions in two dimensional media the right stimulus will also create rotating vortices which radiate such pulses periodically in every direction their investigation in diverse excitable media has dramatically expanded in the usa and ussr since their discovery in both countries in 1970 in three dimensional media the pivot of the rotating structure is not a point but a curved line generically a closed ring rings linked together and knotted constitute topologically distinctive organizing centers for periodic activity throughout the medium it is of interest to understand how stimuli create organizing centers and how organizing centers persist or decay the numerically discovered stable organizing centers exhibit enough symmetries to encourage mathematical attention this paper illustrates organizing centers by computer g

homoclinic tangencies and hyperbolicity for surface diffeomorphisms we prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms the ones exhibiting a homoclinic tangency are c 1 dense this represents a step towards the global understanding of dynamics of surface diffeomorphisms

thermodynamic versus statistical nonequivalence of ensembles for the mean field blume emery griffiths model abstract we illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean field blume emery griffiths beg model as a test model the novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states for this reason it may be viewed as a statistical characterization of nonequivalent ensembles which extends and complements the common thermodynamic characterization of nonequivalent ensembles based on nonconcave anomalies of the microcanonical entropy by computing numerically both the microcanonical and canonical sets of equilibrium distributions of states of the beg model we show that for values of the mean energy where the microcanonical entropy is nonconcave the microcanonical distributions of states are nowhere realized in the canonical ensemble moreover we show that for values of the mean energy where the microcanonical entropy is strictly concave the equilibrium microcanonical distributions of states can be put in one to one correspondence with equivalent canonical equilibrium distributions of states our numerical computations illustrate general results relating thermodynamic and statistical equivalence and nonequivalence of ensembles proved by ellis et al j stat phys 101 2000 999

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

dynamics of spiral rings in the three dimensional ginzburg landau equation in this paper we show that the spiral rings of the 3d ginzburg landau equation shrink out and disappear in finite time in contrast with the 2d case in which the interaction between two spirals of opposite charge is exponential we show analytically and numerically that the spiral rings collapse with a power law due to the continuity of the defect line

complexities of differentiable dynamical systems we define the notion of localizable property for a dynamical system then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems these notions are the fast growth of the number of periodic points the positive entropy and the high emergence we finally propose a dictionary between the previously explained theory on entropy and the ongoing one on emergence

lie symmetries and multiple solutions in reaction diffusion systems lie theory of transformation groups is applied to the study of reaction diffusion systems in two dimensional media our study proves that they are invariant with respect to a five parameter symmetry group multiple types of invariant solutions with physical interest are possible and some of them can be found in the literature applied to particular models

oscillation and pattern formation in a system of self regulating cells abstract in this paper a model of a system of n 2 asymmetrically and nondiffusibly coupled self regulating cells is proposed intended to formally represent well known basic features of biological systems organization the model is discussed in terms of the activation desensitization dynamics in dictyostelium it is shown that the cells in the ensemble are capable of stable oscillations provided the intercellular coupling is strong enough otherwise their stationary values remain asymptotically stable the emergence of oscillations in the one dimensional ring shaped ensemble of size n 2 is due to a bifurcation of naimark sacker type which is proven to exist under rather mild conditions it is shown that a supercritical as well as a subcritical bifurcation branch exist at different parameter values moreover due to the dihedral symmetry of the system the parity of n is shown to determine the multiplicity of the critical pair of eigenvalues crossing the unit circle the model proposed does not represent a reaction diffusion system it is not excitable but exhibits pattern formation under conditions under which a stable limit cycle exists

stabilization of a scroll ring by a cylindrical neumann boundary we study the interaction of phase singularities with homogeneous neumann boundaries in one two and three spatial dimensions for the complex ginzburg landau equation the existence of a boundary induced drift attractor well known for spiral waves in two spatial dimensions is demonstrated for scroll waves in three spatial dimensions we find that a cylindrical neumann boundary can lock a scroll ring thus preventing the collapse of its closed filament

on the interaction of spiral waves in non equilibrium media abstract we show that the asymptotic interaction of spirals in two dimensionally oscillating continuous media falls off exponentially and can be attractive as well as repulsive depending on the parameters of the system but not on the topological charges the exponent is in qualitative agreement with simulations for moderate distances the simulations show that spirals may form a stable bound state pairs with opposite topological charges drift with constant velocity and like charged spirals rotate around each other with constant frequency this is in contrast with the stationary non oscillating case where the interaction of vortices or dislocations is long range and monotonical and depends on topological charges

