on the structure of the cone of positive operators abstract if k 1 is a proper cone in r n 1 and k 2 is a proper cone in r n 2 then as is well known the set k 1 k 2 which consists of all n 2 x n 1 real matrices which take k 1 into k 2 forms a proper cone in the space r n 2 n 1 in this paper a study of this cone is made with particular emphasis on its faces and duality operator a face of k 1 k 2 is called simple if it is composed of all matrices in k 1 k 2 which take some fixed face of k 1 into some fixed face of k 2 maximal faces of k 1 k 2 are characterized as a particular kind of simple faces relations between the duality operator of k 1 k 2 and those of k 1 and k 2 are obtained among many other results it is proved that d k 1 k 2 the duality operator of k 1 k 2 is injective if and only if d k 2 is injective and each face of k 1 k 2 is an intersection of simple faces two open questions are posed

kolmogorov sinai entropy of many body hamiltonian systems the kolmogorov sinai ks entropy is a central measure of complexity and chaos its calculation for many body systems is an interesting and important challenge in this paper the evaluation is formulated by considering n dimensional symplectic maps and deriving a transfer matrix formalism for the stability problem this approach makes explicit a duality relation that is exactly analogous to one found in a generalized anderson tight binding model and leads to a formally exact expression for the finite time ks entropy within this formalism there is a hierarchy of approximations the final one being a diagonal approximation that only makes use of instantaneous hessians of the potential to find the ks entropy by way of a nontrivial illustration the ks entropy of n identically coupled kicked rotors standard maps is investigated the validity of the various approximations with kicking strength particle number and time are elucidated an analytic formula for the ks entropy within the diagonal approximation is derived and its range of validity is also explored

removable singularities for solutions of coupled yang mills dirac equations we prove a removable singularity theorem for solutions of coupled yang mills dirac equations on compact four dimensional manifolds we show that a field satisfying the coupled equations with a point singularity is gauge equivalent to a smooth field if the energy functional is finite the hypotheses are very natural only conformally invariant conditions are placed on force field and particle field interacting with the force field it is noticeable that there is no assumption on the derivative of the particle field

the perron generalized eigenspace and the spectral cone of a cone preserving map a unified treatment is offered to reprove known results on the following four highlights of the combinatorial spectral theory of nonnegative matrices or to extend or partly extend the results to the setting of a linear map preserving a polyhedral proper or proper cone the preferred basis theorem equivalent conditions for equality of the graph theoretic level characteristic and the spectral height characteristic the majorization relation between the two characteristics and the relation between the combinatorial properties of a nonnegative matrix and the positivity of the individual entries in its principal components this is achieved by employing the new concept of spectral cone of a cone preserving map and combining the cone theoretic methods developed in our previous papers on the geometric spectral theory of cone preserving maps with the algebraic analytic method introduced by hartwig neumann and rose and further exploited by neumann and schneider for nonnegative matrices

asymptotic stability of densities in coupled map lattices abstract sufficient conditions for the asymptotic stability strong convergence of density evolution in finite dimensional piecewise monotone map lattices with both constant and state dependent coupling are given these conditions are quite useful in numerical work since they allow one to precisely define where one expects to see numerical signatures of asymptotic stability asymptotic stability is illustrated with several examples it is also shown that in constantly coupled lattices the density formed by collapsing the higher dimensional density into one dimension can be approximated by the evolution of densities under the action of an appropriately perturbed one dimensional map

maximal exponents of polyhedral cones iii article history received 31 may 2009 available online 14 november 2009 submitted by j bastero

bounds on the exponent of primitivity which depend on the spectrum and the minimal polynomial abstract suppose a is an n n nonnegative primitive matrix whose minimal polynomial has degree m we conjecture that the well known bound on the exponent of primitivity n 1 2 1 due to wielandt can be replaced by m 1 2 1 the only case for which we cannot prove the conjecture is when m 5 the number of distinct eigenvalues of a is m 1 or m and the directed graph of a has no circuits of length shorter than m 1 but at least one of its vertices lies on a circuit of length not shorter than m we show that m m 1 is always a bound on the exponent this being an improvement on wielandt s bound when m n for the case in which a is also symmetric the bound which we obtain is 2 m 1 to obtain our results we prove a lemma which shows that for a general nonnegative matrix the number of its distinct eigenvalues is an upper bound on the length of the shortest circuit in its directed graph

computational chaos a prelude to computational instability abstract chaotic behavior sometimes occurs when difference equations used as approximations to ordinary differential equations are solved numerically with an excessively large time increment in two simple examples we find that as increases chaos first sets in when attractor a acquires two distinct points that map to the same point this happens when a acquires slopes of the same sign in a rectifying coordinate system at two consecutive intersections with the critical curve chaotic and quasi periodic behavior may then alternate within a range of before computational instability finally prevails bifurcations to and from chaos and transitions to computational instability are highly scheme dependent even among differencing schemes of the same order systems exhibiting computational chaos can serve as illustrative examples in more general studies of noninvertible mappings

