random catalytic reaction networks abstract we study networks that are a generalization of replicator or lotka volterra equations they model the dynamics of a population of object types whose binary interactions determine the specific type of interaction product such a system always reduces its dimension to a subset that contains production pathways for all of its members the network equation can be rewritten at a level of collectives in terms of two basic interaction patterns replicator sets and cyclic transformation pathways among sets although the system contains well known cases that exhibit very complicated dynamics the generic behavior of randomly generated systems is found numerically to be extremely robust convergence to a globally stable rest point it is easy to tailor networks that display replicator interactions where the replicators are entire self sustaining subsystems rather than structureless units a numerical scan of random systems highlights the special properties of elementary replicators they reduce the effective interconnectedness of the system resulting in enhanced competition and strong correlations between the concentrations

spectral behavior of contractive noise we study the behavior of the spectra corresponding to quantum systems subjected to a contractive noise i e the environment reduces the accessible phase space of the system but the total probability is conserved we find that the number of long lived resonances grows as a power law in h but surprisingly there is no relationship between the exponent of this power law and the fractal dimension of the corresponding classical attractor this is in disagreement with the predictions of the fractal weyl law which has been established for open systems where the probability is lost under the effect of a projective opening

fredholm method for scars a new quasiclassical formula for scars is obtained by using the fredholm method we show that it can be expanded into a formula obtained earlier by agam and fishman the derivation is simple and direct it is also more rigorous and more general than that of agam and fishman it also clarifies the remarkable process of resurgence relating the high order terms based on long orbits to the weyl term whose origin is the zero length orbits

weyl asymptotics from closed to open systems we present microwave experiments on the symmetry reduced five disk billiard studying the transition from a closed to an open system the measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied for the closed system this counting function shows the weyl asymptotic with a leading exponent equal to 2 by opening the system successively this exponent decreases smoothly to a noninteger value for the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed the results can be interpreted as a first experimental indication for the fractal weyl conjecture for resonances

role of short periodic orbits in quantum maps with continuous openings we apply a recently developed semiclassical theory of short periodic orbits to the continuously open quantum tribaker map in this paradigmatic system the trajectories are partially bounced back according to continuous reflectivity functions this is relevant in many situations that include optical microresonators and more complicated boundary conditions in a perturbative regime the shortest periodic orbits belonging to the classical repeller of the open map a cantor set given by a region of exactly zero reflectivity prove to be extremely robust in supporting a set of long lived resonances of the continuously open quantum maps moreover for steplike functions a significant reduction in the number needed is obtained similarly to the completely open situation this happens despite a strong change in the spectral properties when compared to the discontinuous reflectivity case in order to give a more realistic interpretation of these results we compare with a fresnel type reflectivity function

quantum scars of classical closed orbits in phase space the way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x q p through the spectral wigner function w x e this is a sum over wigner functions of eigenstates within a range of energy e in the classical limit w is concentrated on the energy surface and smoothly distributed over it closed orbits provide oscillatory corrections scars for which explicit semiclassical formulae are calculated each scar is a fringe pattern decorating the orbit as x moves off the energy surface the fringes form an airy pattern with spacing of order h as x moves off the closed orbit the fringes form a complex gaussian with spacing h

semiclassical dynamics in the strongly chaotic regime breaking the log time barrier we investigate the behavior of the quantum baker s transformation a system whose classical analogue is completely chaotic for time scales where the classical mechanics generates phase space structures on a scale smaller than planck s constant i e past the log time t ln 1 surprisingly we find that a semiclassical theory can accurately reproduce many features of the quantum evolution of a wave packet in this strongly mixing time regime

experimental versus numerical eigenvalues of a bunimovich stadium billiard a comparison we compare the statistical properties of eigenvalue sequences for a gamma 1 bunimovich stadium billiard the eigenvalues have been obtained in two ways one set results from a measurement of the eigenfrequencies of a superconducting microwave resonator real system and the other set is calculated numerically ideal system we show influence of mechanical imperfections of the real system in the analysis of the spectral fluctuations and in the length spectra compared to the exact data of the ideal system we also discuss the influence of a family of marginally stable orbits the bouncing ball orbits in two microwave stadium billiards with different geometrical dimensions