generic diffeomorphisms with superexponential growth of number of periodic orbits let m be a smooth compact manifold of dimension at least 2 and diff r m be the space of c r smooth diffeomorphisms of m associate to each diffeomorphism f 2 diff r m the sequence pn f of the number of isolated periodic points for f of period n in this paper we exhibit an open set n in the space of diffeomorphisms diff r m such for a baire generic diffeomorphism f 2 n the number of periodic points pnf grows with a period n faster than any following sequence of numbers fangn2zc along a subsequence i e pn f a ni for some ni 1 with i 1 in the cases of surface diffeomorphisms i e dim m d 2 an open set n with a supergrowth of the number of periodic points is a newhouse domain a proof of the man result is based on the gontchenko shilnikov turaev theorem gst a complete proof of that theorem is also presented

stable particle like solutions of multidimensional nonlinear fields abstract stable localized structures of two and three dimensional fields are investigated a theory of the elementary particle interaction the formation of their bound states and dynamics under the action of external perturbations is considered for scalar and complex fields

sources sinks and wavenumber selection in coupled cgl equations and experimental implications for counter propagating wave systems we study the coupled complex ginzburg landau cgl equations for traveling wave systems and show that sources and sinks are the important coherent structures that organize much of the dynamical properties of traveling wave systems we focus on the regime in which sources and sinks separate patches of left and right traveling waves i e the case that these modes suppress each other we present in detail the framework to analyze these coherent structures and show that the theory predicts a number of general properties which can be tested directly in experiments our counting arguments for the multiplicities of these structures show that independently of the precise values of the coefficients in the equations there generally exists a symmetric stationary source solution which sends out waves with a unique frequency and wave number sinks on the other hand occur in two parameter families and play an essentially passive role being sandwiched between the sources these simple but general results imply that sources are important in organizing the dynamics of the coupled cgl equations simulations show that the consequences of the wavenumber selection by the sources is reminiscent of a similar selection by spirals in the 2d complex ginzburg landau equations sources can send out stable waves convectively unstable waves or absolutely unstable waves we show that there exists an additional dynamical regime where both single and bimodal states are unstable the ensuing chaotic states have no counterpart in single amplitude equations a third dynamical mechanism is associated with the fact that the width of the sources does not show simple scaling with the growth rate e this is related to the fact that the standard coupled cgl equations are not uniform in e in particular when the group velocity term dominates over the linear growth term no stationary source can exist however sources displaying nontrivial dynamics can often survive here our results for the existence multiplicity wavelength selection dynamics and scaling of sources and sinks and the patterns they generate are easily accessible by experiments we therefore advocate a study of the sources and sinks as a means to probe traveling wave systems and compare theory and experiment in addition they bring up a large number of new research issues and open problems which are listed explicitly in the concluding section

dynamics of localized structures abstract we describe qualitative behaviour of solutions of the gross pitaevskii equation in 2d in terms of motion of vortices and radiation to this end we introduce the notion of the intervortex energy we develop a rather general adiabatic theory of motion of well separated vortices and present the method of effective action which gives a fairly straightforward justification of this theory finally we mention briefly two special situations where we are able to obtain a rather detailed picture of the vortex dynamics our approach is rather general and is applicable to a wide class of evolution nonlinear equation which exhibit localized stable static solutions it yields description of general time dependent solutions in terms of dynamics of those static solutions glued together

periodic travelling waves in a family of deterministic cellular automata abstract reaction diffusion equations whose kinetics contain a stable limit cycle are an established class of models for a range of biological and chemical systems in this paper i construct a family of deterministic cellular automata with nine states which are qualitatively similar to oscillatory reaction diffusion equations in that their rules reflect both local oscillations and spatial diffusion the automata can be crudely interpreted as models of predator prey interactions and i show that the behaviour following local perturbation of the prey only state in one space dimension is very similar in the automata and in standard reaction diffusion models for predator prey systems in particular in many cases invasion of prey by predators leaves behind periodic travelling waves in the wake of invasion i study in detail these periodic plane waves in the automaton by explicitly investigating periodic solutions of the difference equation governing travelling waves i show that the automaton has many different periodic wave solutions and i compare their properties with those of periodic wave solutions of reaction diffusion systems the basic conclusion is that included amongst the periodic waves in the automaton are a family of solutions which mimic quite closely the properties of reaction diffusion periodic waves

bounds on complexity in reaction diffusion systems under general assumptions we give upper bounds proportional to the volume of the domain on the temporal and spatial complexity of solutions to systems of reaction diffusion equations the notions of complexity measure instantaneous rather than asymptotic complexity but time averaged versions of the estimates give bounds on entropy and hausdorff dimension of any attracting set the techniques involve the use of quantum mechanical estimates to bound the growth rates of small perturbations

subcritical bifurcation to infinitely many rotating waves we consider the equationu 1 r u k2 r2 u u au u 2onr r withk n a c re 0 re a and im im a 1 bounded solutions possess an interesting interpretation as rotating wave solutions to reaction diffusion systems in the plane our main results claim that there are countably many solutions which are decaying to zero at infinity the proofs rely on nodal properties of the equation and a melnikov analysis