noisy collective behaviour in deterministic cellular automata we investigate cellular automata in four and five dimensions for which chate and manneville recently have found nontrivial collective behaviour more precisely though being fully deterministic the average magnetization seems to be periodic respectively quasiperiodic with superimposed noise whose amplitude decreases with system size we confirm this behaviour on very large systems and over very large times we analyse in detail the statistical properties of the noise systems on small lattices and or subject to additional external noise are metastable arguments by grinstein et al suggest that in the periodic case the infinite deterministic systems should be metastable too these arguments are generalized to quasiperiodic systems we find evidence that they do indeed apply but we find no direct evidence for metastability of large systems

homoclinic tangles for noninvertible maps not all naturally occurring iterated systems have the property that time is reversible in which case the dynamics must be modelled by a noninvertible map rather than by a di eomorphism while one dimensional theory describes dynamics of noninvertible maps higher dimensional theory has focussed on di eomorphisms though there are many higher dimensional examples of natural systems modelled by noninvertible maps for example noninvertible maps occur in the study of population dynamics 2 time one maps of delay equations 25 control theory algorithms 1 5 6 neural networks 16 and iterated di erence methods 14 in order to use dynamical systems theory in applications for which maps are noninvertible it is important to know how the theory di ers from the di eomorphism case one of the principal indications of chaotic dynamics in deterministic systems is the existence of transverse homoclinic orbits this paper examines the existence of homoclinic tangles and the associated dynamic behavior for noninvertible maps in particular we give examples of maps which have transverse intersection of stable and unstable manifolds but no horseshoe furthermore we show that this is a codimension two phenomenon papers by steinlein and walther 24 25 and lerman and shil nikov 13 both establish conditions for a transverse homoclinic point of a noninvertible map to be contained in a horseshoe this is discussed further in section 4 the work of laura gardini 10 addressed transverse homoclinic orbits for planar maps however she missed the subtle distinction for noninvertible maps between a transverse homoclinic point and

on the distinguished eigenvalues of a cone preserving map abstract we generalize many known results on a nonnegative matrix concerning linear inequalities collatz wielandt sets and generalized eigenvectors to the setting of a matrix preserving a finite dimensional proper cone a simple cone theoretic proof is given for the nonnegative basis theorem for the algebraic eigenspace of a nonnegative matrix the result is also extended to a matrix preserving a polyhedral cone given proper cones k 1 and k 2 in different euclidean spaces a necessary and sufficient condition is also obtained for the existence of a nonzero matrix x which takes k 2 into k 1 and satisfies ax xb where a b are given matrices preserving k 1 and k 2 respectively this extends and answers a recent open question posed by hartwig

using advection to control the velocity of patterns in rings of maps traveling patterns are well known features of rings of symmetrically coupled maps which however propagate with rather small velocities of the order of 10 3 sites per step we show that it is easy to produce traveling patterns with velocities tunable over three orders of magnitude by simply breaking the symmetry of the coupling between neighbors this asymmetry arises naturally when the usual model of coupled map lattices is generalized to also include advection in addition asymmetries change the wavelength of waves traveling on the lattice 2002 elsevier science b v all rights reserved

asymptotic periodicity and banded chaos abstract treating one dimensional maps as dynamical systems we examine their evolution in terms of density flow in particular a type of density evolution known as asymptotic periodicity is studied unlike statistically stable or exact systems asymptotically periodic systems do not in general evolve to an invariant density even though they possess one consequently a nonequilibrium formalism using several metastable states rather than one invariant density is examined as an alternative to study the physical properties of asymptotically periodic systems asymptotic periodicity is demonstrated for the hat map and the quadratic map at the parameters where these maps generate banded chaos or quasiperiodicity using the ergodic properties of asymptotic periodicity a compact expression for the time correlation function is obtained that does not make any assumptions about decoupling the periodic and stochastic components of c t finally a generalization to the boltzmann gibbs entropy known as the conditional entropy is studied as an index characterizing the asymptotic density sequence that emerges in asymptotically periodic systems

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

on the stability of periodic orbits of two dimensional mappings we present a closed form stability criterion for the periodic orbits of two dimensional conservative as well as dissipative mappings which are analogous to the poincare maps of dynamical systems our stability criterion has a particularly simple form involving a finite symmetric nearly tridiagonal determinant its derivation is based on an extension of the stability analysis of hill s differential equation to difference equations we apply our criterion and derive a sufficient stability condition for a large class of periodic orbits of the widely studied standard mapping describing a periodically kicked free rotator as another example we also obtain explicitly and in closed form the intervals of bounded and unbounded solutions of a discrete schrodinger equation for the kronig and penney crystal model

spatio temporal intermittency in coupled map lattices abstract the transition to turbulence in a one dimensional array of maps coupled by diffusion is shown to display critical properties resembling those of directed percolation the analogy is supported by the reconstruction of a probabilistic cellular automation with closely similar statistical properties numerical results suggest however that spatio temporal intermittency does not belong to the same universality class as directed percolation

bifurcation structure of the c type period doubling transition abstract the period doubling transition to chaos along the edge of an arnold tongue is known to display unusual organization and scaling behavior kuznetsov et al 2005 7 it is also known that forced period doubling systems may be associated with the appearance of so called period doubled tori arneodo et al 1983 15 using the rossler system as an example we present a detailed analysis of the bifurcation structure associated with the forcing of a three dimensional period doubling system we explain how this structure is related to the recently discovered phenomenon of multi layered tori and discuss different bifurcation scenarios that transform a resonance torus into a period doubled ergodic torus similar bifurcation phenomena have recently been observed in a biologically relevant model of kidney blood flow regulation in response to fluctuations in arterial pressure