semiclassical wavefunctions in chaotic scattering systems we have computed quantum wavefunctions in the high energy semiclassical regime in a system the stadium billiard with leads allowing particles to enter and escape exemplifying chaotic scattering the results exhibit a structure associated with classical paths that is dramatically more pronounced than the scars due to periodic orbits seen in bound systems moreover this structure is seen at all energies our results differ significantly from those previously conjectured on the basis of computations in the low energy regime the dominant role played by short classical paths is explained by a semiclassical theory based on the analytic structure of the green function this is verified by a direct semiclassical computation of wavefunctions using classical scattering trajectories

acquisition of pn sequences in chip synchronous ds ss systems using a random sequence model and the sprt the use of a sequential probability ratio test sprt for the acquisition of pseudonoise pn sequences in chip syn chronous direct sequence spread spectrum ds ss systems is considered the out of phase sequence is modeled as a random sequence and the probabilities of error and expected sample sizes for the corresponding test are derived a different and very commonly used test is obtained if the out of phase sequence is modeled as a zero sequence the probabilities of error and the expected sample sizes of both sprt s are compared and it is shown that the latter test has a significantly larger probability of type i error numerical evaluation of the performance of both tests applied to a pn sequence of period 21 1 gives results in agreement with the analytical results we conclude that a random sequence is an excellent model for a pn sequence and that significant degradation in performance can be expected if the test design is based on the zero sequence model rather than on the random sequence model

fractal weyl law for quantum fractal eigenstates the properties of the resonant gamow states are studied numerically in the semiclassical limit for the quantum chirikov standard map with absorption it is shown that the number of such states is described by the fractal weyl law and their husimi distributions closely follow the strange repeller set formed by classical orbits nonescaping in future times for large matrices the distribution of escape rates converges to a fixed shape profile characterized by a spectral gap related to the classical escape rate

random analytic chaotic eigenstates the statistical properties of random analytic functions z are investigated as a phase space model for eigenfunctions of fully chaotic systems we generalize to the plane and to the hyperbolic plane a theorem concerning the equidistribution of the zeros of z previously demonstrated for a spherical phase space su 2 polynomials for systems with time reversal symmetry the number of real roots is computed for the three geometries in the semiclassical regime the local correlation functions are shown to be universal independent of the system considered or the geometry of phase space in particular the autocorrelation function of is given by a gaussian function the connections between this model and the gaussian random function hypothesis as well as the random matrix theory are discussed

fractal weyl law for open quantum chaotic maps we study a semiclassical quantization of poincar e maps arising in scattering problems with fractal hyperbolic trapped sets the main application is the proof of a fractal weyl upper bound for the number of resonances scattering poles in small domains near the real axis this result includes the case of several convex hard obstacles satisfying a no eclipse condition

no quantum ergodicity for star graphs we investigate statistical properties of the eigenfunctions of the schrodinger operator on families of star graphs with incommensurate bond lengths we show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average we further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds we describe how to construct such subsequences explicitly these structures are analogous to scars on short unstable periodic orbits

localization of wave patterns on classical periodic orbits in a square billiard the connection between wave functions and classical periodic trajectories in a square billiard is analytically constructed by using the representation of su 2 coherent states the analytical function form is modified to show that the wave patterns can be apparently localized on the classical periodic trajectories by superposing a few nearly degenerate eigenfunctions based on the analogy between the schrodinger and helmholtz equations the features of wave functions are experimentally studied from the transverse pattern formation in a laterally confined microcavity laser the experimental transverse pattern in a square shaped microcavity agrees very well with the constructed wave pattern concentrated along classical periodic orbits

indices in classical mechanics the authors discuss the linearization of the equations of motion around a periodic orbit in a conservative two degree of freedom hamiltonian system with an emphasis on computationally useful formulae indices are defined using singularities in projections conjugate points self conjugate points and the possibility of assigning a symbolic code are discussed

correlations of chaotic eigenfunctions a semiclassical analysis we derive a semiclassical expression for an energy smoothed autocorrelation function defined on a group of eigenstates of the schr odinger equation the system we consider is an energy conserved hamiltonian system possessing time invariant symmetry the energy smoothed autocorrelation function is expressed as a sum of three terms the first one is analogous to berry s conjecture which is a bessel function of the zeroth order the second and the third terms are trace formulae made from special trajectories the second term is found to be direction dependent in the case of spacing averaging which agrees qualitatively with previous numerical observations in high lying eigenstates of a chaotic billiard