dynamics of homoclinic tangles in periodically perturbed second order equations we obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second order ordinary differential equations with dissipation let be the size of perturbation and be the entire homoclinic tangle we prove in particular that i for infinitely many disjoint open sets of contains nothing else but a horseshoe of infinitely many branches ii for infinitely many disjoint open sets of contains nothing else but one sink and one horseshoe of infinitely many branches and iii there are positive measure sets of so that admits strange attractors with sinai ruelle bowen measure we also use the equation d2qdt2 q2 dqdt q q2 q2sin t to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation

spiral waves in the belousov zhabotinskii reaction abstract the beautiful spiral waves of oxidation in the belousov zhabotinskii reaction are the source of many interesting and important questions about periods structures in excitable media it has long been known that these spirals are similar to involutes of circles at least some distance from the center but until now no way has been known to determine the correct wavelength and frequency in this paper we show that the parameters of a spiral wave can be viwed s eigenvalues of a problem with unique solution the critical ingredients of the theory are the effects of curvature on the propagation of wavefronts in two dimensional media and the dispersion of plane waves our analytical results are shown to be in good agreement with experimental data for the belousov zhabotinskii reagent

stretched exponential estimates on growth of the number of periodic points for prevalent diffeomorphisms i for diffeomorphisms of smooth compact finite dimensional manifolds we consider the problem of how fast the number of periodic points with period n grows as a function of n in many familiar cases e g anosov systems the growth is exponential but arbitrarily fast growth is possible in fact the first author has shown that arbitrarily fast growth is topologically baire generic for c 2 or smoother diffeomorphisms in the present work we show that by contrast for a measure theoretic notion of genericity we call prevalence the growth is not much faster than exponential specifically we show that for each rho delta 0 there is a prevalent set of c 1 rho or smoother diffeomorphisms for which the number of periodic n points is bounded above by exp c n 1 delta for some c independent of n we also obtain a related bound on the decay of hyperbolicity of the periodic points as a function of n and obtain the same results for 1 dimensional endomorphisms the contrast between topologically generic and measure theoretically generic behavior for the growth of the number of periodic points and the decay of their hyperbolicity show this to be a subtle and complex phenomenon reminiscent of kam theory here in part i we state our results and describe the methods we use we complete most of the proof in the 1 dimensional c 2 smooth case and outline the remaining steps deferred to part ii that are needed to establish the general case the novel feature of the approach we develop in this paper is the introduction of newton interpolation polynomials as a tool for perturbing trajectories of iterated maps

one dimensional markovian field ising model physical properties and characteristics of the discrete stochastic mapping the zero temperature physical properties of an ising chain in a markovian field taking only two values with non zero mean are evaluated and a related discrete stochastic mapping with the theory of finite markov chains is investigated a discontinuous behaviour of the magnetisation and the residual entropy dependent on both mean field and exchange is found which can be related to flips of microscopic spin clusters for non zero temperature the mapping is characterised by the fractal properties of its attractor and by the lyapunov exponent an explicit expression for the measure in the nth iteration of the chapman kolmogorov equation is obtained

structure and dynamics of defects in 2d complex vector field abstract minimal orbits of the energy functional of a complex 2d vector field may belong to the strata with different topology depending on the sign of nonlinear terms breaking the su 2 symmetry this gives rise to either linearly or circularly polarized states lp and cp which possess different sets of defects the cp vortices have two alternative core structures with either vanishing amplitude or reversed polarization in the inner core in the lp state there are two topological charges associated with the rotation of either argument or director vortices carrying a unit argument or director charge have a punched core with a vanishing amplitude while vortices carrying two half unit charges have a circularly polarized core the punched core structure is unstable when the symmetry breaking is not sufficiently strong in the cp state this leads to transition to the alternative repolarized structure while in the lp state it signals splitting of unit charged vortices into a pair of half unit charged vortices vortices separated by distances large compared to the core size interact in the same way as vortices in a complex scalar field and interactions due to argument and director charges are independent to the leading order

a bound on the decay of defect defect correlation functions in two dimensional complex order parameter equations abstract motivated by generic scale invariance we examine the behavior of topological defect defect correlation functions in two dimensional systems driven out of equilibrium to regimes where they exhibit defect chaos using the topological nature of the defects we show that these defect defect correlations cannot decay as slowly as predicted by generic scale invariance we also provide numerical calculations that yield defect defect correlation functions in the defect turbulence regime of the two dimensional anisotropic complex ginzburg landau equation these numerical results which test the specific regime of broken square symmetry do not appear to decay as slowly as predicted by the ideas of generic scale invariance these results are in agreement with the analytical predictions

how often do simple dynamical processes have infinitely many coexisting sinks this work concerns the nature of chaotic dynamical processes sheldon newhouse wrote on dynamical processes depending on a parameter xx 1 t xn wherex is in the plane such as might arise when studying poincare return maps for autonomous differential equations in ir3 he proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane and that such parameter values would be dense in some interval the fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes we prove for an apparently typical situation that newhouse s construction yields only a set of parameter values of measure zero