bifurcations of tori and phase locking in a dissipative system of differential equations abstract in a system of ordinary differential equations obtained through a seven mode truncation of the plane incompressible navier stokes equations a two dimensional torus undergoes first two period doubling bifurcations and then a transition to a strange attractor this strange attractor of liapunov dimension larger than three in a wide parameter interval is characterized by a power spectrum which retains the two fundamental frequencies of the original torus superimposed on a broad jagged background as the liapunov dimension goes down towards two an interesting phenomenon of phase locking occurs which gives rise to an alternation of chaotic and periodic behavior

bifurcation scenarios and quasiperiodicity in coupled maps abstract the appearance of coherent structures in a model of two coupled logistic maps is studied analytical results for the existence of multiple attractors and transitions from periodic to quasiperiodic motion are presented the theoretical prediction for the hopf radius agrees well with computer experiment throughout the quasiperiodic region

comment on removable singularities for solutions of coupled yang mills dirac equations j math phys 47 103502 2006 a lemma from elliptic theory is used to improve a recent result by li j math phys 47 103502 2006 concerning the removability of an isolated point singularity from solutions of the coupled yang mills dirac equations

on the spectrum of a leslie matrix with a near periodic fecundity pattern abstract leslie matrices are square and nonnegative and arise in the classical discrete age dependent model of population growth their eigenvalues are important in determining the asymptotic behavior of the age distributions in the model denoting the top row of the leslie matrix by m 1 m 2 m n it is well known that if d gcd i m i 0 2 which corresponds to a periodic fecundity pattern then the matrix has d eigenvalues with moduli equal to its spectral radius in this paper we consider leslie matrices with a near periodic fecundity pattern roughly speaking m i 0 only if i is close to a multiple of some d 1 and show that such matrices have at least two nonreal eigenvalues with moduli close to the spectral radius we discuss a specific example of such a leslie matrix which appears in the demographic literature and give a numerical example to show that the age distributions in the model can also exhibit near periodic behavior

fluctuations of finite time stability exponents in the standard map and the detection of small islands some statistical properties of finite time stability exponents in the standard map can be estimated analytically the mean exponent averaged over the entire phase space behaves quite differently from all the other cumulants whereas the mean carries information about the strength of the interaction and only indirect information about dynamical correlations the higher cumulants carry information about dynamical correlations and essentially no information about the interaction strength in particular the variance and higher cumulants of the exponent are very sensitive to dynamical correlations and easily detect the presence of very small islands of regular motion via their anomalous time scalings the average of the stability matrix inverse trace is even more sensitive to the presence of small islands and has a seemingly fractal behavior in the standard map parameter the usual accelerator modes and the small islands created through double saddle node bifurcations which come halfway between the positions in interaction strength of the usual accelerator modes are clearly visible in the variance whose time scaling is capable of detecting the presence of islands as small as 0 01 of the phase space we study these quantities with a local approximation to the trace of the stability matrix which significantly simplifies the numerical calculations as well as allows for generalization of these methods to higher dimensions we also discuss the nature of this local approximation in some detail

remarks on the quasi classical propagator of area preserving maps abstract the path integral like expression for the quantum propagator of discrete time area preserving maps is evaluated approximately by neglecting higher than second order terms in an expansion of the action about the classical paths in the resulting quasi classical approximation for the propagator special attention is paid to the recursive nature of its amplitude and the possible appearance of maslov like phases using a further approximation for the amplitude we arrive at explicit expressions which clearly show the differences between the contributions of stable and unstable classical paths an estimate for the range of validity for the quasi classical approximation is also given

on the bifurcation of marotto s map and its application in image encryption abstract the aim of this paper is to address the codimension one bifurcation of marotto s map and its utility in image encryption first of all local stability analysis and local bifurcation analysis of fixed points of the considered map are investigated in details according to the classical bifurcation theory and the center manifold theorem the map exhibits various bifurcation types such as transcritical flip and neimark sacker bifurcations second of all the map is proven to be chaotic in the sense of marotto since image encryption based on chaotic maps is very promising for cryptography marotto s map compound chaos and spatiotemporal chaos are combined to encrypt and decrypt images numerical simulations agree with the analytical framework for the complex dynamics of the map furthermore different test images are used to demonstrate the effectiveness of the method implemented for encryption

some symmetric two dimensional dissipative maps abstract a class of simple symmetric two dimensional maps have been studied theoretically and one finds periode n doubling at values of a parameter of the form cos k n a computer simulation confirms this and it is also found that in general there are only a finite number of successive period doubling generations finite jump first order transitions are shown to take place

naimark sacker bifurcations in linearly coupled quadratic maps abstract we report exact analytical expressions locating the 0 1 1 2 and 2 4 bifurcation curves for a prototypical system of two linearly coupled quadratic maps of interest is the precise location of the parameter sets where naimark sacker bifurcations occur starting from a non diagonal period 2 orbit this result is the key to understand the onset of synchronization in networks of quadratic maps

periodic behavior of cellular automata i propose an explanation of the observation of a globally synchronized behavior of deterministic cellular automata and coupled map lattices together with local fluctuations