fringes decorating anticaustics in ergodic wavefunctions the probability density ii is calculated for quantum eigenstates near spatial boundaries of classically chaotic regions by contrast with integrable systems for which the classical ii diverges on classical boundaries which are caustics in chaotic systems the classical ii does not diverge but vanishes abruptly in a way that depends on the number of freedoms n the boundaries are anticaustics quantum mechanics softens anticaustics to give ii in terms of a set of canonical diffraction patterns one for each n these are studied in detail the appropriate definition of ii involves averaging over eigenstates in an energy range larger than o h but smaller than o h frac 2 3 where h is planck s constant that is over a range of delta n states near the nth where n 1 1 n ll delta n ll n 1 2 3n

survey of the eigenfunctions of a billiard system between integrability and chaos we study numerically the eigenfunctions and their wigner phase space distributions of the two dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by robnik 1983 this system is a generic kam system and displays a transition from integrability to almost ergodicity as the billiard shape changes we clearly identify two classes of states the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions whilst the mixed type states were not found in support of percival s conjecture 1973 we confirm the existence of extremely intense scars in the classically chaotic regions and demonstrate their association with classical periodic orbits three classes of scars are revealed one orbit scars many orbit one family scars of statistically similar orbits in the homoclinic neighbourhood and many orbit many family scars we argue that it is impossible to find an a priori semiclassical theory of individual eigenstates but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates

phase transition in random catalytic networks the notion of auto catalytic networks has become a cornerstone in understanding the possibility of a sudden dramatic increase of diversity in biological evolution as well as in the evolution of social and economical systems here we study catalytic random networks with respect to the final outcome diversity of products we show that an analytical treatment of this long standing problem is possible by mapping the problem onto a set of nonlinear recurrence equations the solution of these equations shows a crucial dependence of the final number of products on the initial number of products and the density of catalytic production rules for a fixed density of rules we can demonstrate the existence of a phase transition from a practically unpopulated regime to a fully populated and diverse one the order parameter is the number of final products we are able to fully understand the origin of this phase transition as a crossover from one set of solutions from a quadratic equation to the other we observe a remarkable similarity of the solution of the system and the pvt diagrams in standard thermodynamics

semiclassical mechanics of symmetry reduction the author discusses semiclassical approximations that are adapted to given symmetry classes in quantum mechanics arbitrary abelian symmetries and also rotational symmetry are treated semiclassical approximations are derived for the projected propagator and energy dependent green s function associated with a given irreducible representation of the symmetry group from these they derive trace formulae analogous to the usual trace formula that determine the energy levels in a given symmetry class in terms of classical orbits

semiclassical limit of chaotic eigenfunctions a generic chaotic eigenfunction has a non universal contribution consisting of scars of short periodic orbits this contribution which cannot be predicted by a model of random universal waves survives the semiclassical limit when goes to zero in this limit the sum of scarred intensities only depends on f 1 2i 1 2 ht with f the degrees of freedom i the set of positive lyapunov exponents and ht the topological entropy moreover taking into account that relative fluctuations of the scarred intensities tend to zero as 1 ln we are able to provide a detailed description of a generic chaotic eigenfunction in the semiclassical limit our conclusions were verified in the bunimovich stadium billiard

fractal weyl law behavior in an open hamiltonian system we numerically show fractal weyl law behavior in an open hamiltonian system that is described by a smooth potential and which supports numerous above barrier resonances this behavior holds even relatively far away from the classical limit the complex resonance wave functions are found to be localized on the fractal classical repeller

towards a physics of evolution critical diversity dynamics at the edges of collapse and bursts of diversification systems governed by the standard mechanisms of biological or technological evolution are often described by catalytic evolution equations we study the structure of these equations and find an analogy with classical thermodynamic systems in particular we can demonstrate the existence of several distinct phases of evolutionary dynamics a phase of fast growing diversity one of stationary finite diversity and one of rapidly decaying diversity while the first two phases have been subject to previous work here we focus on the destructive aspects in particular the phase diagram of evolutionary dynamics the main message is that within a critical region massive loss of diversity can be triggered by very small external fluctuations we further propose a dynamical model of diversity which captures spontaneous creation and destruction processes fully respecting the phase diagrams of evolutionary systems the emergent time series show rich diversity dynamics including power laws as observed in actual economical data e g firm bankruptcy data we believe the present model presents a possibility to cast the famous qualitative picture of schumpeterian economic evolution into a quantifiable and testable framework