computation of the response functions of spiral waves in active media rotating spiral waves are a form of self organization observed in spatially extended systems of physical chemical and biological natures a small perturbation causes gradual change in spatial location of spiral s rotation center and frequency i e drift the response functions rfs of a spiral wave are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues lambda 0 i omega the rfs describe the spiral s sensitivity to small perturbations in the way that a spiral is insensitive to small perturbations where its rfs are close to zero the velocity of a spiral s drift is proportional to the convolution of rfs with the perturbation here we develop a regular and generic method of computing the rfs of stationary rotating spirals in reaction diffusion equations we demonstrate the method on the fitzhugh nagumo system and also show convergence of the method with respect to the computational parameters i e discretization steps and size of the medium the obtained rfs are localized at the spiral s core

thermodynamic limits of dynamic cooling we study dynamic cooling where an externally driven two level system is cooled via reservoir a quantum system with initial canonical equilibrium state we obtain explicitly the minimal possible temperature t min 0 reachable for the two level system the minimization goes over all unitary dynamic processes operating on the system and reservoir and over the reservoir energy spectrum the minimal work needed to reach t min grows as 1 t min this work cost can be significantly reduced though if one is satisfied by temperatures slightly above t min our results on t min 0 prove unattainability of the absolute zero temperature without ambiguities that surround its derivation from the entropic version of the third law we also study cooling via a reservoir consisting of n 1 identical spins here we show that t min 1 n and find the maximal cooling compatible with the minimal work determined by the free energy finally we discuss cooling by reservoir with an initially microcanonic state and show that although a purely microcanonic state can yield the zero temperature the unattainability is recovered when taking into account imperfections in preparing the microcanonic state

two dimensional spectrophotometry of spiral wave propagation in the belousov zhabotinskii reaction i experiments and digital data representation abstract the spiral shaped wave of chemical activity propagating in a 1 mm layer of an excitable solution of the belousov zhabotinskii reaction is analysed quantitatively for a 5 min time interval in a 4 5 4 5 mm area the spatial distribution of the reaction catalyst and indicator ferroin is measured by light absorption at 490 nm with a video and computer based two dimensional spectrophotometer digital images with 10 m spatial and 256 digital units intensity resolution are acquired at a frequency of 20 per minute initially the spiral has a pitch of 1 18 mm a revolution time of 17 2 s and the wavefronts propagate outwards with a velocity of 69 m s after 5 min these quantities have changed by up to 5 three dimensional graphic procedures and image analysis reveal that the wave profiles are symmetric close to the spiral center and become highly asymmetric during their outward displacement the core of the spiral has a radius of 0 35 mm its center is a singular site diameter

prevalence a translation invariant almost every on infinite dimensional spaces we present a measure theoretic condition for a property to hold almost everywhere on an infinite dimensional vector space with particularemphasis on function spaces such as c k and l p like the concept of lebesgue almost every on an infinite dimensional spaces our notion of prevalence is translation invariant instead of using a specific measure on the entire space we define prevalence in terms of the class of all probability measures with compact support prevalence is a more appropriate condition than the topological concepts of open and dense or generic when one desires a probabilistic result on the likelihood of a given property on a function space

different asymptotic behavior versus same dynamical complexity recurrence ir regularity abstract for any dynamical system t x x of a compact metric space x with g almost product property and uniform separation property under the assumptions that the periodic points are dense in x and the periodic measures are dense in the space of invariant measures we distinguish various periodic like recurrences and find that they all carry full topological entropy and so do their gap sets in particular this implies that any two kind of periodic like recurrences are essentially different moreover we coordinate periodic like recurrences with ir regularity and obtain lots of generalized multi fractal analyses for all continuous observable functions these results are suitable for all shifts 1 topological mixing subshifts of finite type topological mixing expanding maps or topological mixing hyperbolic diffeomorphisms etc roughly speaking we combine many different eyes i e observable functions and periodic like recurrences to observe the dynamical complexity and obtain a refined dynamical structure for recurrence theory and multi fractal analysis

dynamics of phase singularities in two dimensional oscillating systems abstract we study the dynamics of vortices that arise in two dimensional nonequilibrium systems above a hopf bifurcation to uniform temporal oscillations evolution equations for the vortex positions and for a global phase field are derived and used to study vortex interactions we suggest that the interactions need not be purely attractive or repulsive bound vortex pairs can exist which either precess around fixed points in the plane or drift in directions perpendicular to the inter vortex axes

stationary self propagating fronts in potential flow abstract we analyze the problem of stationary self propagating fronts in potential flow the issues of local existence and uniqueness for solutions of the ode describing stationary fronts multiplicity of solutions and linearized stability of a stationary front as a solution of the hyperbolic evolution equation are addressed the results are illustrated in the case of the dual source system which is a simple model of a combustion system in which local extinction may arise model extensions for combustion applications are presented