desynchronization of chaos in coupled logistic maps when identical chaotic oscillators interact a state of complete or partial synchronization may be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state remains attracting on the average considering a system of two coupled logistic maps we show that the transition to riddling will be soft or hard depending on whether the first orbit to lose its transverse stability undergoes a supercritical or subcritical bifurcation a subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state a supercritical bifurcation on the other hand is associated with the formation of a so called mixed absorbing area that stretches along the synchronized chaotic state and from which trajectories cannot escape this gives rise to locally riddled basins of attraction we present three different scenarios for the onset of riddling and for the subsequent transformations of the basins of attraction each scenario is described by following the type and location of the relevant asynchronous cycles and determining their stable and unstable invariant manifolds one scenario involves a contact bifurcation between the boundary of the basin of attraction and the absorbing area another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle produced in the riddling bifurcation and ending in a boundary crisis where the stability of an asynchronous chaotic state is destroyed finally a phase diagram is presented to illustrate the parameter values at which the various transitions occur s1063 651x 99 04509 2

a bound on the exponent of primitivity in terms of diameter abstract an n n nonnegative matrix a is called primitive if for some positive integer k every entry in ak is positive ak 0 the exponent of primitivity of a is defined to be a min k z ak 0 where z denotes the set of positive integers two conjectures due to hartwig neumann and lin in 1984 or so are that a m 1 2 1 and a d2 1 where m is the degree of the minimal polynomial of a and d is the diameter of the directed graph of a it is well known that the latter is stronger than the former because d m 1 in a recent paper we have proved a m 1 2 1 in this paper we prove the conjecture a d2 1 on that basis

the emergence of coherent structures in coupled map lattices we show how increasing spatial interaction leads to the merging of coherent structures from chaos in some systems of coupled map lattices this phenomenon reflects the arising of new ground states in the corresponding model of statistical mechanics if we further increase the coupling then new ground states appear showing the coexistence of a large scale coherent structure with a small scale chaotic motion this allows us to propose a generalization of the notion of spatial intermittency

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

torus breakdown in noninvertible maps we propose a criterion for the destruction of a two dimensional torus through the formation of an infinite set of cusp points on the closed invariant curves defining the resonance torus this mechanism is specific to noninvertible maps the cusp points arise when the tangent to the torus at the point of intersection with the critical curve l 0 coincides with the eigendirection corresponding to vanishing eigenvalue for the noninvertible map further parameter changes lead typically to the generation of loops self intersections of the invariant manifolds followed by the transformation of the torus into a complex chaotic set

stability of wavelengths and spatiotemporal intermittency in coupled map lattices abstract in relation to spatiotemporal intermittency as it can be observed in coupled map lattices we study the stability of different wavelengths in competition introducing a two dimensional map we compare its dynamics with the one of the whole lattice we conclude that a good agreement between the two exists the reduced model also allows to introduce an order parameter which combines the diffusion parameter and the spatial wavelength under consideration

pattern dynamics in spatiotemporal chaos pattern selection diffusion of defect and pattern competition intermettency abstract a class of coupled map lattices is investigated as a model for the spatiotemporal chaos as the parameters are changed phase transitions among the patterns are observed pattern selection through the chaotic motion of domain is observed chaos is suppressed by the selection selection of the zigzag pattern is studied where localized defects which separate the two anti phase domains are found they change chaotically in time and move randomly in space the diffusion of defect is related with its chaotic motion collapse of zigzag pattern and the defect turbulence are studied in connection with the crisis in high dimensional dynamical systems pattern selection with longer wavelengths is found in the stronger coupling regime transition from pattern selection phase to fully developed turbulence occurs via the intermittency selective flicker like noise and the pareto distribution of lifetime of pattern are observed at the intermittency regime for the quantitative characterization of each phase pattern distributions pattern transition matrix static and dynamical entropies lifetime distribution of pattern spatiotemporal power spectra and lyapunov spectra are calculated phase transition from the pattern selection to fully developed turbulence is investigated with the use of these quantifiers where some order parameters and critical indeces are introduced

period multiplying bifurcations and multifurcations in conservative mappings the authors have investigated numerically and analytically the period doubling bifurcations and multifurcations of the periodic orbits of the conservative sine gordon mappings they have derived a general equation for the appearance of multifurcations in conservative mappings in agreement with many recent studies they also find evidence that such mappings possess universality properties they also discuss the role of multifurcations in conservative mappings exhibiting chaotic behaviour

theory of cones abstract this survey deals with the aspects of archimedian partially ordered finite dimensional real vector spaces and order preserving linear maps which do not involve spectral theory the first section sketches some of the background of entrywise nonnegative matrices and of systems of inequalities which motivate much of the current investigations the study of inequalities resulted in the definition of a polyhedral cone k and its face lattice f k in section ii a the face lattice of a not necessarily polyhedral cone k in a vector space v is investigated in particular the interplay between the lattice properties of f k and geometric properties of k is emphasized section ii b turns to the cones k in the space of linear maps on v recall that k is the cone of all order preserving linear maps of particular interest are the algebraic structure of k as a semiring and the nature of the group aut k of nonsingular elements a k for which a 1 k as well in a short final section the cone p n of n n positive semidefinite matrices is discussed a characterization of the set of completely positive linear maps is stated the proofs will appear in a forthcoming paper