quantum phase space structures in classically mixed systems a scattering approach the structure of quantum wavefunctions is discussed for a billiard system with mixed classical dynamics using a scattering formalism we introduce a husimi like distribution for s matrix eigenvectors at arbitrary wavenumbers k not necessarily eigenenergies of the system this constitutes probability density plots on the poincare surface of section at any k many eigenvectors are structured by classical objects such as invariant tori the intermediate layer between regular and chaotic motion and unstable periodic orbits scars we find that eigenvector structure is stable over significant energy ranges while the associated eigenphases increase linearly with k at a rate determined by the underlying classical objects additionally we observe that in the presence of time reversal symmetry eigenvectors scarred by non self retracing periodic orbits form doublets which are closely spaced in eigenphase our results provide a new perspective on the quantization of structured states quantization is determined by the linearly increasing eigenphases and occurs whenever an eigenphase is equal to while structure is encoded in the eigenvectors it is present at all energies and varies only slowly this allows one to predict many quantized eigenfunctions over an entire energy range by studying the eigenparameters of at a single intermediate energy and identifying the underlying classical objects

adaptive evolution on a continuous lattice model in the current work we investigate the evolutionary dynamics of a spatially structured population model defined on a continuous lattice in the model individuals disperse at a constant rate v and competition is local and delimited by the competition radius r due to dispersal the neighborhood size number of individuals competing for reproduction fluctuates over time here we address how these new variables affect the adaptive process while the fixation probabilities of beneficial mutations are roughly the same as in a panmitic population for small fitness effects s a dependence on v and r becomes more evident for large s these quantities also strongly influence fixation times but their dependencies on s are well approximated by s ensuremath 1 2 which means that the speed of the genetic wave front is proportional to sqrt s most important is the observation that the model exhibits a dual behavior displaying a power law growth for the fixation rate and speed of adaptation with the beneficial mutation rate as observed in other spatially structured population models while simultaneously showing a nonsaturating behavior for the speed of adaptation with the population size n as in homogeneous populations

optimal and suboptimal coherent acquisition schemes for pn sequences with data modulation almost all previous acquisition schemes for direct sequence spread spectrum systems assume that there is no data modulation however in some applications there is need for acquisition systems with the presence of data modulation which is what we consider in this paper first we derive an optimal scheme using maximum likelihood estimate by treating the phase of the incoming sequence as a signal parameter to be estimated two practical parallel search schemes as approximations to the optimal scheme are presented we also consider serial and hybrid search schemes numerical results show that significant degradation of the standard coherent acquisition schemes occurs when there is data modulation using the proposed search schemes performance is restored to the level which is almost equal to that of the standard schemes without data modulation

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

weak quantum ergodicity abstract we examine the consequences of classical ergodicity for the localization properties of individual quantum eigenstates in the classical limit we note that the well known schnirelman result is a weaker form of quantum ergodicity than the one implied by random matrix theory this suggests the possibility of systems with non gaussian random eigenstates which are nonetheless ergodic in the sense of schnirelman and lead to ergodic transport in the classical limit these we call weakly quantum ergodic indeed for a class of slow ergodic classical systems it is found that each eigenstate becomes localized to an ever decreasing fraction of the available state space in the semiclassical limit nevertheless each eigenstate in this limit covers phase space evenly on any classical scale and long time transport properties between individual quantum states remain ergodic due to the diffractive effects which dominate quantum phase space exploration

lyapunov stability for a generalized henon heiles system in a rotating reference frame in this paper we focus on a generalized henon heiles system in a rotating reference frame in such a way that lagrangian like equilibrium points appear our goal is to study their nonlinear stability properties to better understand the dynamics around these points we show the conditions on the free parameters to have stability and we prove the superstable character of the origin for the classical case it is a stable equilibrium point regardless of the frequency value of the rotating frame

quantum eigenfunctions in terms of periodic orbits of chaotic systems a resummed formula for the wigner function corresponding to an eigenfunction of a chaotic system in terms of periodic orbits is developed the infinite sum over periodic orbits is effectively truncated with the help of an extension of a method that was applied to the spectral determinant by berry and keating 1992 in principle the formula enables the computation of eigenstates and the probability density of wavefunctions from classical periodic orbits the conditions for appearance of scars are discussed