two results on entire solutions of ginzburg landau system in higher dimensions in this short article we prove two results on the ginzburg landau system of equations u u u 2 1 where u c2 rn rm n m 1 first we prove a liouville type theorem which asserts that every solution u satisfying rn u 2 1 2 is constant and of unit norm provided n 4 here m 1 in our second result we give an answer to a question raised by brezis open problem 3 of proceedings of the symposium on pure mathematics vol 65 american mathematical society providence ri 1999 about the symmetry for the ginzburg landau system in the case n m 3 we also formulate three open problems concerning the classification of entire solutions of the ginzburg landau system in any dimension

dynamics of defects in the complex ginzburg landau equation abstract we review the derivation of basic equations for the spiral dynamics proposed by us before these coupled equations involving the trajectories and a global phase are solved here in various situations of physical interest

time periodic spatial chaos in the complex ginzburg landau equation the phenomenon of time periodic evolution of spatial chaos is investigated in the frames of one and two dimensional complex ginzburg landau equations it is found that there exists a region of the parameters in which disordered spatial distribution of the field behaves periodically in time the boundaries of this region are determined the transition to the regime of spatiotemporal chaos is investigated and the possibility of describing spatial disorder by a system of ordinary differential equations is analyzed the effect of the size of the system on the shape and period of oscillations is investigated it is found that in the two dimensional case the regime of time periodic spatial disorder arises only in a narrow strip the critical width of which is estimated the phenomenon investigated in this paper indicates that a family of limit cycles with finite basins exists in the functional phase space of the complex ginzburg landau equation in finite regions of the parameters

spiral wave solutions for reaction diffusion equations in a fast reaction slow diffusion limit abstract a two species reaction diffusion equation is studied in a distinguished limit of fast reaction and slow diffusion asymptotic matching yields a spiral solution whose rotation frequency is a unique function of the physical parameters and the constant pitch of the spiral using an idea of fife the problem is first reduced to a free boundary problem the solution is obtained by matching the core where diffusion balances the curvature of the free boundary to the far field where the effects of diffusion are small the equations of motion are averaged over the rotation frequency to yield steady equations in the two regions helically symmetric solutions to the averaged equation reduce to solving a fourth order ode in the core region matched to a second order ode in the far field speculations on the dynamics of the system and coupling between the core and far field are presented

the structure of spiral domain patterns and shocks in the 2d complex ginzburg landau equation abstract cellular patterns appear spontaneously in a number of nonequilibrium systems governed by the dynamics of a complex field in the case of the complex ginzburg landau equation disordered cells of effectively frozen spirals appear separated by thin walls shocks on a scale much larger than the basic wavelength of the spirals we show that these structures can be understood in very simple terms in particular we show that the walls are to a good approximation segments of hyperbolae and this allows us to construct the wall pattern given the vortex centers and a phase constant for each vortex the fact that the phase is only defined up to an integer multiple of 2 introduces a quantization condition on the sizes of the smallest spiral domains the transverse structure of the walls is analyzed by treating them as heteroclinic connections of a system of ordinary differential equations the structure depends on the angle the wall makes with the local phase contours and the behavior can be either monotonic or oscillatory depending on the parameters

stability limits of traveling waves and the transition to spatiotemporal chaos in the complex ginzburg landau equation abstract due to the convective nature of the eckhaus instability in oscillatory media there exists a conventively unstable wavenumber band in the benjamin feir unstable region this appears to explain at least qualitatively some features of spatiotemporal chaos found recently in the ginzburg landau theory

dynamics of spiral waves in non equilibrium media in this paper the spiral waves in two dimensional nonequilibrium media are studied on the basis of the two dimensional ginzburg landau equation in the general case the perturbation of origin spiral waves leads to the spiral rotation and to the motion of spiral cores motion in media with hard and soft excitation is analysed separately

sources and sinks of wave patterns abstract we analyze in this paper the behavior of sources and sinks of traveling waves in one dimensional systems it is shown that the presence of these defects induces a selection of the wavenumber in the bulk of the pattern itself this selection mechanism can drive the pattern into the benjamin fier eckhaus unstable regime generating spatio temporal chaos with space time dislocations in the convectively unstable regime the source of group perturbation becomes a noise amplifier

a mathematical theorem as the basis for the second law thomson s formulation applied to equilibrium abstract there are several formulations of the second law and they may in principle have different domains of validity here a simple mathematical theorem is proven which serves as the most general basis for the second law namely the thomson formulation cyclic changes cost energy applied to equilibrium this formulation of the second law is a property akin to particle conservation normalization of the wave function it has been strictly proven for a canonical ensemble and made plausible for a micro canonical ensemble as the derivation does not assume time inversion invariance it is applicable to situations where persistent currents occur this clear cut derivation allows to revive the no perpetuum mobile in equilibrium formulation of the second law and to criticize some assumptions which are widespread in literature the result puts recent results devoted to foundations and limitations of the second law in proper perspective and structurizes this relatively new field of research