maximal exponents of polyhedral cones i abstract let k be a proper i e closed pointed full convex cone in r n an n n matrix a is said to be k primitive if there exists a positive integer k such that a k k 0 int k the least such k is referred to as the exponent of a and is denoted by a for a polyhedral cone k the maximum value of a taken over all k primitive matrices a is called the exponent of k and is denoted by k it is proved that if k is an n dimensional polyhedral cone with m extreme rays then for any k primitive matrix a a m a 1 m 1 1 where m a denotes the degree of the minimal polynomial of a and the equality holds only if the digraph e p a k associated with a as a cone preserving map is equal to the unique up to isomorphism usual digraph associated with an m m primitive matrix whose exponent attains wielandt s classical sharp bound as a consequence for any n dimensional polyhedral cone k with m extreme rays k n 1 m 1 1 our work answers in the affirmative a conjecture posed by steve kirkland about an upper bound of k for a polyhedral cone k with a given number of extreme rays

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

linear equations over cones and collatz wielandt numbers let k be a proper cone in r n let a be an n x n real matrix that satisfies ak k let b be a given vector of k and let be a given positive real number the following two linear equations are considered in this paper i i n a x b x k and ii a l n x b x k we obtain several equivalent conditions for the solvability of the first equation for the second equation we give an equivalent condition for its solvability in case when b a and we also find a necessary condition when b a and also when b a sufficiently close to b a where b a denotes the local spectral radius of a at b with fixed we also consider the questions of when the set a l n k k equals 0 or k and what the face of k generated by the set is then we derive some new results about local spectral radii and collatz wielandt sets or numbers associated with a cone preserving map and extend a known characterization of m matrices among z matrices in terms of alternating sequences

a diameter bound on the exponent of a primitive directed graph abstract a directed graph g is primitive if there exists a positive integer k such that for every pair of vertices u v g there is a walk from u to v of length k the least such k is called the exponent of g we define g k to be the directed graph having the same vertex set as g and arcs u v if and only if there is a walk in g of length k from vertex u to vertex v a well known upper bound for the exponent of a primitive directed graph g of order n is n 1 2 1 due to h wielandt in 1950 our main result is the following refinement of the wielandt bound if g is a primitive directed graph with diameter d then the exponent of g is at most d 2 1 we construct the primitive graphs for which equality is attained and we generalize the bound to the class of irreducible matrices in the course of proving the main result we find the following which is interesting in its own right if g is a primitive directed graph with diameter d then the diameter of g k is at most d for all positive integers k

coupling induced statistical cycling in two diffusively coupled maps abstract this paper discusses the effects of diffusively coupling two identical one dimensional maps attention is focused on situations where the local isolated maps are statistically stable but where the coupled system is not a biologically motivated map and the quadratic map are numerically shown to display this behavior the piecewise linear tent map is then investigated analytically and we give a phase diagram of this system which displays the location of nonequilibrium phase transitions it is conjectured that the diffusive coupling of two chaotic but statistically stable maps i e with asymptotically stable perron frobenius operators can yield a two dimensional system which is not statistically stable whose associated perron frobenius operator is asymptotically periodic

proof of a conjecture about the exponent of primitive matrices abstract an n n nonnegative matrix a is called primitive if for some positive integer k every entry in the matrix ak is positive ak 0 the exponent of primitivity of a is defined to be a min k z ak 0 where z denotes the set of positive integers the upper bound on a due to wielandt is a n 1 2 1 and a better bound for a due to hartwig and neumann is a m m 1 where m is the degree of the minimal polynomial of a also hartwig and neumann conjecture that a m 1 2 1 which had been suggested in 1984 in this paper we prove this conjecture

loss of coherence in a system of globally coupled maps we study the formation of symmetric i e equally sized or nearly symmetric clusters in an ensemble of globally coupled identical chaotic maps it is shown that the loss of synchronization for the coherent state and the emergence of subgroups of oscillators with synchronized behavior are two distinct processes and that the type of behavior that arises after the loss of total synchronization depends sensitively on the dynamics of the individual map for our system of globally coupled logistic maps symmetric two cluster formation is found to proceed through a periodic state associated with the stabilization either of an asynchronous period 2 cycle or of an asynchronous period 4 cycle with further reduction of the coupling strength each of the principal clustering states undergoes additional bifurcations leading to cycles of higher periodicity or to quasiperiodic and chaotic dynamics if desynchronization of the coherent chaotic state occurs before the formation of stable clusters becomes possible high dimensional chaotic motion is observed in the intermediate parameter interval

on the invariant faces associated with a cone preserving map for an n n nonnegative matrix p an isomorphism is obtained between the lattice of initial subsets of 1 n for p and the lattice of p invariant faces of the nonnegative orthant r motivated by this isomorphism we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone preserving map on a polyhedral cone formulated in terms of its invariant faces in particular we obtain the following extension of the famous rothblum index theorem for a nonnegative matrix if a leaves invariant a polyhedral cone k then for each distinguished eigenvalue ofa fork there is a chain of m distinct a invariant join irreducible faces of k each containing in its relative interior a generalized eigenvector of a corresponding to referred to as semi distinguished a invariant faces associated with where m is the maximal order of distinguished generalized eigenvectors of a corresponding to but there is no such chain with more than m members we introduce the important new concepts of semi distinguished a invariant faces and of spectral pairs of faces associated with a cone preserving map and obtain several properties of a cone preserving map that mostly involve these two concepts when the underlying cone is polyhedral perfect or strictly convex and or smooth or is the cone of all real polynomials of degree not exceeding n that are nonnegative on a closed interval plentiful illustrative examples are provided some open problems are posed at the end 1991 mathematics subject classification primary 15a48 secondary 47b65 47a25 46b42