physics of evolution selection without fitness abstract traditionally evolution is seen as a process where from a pool of possible variations of a population e g biological species or industrial goods a few variations get selected which survive and proliferate whereas the others vanish survival probabilities and proliferation rates are typically associated with the fitness of particular variations in this paper we argue that the notion of fitness is an a posteriori concept in the sense that one can assign higher fitness to species that survive but one can generally not derive or predict fitness per se proliferation rates can be measured whereas fitness landscapes i e the inter dependence of proliferation rates cannot for this reason we think that in a physical theory of evolution such notions should be avoided in this spirit here we propose a random matrix model of evolution where selection mechanisms are encoded in interaction matrices of species thereby extending the previous work of ours by a control parameter describing suppressors in the system we are able to recover some key facts of evolution dynamics endogenously such as punctuated equilibrium i e the existence of intrinsic large extinction events and at the same time periods of dramatic diversification as known e g from the fossil record further we comment on two fundamental technical problems of a physics of evolution the non closedness of its phase space and the problem of co evolving boundary conditions apparent in all systems subject to evolution

on the distribution of zeros of chaotic wavefunctions wavefunctions in a phase space representation can be expressed as entire functions of their zeros if the phase space is compact these zeros seem to hide a lot of relevant and explicit information about the underlying classical dynamics besides an understanding of their statistical properties may prove to be useful in the analytical calculations of the wavefunctions in quantum chaotic systems this motivates us to pursue the present study which by a numerical statistical analysis shows that both long range correlations as well as short range correlations exist between zeros while the latter turn out to be universal and parametric independent the former seem to be system dependent and are significantly affected by various parameters i e symmetry localization etc furthermore for the delocalized quantum dynamics the distribution of these zeros seem to mimic that of the zeros of the random functions as well as random polynomials

using basis sets of scar functions we present a method to efficiently compute the eigenfunctions of classically chaotic systems the key point is the definition of a modified gram schmidt procedure which selects the most suitable elements from a basis set of scar functions localized along the shortest periodic orbits of the system in this way one benefits from the semiclassical dynamical properties of such functions the performance of the method is assessed by presenting an application to a quartic two dimensional oscillator whose classical dynamics are highly chaotic we have been able to compute the eigenfunctions of the system using a small basis set an estimate of the basis size is obtained from the mean participation ratio a thorough analysis of the results using different indicators such as eigenstate reconstruction in the local representation scar intensities participation ratios and error bounds is also presented

full characterization of a strange attractor chaotic dynamics in low dimensional replicator system abstract two chaotic attractors observed in lotka volterra equations of dimension n 3 are shown to represent two different cross sections of one and the same chaotic regime the strange attractor is studied in the equivalent four dimensional catalytic replicator network analytical expressions are derived for the lyapunov exponents of the flow in the centre of the chaotic regime the strange attractor was characterized by the lyapunov dimension dl 2 06 0 02 and the renyi fractal dimensions the set corresponding to the attractor represents a multifractal its singularity spectrum is computed one route in parameter space leading into the chaotic regime and crossing it was studied in detail it leads to a feigenbaum type cascade of bifurcations before the chaos is fully developed the dynamic system passes an internal crisis through a period two tangent bifurcation then a second sequence of period doubling bifurcations leading to another chaotic regime is observed which eventually ends in a crisis and then after a rather complicated periodic regime the attractor finally disappears trajectories in the intermediate periodic regime show transient chaotic bursts of intermittency type a series of one dimensional maps is derived from a properly chosen poincare cross section which illustrates structural changes in the attractor mutations are included in the catalytic replicator network and the changes in the dynamics observed are compared with the predictions of an approach based on perturbation theory the most striking result is the gradual disappearance of complex dynamics with increasing mutation rates the parameter space spanned by the mutation rate and selected parameters of the replication network is split in regions of complex periodic behaviour near the chaotic regime fractal boundaries of these regions are likely to occur but the data available do not allow definitive conclusions yet

smoothed wave functions of chaotic quantum systems abstract it is shown that wave functions of quantum systems as 0 have an extra density near unstable periodic trajectories of the classical problem the averaged wave function square is represented as the sum over a finite number of periodic trajectories the contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory the results are compared with the numerical calculations of the wave functions for the stadium billiard

noncoherent sequential acquisition of pn sequences for ds ss communications with without channel fading studies sequential acquisition schemes of m sequences based on the sequential probability ratio test sprt and a truncated sprt tsprt with noncoherent demodulation most reported results on sequential acquisition schemes assume that independent samples are available for the decision process the assumption of independent and identically distributed i i d samples requires the integrator in the receiver to be reset periodically this introduces loss in the effective signal to noise ratio degrading the performance in the present paper on the contrary the authors two sequential schemes use continuous integration it can be shown that the likelihood ratio is monotonic consequently the tests can be easily implemented in realtime methods are proposed for designing the decision thresholds to achieve the desired false alarm and miss probabilities performances of the proposed schemes are obtained and they suggest that the tsprt is more desirable the effect of slowly varying channel fading is also investigated results show that fading does not affect the false alarm probabilities but it can drastically reduce the probability of detecting the alignment of the two pn sequences especially when the fading is severe