pattern generation in space and aspect a survey is presented of theories for the generation and maintenance of spatial pattern in reaction diffusion equations and their generalizations applications are selected from the biological sciences and physical chemistry special emphasis is placed on nonlocal interaction as manifested by the inclusion of terms involving higher derivatives or integrals it is stressed that traditional ideas of spatial pattern generation can usefully be extended to the study of pattern in general descriptive aspect variables particularly in understanding ecological diversity and heterogeneity

some remarks on the linearized operator about the radial solution for the ginzburg landau equation abstract we consider the linearized operators denoted l d 1 of the ginzburg landau operator u u 1 u 2 in r2 about the radial solutions ud 1 x fd r eid for all d 1 we state the correspondence between the real vector space of the bounded solutions of the equation l d 1 w 0 and the eigenvalues of the linearized operators of the equations u 1 e2u 1 u 2 0 in b 0 1 about the radial solutions ud e x fd r e eid that tend to 0 as e tends to 0

rotating spiral waves in λ ω systems on circular domains abstract systems are reaction diffusion systems whose reaction kinetics admit a stable limit cycle it is known that systems can possess various types of solutions among them spiral waves are the most fascinating pattern however the effects of the diffusivity the sizes of the domains and the reaction kinetics on spiral waves are largely unknown in this paper we investigate how these quantities affect the properties of m armed spiral waves in a generalized class of system on a circular disk with no flux boundary condition first we derive a criterion for the existence of m armed spiral waves specifically we show that m armed spiral waves do not exist for d 0 r 2 j m 2 while for d 0 0 r 2 j m 2 there exists an m armed spiral wave if the twist parameter q is small here d is the diffusivity for the system r is the radius of the circular disk 0 is the value of the function a at a 0 and j m is the first positive zero of the first derivative of the bessel function of the first kind of order m we also show that the critical diffusivity d 0 r 2 j m 2 is a bifurcation point next we use the numerical simulation to show that for small twist parameter the rotational frequency increases with increasing domain size while for large twist parameter the dependence of the rotational frequency on the domain size is not monotonic moreover small circular domains may change the properties of spiral waves drastically these numerical results are in contrast to those in excitable media finally the stability of spiral waves is investigated numerically

globally coupled noisy oscillators with inhomogeneous periodic forcing we study the collective properties of an array of nonlinear noisy oscillators driven by nonidentical periodic signals we consider the case of a globally coupled array of harmonically forced weakly nonlinear oscillators where there is a constant difference between the phases of the forcing signals applied to adjacent oscillators this system is a prototypical model of a nonlinear phased array receiver we derive analytical results for the array output in the limit of a large number of oscillators for the noise free and noisy cases numerical simulations show good agreement with the theoretical analysis

performance of autonomous quantum thermal machines hilbert space dimension as a thermodynamical resource multilevel autonomous quantum thermal machines are discussed in particular we explore the relationship between the size of the machine captured by hilbert space dimension and the performance of the machine using the concepts of virtual qubits and virtual temperatures we show that higher dimensional machines can outperform smaller ones for instance by considering refrigerators with more levels lower temperatures can be achieved as well as higher power we discuss the optimal design for refrigerators of a given dimension as a consequence we obtain a statement of the third law in terms of hilbert space dimension reaching absolute zero temperature requires infinite dimension these results demonstrate that hilbert space dimension should be considered a thermodynamic resource

on the mean field spherical model exact solutions are obtained for the mean field spherical model with or without an external magnetic field for any finite or infinite number n of degrees of freedom both in the microcanonical and in the canonical ensemble the canonical result allows for an exact discussion of the loci of the fisher zeros of the canonical partition function the microcanonical entropy is found to be nonanalytic for arbitrary finite n the mean field spherical model of finite size n is shown to be equivalent to a mixed isovector isotensor model on a lattice of two sites partial equivalence of statistical ensembles is observed for the mean field spherical model in the thermodynamic limit a discussion of the topology of certain state space submanifolds yields insights into the relation of these topological quantities to the thermodynamic behavior of the system in the presence of ensemble nonequivalence

abundance of c 1 robust homoclinic tangencies a diffeomorphism f has a c 1 robust homoclinic tangency if there is a c 1 neighbourhood u of f such that every diffeomorphism in g u has a hyperbolic setg depending contin uously on g such that the stable and unstable manifolds ofg have some non transverse intersection for every manifold of dimension greater than or equal to three we exhibit a lo cal mechanism blender horseshoes generating diffeomorphisms with c 1 robust homoclinic tangencies using blender horseshoes we prove that homoclinic classes of c 1 generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings of appropriate dimensions display c 1 robust homoclinic tangencies