a route to computational chaos revisited noninvertibility and the breakup of an invariant circle abstract in a one parameter study of a noninvertible family of maps of the plane arising in the context of a numerical integration scheme lorenz studied a sequence of transitions from an attracting fixed point to computational chaos as part of the transition sequence he proposed the following as a possible scenario for the breakup of an invariant circle ic the ic develops regions of increasingly sharper curvature until at a critical parameter value it develops cusps beyond this parameter value the ic fails to persist and the system exhibits chaotic behavior on an invariant set with loops computational chaos a prelude to computational instability physica d 35 1989 299 we investigate this problem in more detail and show that the ic is actually destroyed in a global bifurcation before it has a chance to develop cusps instead the global unstable manifolds of saddle type periodic points are the objects which develop cusps and subsequently loops or antennae the one parameter study is better understood when embedded in the full two parameter space and viewed in the context of the two parameter arnold horn structure certain elements of the interplay of noninvertibility with this structure the associated ics periodic points and global bifurcations are examined

removable singularities in yang mills fields we show that a field satisfying the yang mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite we obtain the result that every yang mills field overr4 with bounded functional l2 norm may be obtained from a field ons4 r4 hodge or coulomb gauges are constructed for general small fields in arbitrary dimensions including 4

from mandelbrot like sets to arnold tongues a transition from mandelbrot like sets to arnold tongues is characterized via a coupling of two non identical quadratic maps proposed by us a two dimensional parameter space considering the parameters of the individual quadratic maps was used to demonstrate numerically the event the location of the parameter sets where naimark sacker bifurcations occur which is exactly the place where arnold tongues of arbitrary periods are born was computed analytically

bifurcation and chaos in a periodically stimulated cardiac oscillator abstract periodic stimulation of an aggregate of spontaneously beating cultured cardiac cells displays phase locking period doubling bifurcations and aperiodic chaotic dynamics at different values of the stimulation parameters this behavior is analyzed by considering an experimentally determined one dimensional poincare or first return map a simplified version of the experimentally determined poincare map is proposed and several features of the bifurcations of this map are described

chaotic sub dynamics in coupled logistic maps abstract we study the dynamics of laplacian type coupling induced by logistic family f x x 1 x where 0 4 on a periodic lattice that is the dynamics of maps of the form f x y 1 e f x e f y 1 e f y e f x where e 0 determines strength of coupling our main objective is to analyze the structure of attractors in such systems and especially detect invariant regions with nontrivial dynamics outside the diagonal in analytical way we detect some regions of parameters for which a horseshoe is present and using simulations global attractors and invariant sets are depicted

on the collatz wielandt sets associated with a cone preserving map abstract let k be a closed pointed full cone in r n in their treatment of perron frobenius theory for a linear map a preserving k by wielandt s approach barker and schneider introduced four sets namely 1 and 1 the collatz wielandt sets associated with a we determine the greatest lower bound and the least upper bound of these sets some known results for a nonnegative matrix relating its spectral radius to its upper and lower collatz wielandt number are extended to the setting of a cone preserving map applications of our results to the nonnegativity of solutions of linear inequalities associated with a nonnegative matrix are also considered

gauge theories on four dimensional riemannian manifolds this paper develops the riemannian geometry of classical gauge theories yang mills fields coupled with scalar and spinor fields on compact four dimensional manifolds some important properties of these fields are derived from elliptic theory regularity an energy gap theorem the manifold structure of the configuration space and a bound for the supremum of the field in terms of the energy it is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities this extends a theorem of k uhlenbeck

cross positive matrices revisited abstract for a closed pointed n dimensional convex cone k in r n let k denote the set of all n n real matrices a which as linear operators map k into itself let k denote the set of all n n matrices that are cross positive on k and l k k k the lineality space of k let r i the set of all real multiples of the n n identity matrix i then k k l k cl k k the final equality was proved in 1970 by schneider and vidyasagar who showed also that k k when k is polyhedral but not when k is a three dimensional circular cone they asked for a general characterization of those k for which the equality holds it is shown here that if n 3 and the cone k is strictly convex or smooth then k k hence for n 3 the equality fails for almost all k in the sense of baire category however the equality does hold for some nonpolyhedral k as was shown by a construction that appeared in the third author s 1977 dissertation and is explained here in more detail in 1994 it was shown by stern and wolkowicz that the weaker equality k l k k holds for all ellipsoidal as well as for all polyhedral k and they wondered whether this equality holds for all k however their equality certainly fails for all strictly convex or smooth k such that l k and it is shown here that this also includes almost all k when n 3

a note on the eigenvalues of a primitive matrix with large exponent abstract let a be a primitive stochastic matrix of order n 7 and exponent at least n 1 2 1 2 2 we describe the general form of the characteristic polynomial of a and prove that a must have at least 2 n 4 4 complex eigenvalues of modulus at greater than 1 2 sin n 1 2 n 1 observe that this last quantity tends to 1 as n both combinatorial and algebraic arguments are used to establish the result