evolutionary dynamics from a variational principle we demonstrate with a thought experiment that fitness based population dynamical approaches to evolution are not able to make quantitative falsifiable predictions about the long term behavior of some evolutionary systems a key characteristic of evolutionary systems is the ongoing endogenous production of new species these novel entities change the conditions for already existing species even darwin s demon a hypothetical entity with exact knowledge of the abundance of all species and their fitness functions at a given time could not prestate the impact of these novelties on established populations we argue that fitness is always a posteriori knowledge it measures but does not explain why a species has reproductive success or not to overcome these conceptual limitations a variational principle is proposed in a spin model like setup of evolutionary systems we derive a functional which is minimized under the most general evolutionary formulation of a dynamical system i e evolutionary trajectories causally emerge as a minimization of a functional this functional allows the derivation of analytic solutions of the asymptotic diversity for stochastic evolutionary systems within a mean field approximation we test these approximations by numerical simulations of the corresponding model and find good agreement in the position of phase transitions in diversity curves the model is further able to reproduce stylized facts of timeseries from several man made and natural evolutionary systems light will be thrown on how species and their fitness landscapes dynamically coevolve

mean field approximation for two and three person prisoner s dilemmas abstract we consider the mean field approximation mfa to the system of interacting agents playing two and three person prisoner s dilemma games 2 pd and 3 pd the agents have three available strategies all c all d and a third one which we choose in various ways long time distribution of the strategies is compared for both games the latter game admits coexistence of larger number of strategies in the long run in the case of external pressure for cooperation more pressure is in general necessary for 3 pd than for 2 pd to guarantee cooperation for almost all initial compositions of the strategies results for different types of tit for tat strategies for 3 pd are discussed unlike the 2 pd case the more forgiving tit for tat strategy can survive in the long run in 3 pd

dynamics of the eigen and the crow kimura models for molecular evolution we introduce an alternative way to study molecular evolution within well established hamilton jacobi formalism showing that for a broad class of fitness landscapes it is possible to derive dynamics analytically within the 1n accuracy where n is the genome length for a smooth and monotonic fitness function this approach gives two dynamical phases smooth dynamics and discontinuous dynamics the latter phase arises naturally with no explicite singular fitness function counterintuitively the hamilton jacobi method yields straightforward analytical results for the models that utilize fitness as a function of hamming distance from a reference genome sequence we also show the way in which this method gives dynamical phase structure for multipeak fitness

exact theory for the quantum eigenstates of a strongly chaotic system we present an exact theory for the quantum eigenstates of a strongly chaotic system which consists of a point particle sliding freely on a two dimensional compact surface of constant negative curvature the main result is a general sum rule which relates an almost arbitrarily smoothed sum over the quantum wavefunctions to a sum over the lengths of the classical orbits as an example we apply this formula to the gaussian smoothed sum over wavefunctions the question of scars as imprints of individual closed orbits is discussed and the connection with bogomolny s semiclassical theory is explicitly worked out the investigation of the amplitude distribution p reveals that the higher excited wavefunctions behave as gaussian random waves

kam tori are very sticky rigorous lower bounds on the time to move away from an invariant lagrangian torus with linear flow abstract hamiltonian perturbation theory is used to derive rigorous nekhoroshev type lower bounds on the time it takes to move away from an invariant lagrangian torus when the flow on the torus is conjugate to a highly non resonant linear flow in particular these results give bounds on the time to move away from a kam torus if r is the distance from the torus then the lower bound on the time to double the distance from the torus is c r e k r where c and k are constants 1 n 2 n is the number of degrees of freedom and is a measure of the irrationality of the flow we thus prove that nearby trajectories take an extremely long time to move away from such an invariant torus and that the more irrational the linear flow the stickier the torus will be these estimates do not require the steepness or quasi convexity assumptions of nekherosev s theorem

wavefunctions expectation values and scars on poincare sections a scattering approach using the scattering approach to quantization we define an appropriate representation of the billiard wavefunction on the scattering poincar section the expectation values of smooth operators in terms of these poincar section wavefunctions are expressed as sums over periodic orbits a special operator is used to define scars on the section and the relation to scars in configuration space is discussed and demonstrated numerically