symmetries periodic plane waves and blow up of l ω systems abstract nonclassical symmetries of one dimensional reaction diffusion equations of the type have been studied the functional forms of and for which the system admits nonclassical symmetries have been determined and the corresponding reduced systems have been obtained some of these reduced systems admit symmetries which lead to further reductions among the several classes of exact solutions that have been obtained asymptotically periodic plane waves appear as similarity solutions of systems we also have obtained a family of solutions that exhibit a blow up process

irregular wakes in reaction diffusion waves abstract reaction diffusion equations have proved to be highly successful models for a wide range of biological and chemical systems but chaotic solutions have been very rarely documented we present a new mechanism for generating apparently chaotic spatiotemporal irregularity in such systems by analysing in detail the bifurcation structure of a particular set of reaction diffusion equations on an infinite one dimensional domain with particular initial conditions we show that possible solutions include travelling fronts which leave behind either regular or irregular spatiotemporal oscillations using a combination of analytical and numerical analysis we show that the irregular behaviour arises from the instability of oscillations induced by the passage of the front finally we discuss the generality of this mechanism as a way in which spatiotemporal irregularities can arise naturally in reaction diffusion systems

curl free ginzburg landau vortices for certain nonlinear elliptic pde problems in two dimensions the classical isoperimetric inequality produces a sharp inequality that violates a pohozaev identity except for radial symmetric decreasing solutions a generalized version of this technique is used here to prove radial symmetry of curl free ginzburg landau vortices

the dynamics of three dimensional scroll waves in excitable media abstract we investigate the motion of the central filament of a scroll wave in excitable medium we assume that two dimensional spiral solutions are known and use these solutions to produce slowly varying scroll waves in three dimensional space using standard ideas from perturbation theory multiscale techniques we find equations governing the motion of the central filament of the scroll and the twist of the scroll wave these equations are specialized to the case of a circular filament with uniformly distributed twist to determine the collapse or expansion and vertical drift of the spiral filament and to a helical filament with drift the predictions of these equations are in good qualitative agreement with the numerical experiments

dynamics of wave fronts in excitable media abstract the dynamics of wave fronts in two dimensional excitable media is analytically studied by means of the asymptotic perturbation method for wave fronts in reaction diffusion systems we derive a space derivative burgers equation which governs the shape of boundary layers and their propagation application of lie group analysis of differential equations yields group invariant solutions of this equation and their classification some classes of the group invariant solutions determine the remarkable patterns expanding circle waves rotating spiral waves initial value problems for the burgers equation are also studied and a spiral wave is shown to be asymptotically realized under some proper boundary conditions

synchronized disorder in a 2d complex ginzburg landau equation the phenomenon of spiral pair synchronization by oscillating strips nozaki bekki solution in a 2d cgle is investigated analytically the equations describing the interaction of the strips with one another and with spirals are derived analysis of the equations shows that under certain conditions the strips lead to frequency and phase locking of the spirals in this case the spiral pair dipole is aligned parallel to the strips with the position along the strips being arbitrary thus the interaction with strips may transform the spatio temporal chaos of spirals to the regime of periodically oscillating spatial disorder the dynamics of circular strips is investigated and their lifetime is estimated the behavior of the spirals bounded by circular stript is analysed

spatially structured activity in synaptically coupled neuronal networks i traveling fronts and pulses we consider traveling front and pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions this is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed our second goal is to establish the existence of traveling pulse solutions using more rigorous methods two techniques are presented the first a shooting argument reduces the problem from finding a specific solution to an integro differential equation system to finding any solution to an ode system the second a singular perturbation argument provides a construction of traveling pulse solutions un

abundance of fast growth of the number of periodic points in 2 dimensional area preserving dynamics we prove that there exists an open subset of the set of real analytic hamiltonian diffeomorphisms of the two dimensional torus in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense we also prove that there exists an open subset of the set of smooth area preserving diffeomorphisms of a closed surface in which typical diffeomorphisms exhibit fast growth of the number of periodic points

strong resonances of spatially distributed oscillators a laboratory to study patterns and defects abstract this paper describes phenomena which arise in a continuous system of coupled oscillators subject to a strongly resonant forcing we show that the temporal forcing induces a new type of instability leading to the formation of spatial patterns such as stripes hexagons and spatio temporal intermittent states reminiscent of chemical and magnetical systems we also report the existence of localized states where the instability is confined in the core of a defect leading to an increased mobility and diffusive behaviour of the defect

knotted scroll wave filaments in excitable media abstract it is shown that certain general systems of diffusion reaction equations can have invariant knotted scroll wave solutions the theory is based on the assumption that the system of equations supports permanently rotating spiral solutions and from this assumptions equations governing the motion of the scroll wave filament can be derived then bifurcatilon theory is used to show that the equations of motion have torus knots as invariant solutions analytical methods are used to determine qualitative and quantitative features of these knotted solutions