coexisting tori and torus bubbling in non smooth systems abstract pulse modulated power electronic converters represent an important class of piecewise smooth dynamical systems with a broad range of applications in modern power supply systems the paper presents a detailed investigation of a number of unusual bifurcation phenomena that can occur in power converters with multilevel control in the first example a closed invariant curve arises in a border collision bifurcation as a period 6 saddle cycle collides with a stable fixed point of focus type and transforms it into an unstable focus point the second example involves the formation of a structure of coexisting tori through the interplay between border collision and global bifurcations we examine the behavior of the system in the presence of two coexisting stable resonance tori and finally show how an existing torus can develop heteroclinic bubbles that connect the points of a stable resonance cycle with an external pair of saddle and focus cycles the appearance of these structures is explained in terms of a sequence torus birth bifurcations with pairs of stable and unstable tori folding one over the other

a condition for an area preserving mapping to be in the engel s form we establish explicit expressions of restrictions on the coefficients of nonlinear terms in a two dimensional area preserving polynomial map imposed by the property of area preserving we also establish a necessary and sufficient condition for a two dimensional area preserving generic polynomial map to be in the engel s form the condition is that coefficients of nonlinear terms in the first mapping equation are proportional to the corresponding ones in the second mapping equation as an application of the results we discuss how to regain the area preserving property lost in truncating a two dimensional area preserving map

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

transverse instability and riddled basins in a system of two coupled logistic maps riddled basins denote a characteristic type of fractal domain of attraction that can arise when a chaotic motion is restricted to an invariant subspace of total phase space an example is the synchronized motion of two identical chaotic oscillators the paper examines the conditions for the appearance of such basins for a system of two symmetrically coupled logistic maps we determine the regions in parameter plane where the transverse lyapunov exponent is negative the bifurcation curves for the transverse destabilization of low periodic orbits embedded in the chaotic attractor are obtained and we follow the changes in the attractor and its basin of attraction when scanning across the riddling and blowout bifurcations it is shown that the appearance of transversely unstable orbits does not necessarily lead to an observable basin riddling and that the loss of weak stability when the transverse lyapunov exponent becomes positive does not necessarily destroy the basin of attraction instead the symmetry of the synchronized state may break and the attractor may spread into two dimensional phase space

statistical cycling in coupled map lattices this work describes a class of phase transitions observed in large coupled map lattices cml s with chaotic local maps the phase transitions are seen to occur 1 when the coupling is fixed as other parameters are changed and 2 when the coupling is changed with other parameters fixed the different phases of the lattices of tent and quadratic maps are interpreted as reflecting different spectral properties of the perron frobenius operator for the cml s the spectral characteristics of this operator induced by piecewise linear cml s are investigated analytically using general results from the theory of linear operators sufficient conditions for the cyclical evolution of phase space densities are given this implies that the cml s under consideration can reach equilibrium states in which ensemble statistics are not time independent stationary but time periodic

period doubling in six dimensional symmetric volume preserving maps period doubling in three symmetrically coupled two dimensional area preserving maps is numerically studied it is found that there is a two dimensional manifold on which all period 2n n 0 1 2 orbits of a period doubling bifurcation sequence lie on this manifold the universality classes of period doubling are just the three classes for two symmetrically coupled two dimensional area preserving maps each characterised by its own feigenbaum constants reported by mao and helleman phys rev a vol 35 p 1847 1987 it is also reported that for three non symmetrically coupled two dimensional area preserving maps three maps which are fixed under the renormalisation operator have been found the relevant eigenvalues of perturbation around the fixed maps are again the same as those found for two non symmetrically coupled two dimensional area preserving maps by mao and greene phys rev a vol 35 p 3911 1987 the above mentioned numerical and renormalisation results for the six dimensional maps agree with each other

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

statistical approach to nonhyperbolic chaotic systems spectral properties of the evolution operator for probability densities are obtained for unimodal maps for which all periodic orbits are unstable and the lyapunov exponent calculated from the first iterate of the critical point converges to a positive constant the method is applied to the logistic map both for parameter values at which finite markov partitions can be found as well as for more typical parameter values a universal behavior is found for the spectral gap in the period doubling inverse cascade of chaotic band merging bifurcations full agreement with numerical simulation is obtained textcopyright 1996 the american physical society

graphs for cone preserving maps abstract let k be a closed pointed full cone in a finite dimensional real vector space we associate with a linear map a for which ak k four directed graphs for two of the graphs the vertex set is the collection of all faces of k and for two the vertices are all the extreme rays of k we relate the irreducibility and primitivity of a to the strong connectedness of some of these graphs

maximal exponents of polyhedral cones ii abstract let k be a proper i e closed pointed full convex cone in r n an n n matrix a is said to be k primitive if there exists a positive integer k such that a k k 0 int k the least such k is referred to as the exponent of a and is denoted by a for a polyhedral cone k the maximum value of a taken over all k primitive matrices a is called the exponent of k and is denoted by k it is proved that the maximum value of k as k runs through all n dimensional minimal cones i e cones having n 1 extreme rays is n 2 n 1 if n is odd and is n 2 n if n is even the maximum value of the exponent being attained by a minimal cone with a balanced relation for its extreme vectors the k primitive matrices a such that a attain the maximum value are identified up to cone equivalence modulo positive scalar multiplication

dynamics of the two frequency torus breakdown in the driven double scroll circuit in this work we numerically identify three scenarios for the two frequency torus breakdown to chaos using the driven double scroll circuit with varying driving parameters two of these scenarios follow the curryyorke route to chaos for one scenario we identify the transition to chaos through the onset of a heteroclinic tangle and its heteroclinic points in the other scenario chaos appears via period doubling bifurcations the third scenario is through the type ii intermittency for which a quasiperiodic torus grows in size breaks by touching external saddle points and forms a heteroclinic saddle connection these dynamic scenarios have distinct structure evolutions for the curry yorke route chaos appears softly and alternates with phase locking while for the type ii intermittency chaos appears abruptly and is preserved for a large range of the varying driving parameter s1063 651x 98 05210 6

novel routes to chaos through torus breakdown in non invertible maps the paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non invertible maps by means of detailed numerically calculated phase portraits we first describe how three and five layered tori arise through period doubling and or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus we then describe several different mechanisms for the destruction of five layered tori in a system of two linearly coupled logistic maps one of these scenarios involves the destruction of the two intermediate layers of the five layered torus through the transformation of two unstable node cycles into unstable focus cycles followed by a saddle node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set other scenarios involve different combinations of local and global bifurcations including bifurcations that lead to various forms of homoclinic and heteroclinic tangles we finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two dimensional non invertible maps that have previously been used to study the properties of invariant sets

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

correlations and spectra of periodic chaos generated by the logistic parabola we consider the one parameter family of mappingsfa x 4ax 1 x a x e 0 1 and define an infinite countable set of parameter values a for which the solutions show observable chaos their properties are investigated by means of correlation functions and spectra which can be interpreted and approximated by separating periodic and chaotic components in the solutions and introducing two simple assumptions on the statistics of the chaotic component

transitions from phase locked dynamics to chaos in a piecewise linear map recent work has shown that torus formation in piecewise smooth maps can take place through a special type of border collision bifurcation in which a pair of complex conjugate multipliers for a stable cycle abruptly jump out of the unit circle transitions from an ergodic to a resonant torus take place via border collision fold bifurcations we examine the transition to chaos through torus destruction in such maps considering a piecewise linear normal form map we show that this transition by virtue of the interplay of border collision bifurcations with period doubling and homoclinic bifurcations can involve mechanisms that differ qualitatively from those described by afraimovich and shilnikov

on a coupled logistic map with large strength abstract in this paper a class of diffusive coupled logistic map with large strength is considered depending on the parameter of the logistic family and the strength of coupling we show the existence of a forward invariant curve which determines in some sense the boundary of the set whose points have unbounded orbit

existence and computation of short run equilibria in economic geography the new economic geography literature provides a general equilibrium framework that explains the emergence of economic agglomerations as a trade off between increasing returns at the firm level and transportation costs related to the shipment of goods the existence and uniqueness of short run equilibria of this model has been shown for the case of two regions the proposed approach employs the differential evolution algorithm to obtain estimates of the lipschitz constant and the infinity norm of the function along the boundary and utilizes these values to investigate the existence of solutions of a function and the computational burden of computing the topological degree of this function this approach is employed to investigate the existence of short run equilibria for more than two regions using fixed point and topological degree theory as well as the differential evolution algorithm irrespective of parameter settings the criteria from topological degree theory suggest that the model can have equilibria the differential evolution algorithm identified such equilibria and for none of the parameter settings that were considered more than one equilibria were detected the experimental results obtained also indicate that the computation of such equilibria has an exponential worst case lower bound complexity as the model yields a function that is neither contractive nor nonexpanding finally the computation of the topological degree to identify the number of equilibria also has a very high computational cost

analytic study of chaos of the tent map band structures power spectra and critical behaviors chaotic behaviors of the tent map a piecewise linear continuous map with a unique maximum are studied analytically throughout its chaotic region in terms of the invariant density and the power spectrum as the height of the maximum is lowered successive band splitting transitions occur in the chaotic region and accumulate to the transition point into the nonchaotic region the timecorrelation function of nonperiodic orbits and their power spectrum are calculated exactly at the band splitting points and in the vicinity of these points the method of eigenvalue problems of the frobenius perron operator is used 2m 1 critical modes wherem 1 2 3 are found which exhibit the critical slowing down near the 2m 1 band to 2m band transition point after the transition these modes become periodic modes which represent the cycling of nonperiodic orbits among 2m bands together with the periodic modes generated by the preceding band splittings scaling laws near the transition point into the nonchaotic region are investigated and a new scaling law is found for the total intensity of the periodic part of the spectrum

global bifurcation destroying the experimental torus t 2 we show experimentally the scenario of a two frequency torus t 2 breakdown in which a global bifurcation occurs due to the collision of a torus with an unstable periodic orbit creating a heteroclinic saddle connection followed by an intermittent behavior

phase transitions in networks of chaotic elements with short and long range interactions abstract this paper investigates the statistical properties of networks of chaotic elements modeled by coupled map lattices transitions separating statistically stable and periodic phases are numerically observed in generic models of excitable media similar transitions are studied analytically in lattices of piecewise expanding maps by considering the spectral properties of the perron frobenius operator using the theory of functions of bounded variation in r n