synchrony stability and firing patterns in pulse coupled oscillators we study non trivial firing patterns in small assemblies of pulse coupled oscillatory maps we find conditions for the existence of waves in rings of coupled maps that are coupled bi directionally we also find conditions for stable synchrony in general all to all coupled oscillators surprisingly we find that for maps that are derived from physiological data the stability of synchrony depends on the number of oscillators we describe rotating waves in two dimensional lattices of maps and reduce their existence to a reduced system of algebraic equations which are solved numerically

the effect of boundaries on the asymptotic wavenumber of spiral wave solutions of the complex ginzburg landau equation abstract in this paper we consider an oscillatory medium whose dynamics are modeled by the complex ginzburg landau equation in particular we focus on n armed spiral wave solutions of the complex ginzburg landau equation in a disk of radius d with homogeneous neumann boundary conditions it is well known that such solutions exist for small enough values of the twist parameter q and large enough values of d we investigate the effect of boundaries on the rotational frequency of the spirals which is an unknown of the problem uniquely determined by the parameters d and q we show that there is a threshold in the parameter space where the effect of the boundary on the rotational frequency switches from being algebraic to exponentially weak we use the method of matched asymptotic expansions to obtain explicit expressions for the asymptotic wavenumber as a function of the twist parameter and the domain size for small values of q

numerical study of the vector complex swift hohenberg equation abstract the vector complex swift hohenberg equation is a natural extension of the scalar complex version that is widely used for the description of lasers and other non linear optical systems numerical analysis of this equation reveals a great variety of patterns and structures such as traveling waves spiral waves defects segregation and competition between stable solutions simple analytical arguments are given to explain these behaviours

wavefront geometry in excitable media abstract in a certain kind of continuous medium an isolated fragment of wavefront behaves peculiarly its edge a closed ring remains stationary while the fragment s interior continues its forward propagation curling into a scroll like shape along the edge a scroll ring the edge the axis of the scroll ring is a 1 dimensional singularity in the resulting periodic wave field from numerical simulations in 2 dimensional continua and from laboratory experiments in 3 dimensional continua this seems to be typical behavior for so called excitable media the implications include a bestiary of organizing centers each composed of a topologically distinct arrangement of linked and knotted scroll rings two examples are given exact description of these organizing centers awaits formulation of a tractable equation that embodies the essential features of an excitable medium

planar solitary and spiral waves of the burgers cgl equations for flames governed by a sequential reaction this paper studied the planar solitary and spiral waves of the coupled burgers complex ginzburg landau burgers cgl equations which were derived from the nonlinear evolution of the coupled long scale oscillatory and monotonic instabilities of a uniformly propagating combustion wave governed by a sequential chemical reaction having two flame fronts corresponding to two reaction zones with a finite separation distance between them first some exact solutions including the planar and solitary waves for the one dimensional burgers cgl equations that are obtained by subtle transforms and the hyperbolic tangent function expansion method second some spiral waves for the two dimensional burgers cgl equations are investigated the existence of the spiral waves is proved rigorously by schauder s fixed point theorem applied to a class of functions and the approximate solutions are also obtained by the power series expansion method furthermore some numerical simulations are carried out near 0 r 1 since the

analytical description of a state dominated by spiral defects in two dimensional extended systems abstract we study the state of a diluted gas of spiral defects observed in a recent simulation of a two dimensional extended system governed by the ginzburg landau equation with complex coefficients we show that this state which is responsible for the disorganization of the system can be described by an approximate solution in which the new variables are the positions of the spirals and a new global phase the dynamics of these new variables and the conditions for the existence of the state are determined

singular perturbation theory of traveling waves in excitable media a review abstract waves of chemical or electrical activity traveling through space have been observed in several contexts chemical reaction mixtures in non convecting liquid phase cell suspensions nerve axons and neuromuscular tissues typically wave supporting preparations are excitable that is they respond and sensitivity to perturbations are rapidly damped out but suprathreshold disturbances trigger an abrupt and substantial response the abruptness of the response can be exploited by the methods of singular perturbation theory to obtain a mathematical description of wave propagation in spatially distributed excitable media singular perturbation analysis of propagating waves in one spatial dimension is straightforward and uncontentious but the analysis of propagating waves in one spatial dimension is straightforward and several fundamentally different ways we compare and contrast the approaches taken by greenberg zykov fife krinskii and his collaborators and ourselves with particular emphasis on the case of rotating spiral waves our intention is to bring some order to the important but difficult theory of propagating waves in two dimensional excitable media in conclusion we discuss briefly some possible extensions of the singular perturbation approach to propagating wave surfaces in three dimensional space

on models with non rough poincare homoclinic curves abstract the possibility of an a priori complete description of finite parameter models including systems with structurally unstable poincare homoclinic curves is studied the main result reported here is that systems having a countable set of moduli of equivalence and systems having infinitely many degenerate periodic and homoclinic orbits are dense in the newhouse regions of non stability we discuss the question of correctly setting a problem for the analysis of models of such type

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3