explicit series solutions to nonlinear evolution equations the sine cosine method in this paper a sine cosine method is used to construct many periodic and solitary wave solutions to two nonlinear evolution systems the coupled quadratic nonlinear equations and the coupled klein gordon schrodinger equations under different parameter conditions explicit formulas for some new periodic and solitary wave solutions are successfully obtained the proposed solutions are found to be important for the explanation of some practical physical problems

intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators a theory of intrinsic fluctuations is developed of a phase ordering parameter for large populations of weakly and uniformly coupled limit cycle oscillators with distributed native frequencies in particular it is shown that the intensity as well as the correlation time of fluctuations exhibit power law divergence at the onset of mutual entrainment with critical exponents which depend on whether the coupling strength approaches the threshold from below or above this peculiar feature is demonstrated by numerical simulations mainly through finite size scaling analyses in the course of exploring its origin we encounter a new concept termed a correlation frequency which provides a natural interpretation of the finite size scaling laws a comment is given on a recent theory by kuramoto and nishikawa to clarify why it contradicts our results

chaos in the hamiltonian mean field model rst show that the largest lyapunov exponent remains strictly positive in the innite size limit converging to its asymptotic value with 1 lnn corrections we then elucidate the scaling laws ruling the behavior of this asymptotic value in the critical region separating the ordered clustered phase and the disordered phase present at high energy densities we also show that the full spectrum of lyapunov exponents consists of a bulk component converging to the zero value taken by a test oscillator forced by the mean eld plus subextensive bands of o lnn exponents taking nite values we nally investigate the robustness of these results by studying a 2d extension of the hmf model where each particle is endowed with 4 degrees of freedom thus allowing the emergence of chaos at the level of single particle altogether these results illustrate the subtle eects of global or long range coupling and the importance of the order in which the innite time and innite size limits are taken for an innite size hmf system represented by the vlasov equation no chaos is present while chaos exists and subsists for any nite system size

scaling of lyapunov exponents of coupled chaotic systems we develop a statistical theory of the coupling sensitivity of chaos the effect was first described by daido prog theor phys 72 853 1984 it appears as a logarithmic singularity in the lyapunov exponent in coupled chaotic systems at very small couplings using a continuous time stochastic model for the coupled systems we derive a scaling relation for the largest lyapunov exponent the singularity is shown to depend on the coupling and the systems mismatch generalizations to the cases of asymmetrical coupling and three interacting oscillators are considered too the analytical results are confirmed by numerical simulations

a function transformation method and exact solutions to a generalized sinh gordon equation based on a transformed painleve property and the variable separated ode method a function transformation method is proposed to search exact solutions to some partial differential equations pdes with hyperbolic or exponential functions the new approach provides a more systematical and convenient handling of the solution process for the nonlinear equations its key point is to eradicate the hyperbolic or exponential terms by a transformed painleve property and reduce the given pdes to a variable coefficient ordinary differential equations then we look for solutions to the resulting equations by some methods as an application exact solutions for a generalized sinh gordon equation are formally derived

clustered motion in symplectic coupled map systems clustered motion of particles is found in hamiltonian dynamics of symplectic coupled map systems particles assemble and move with strong correlation the motion is chaotic but is distinguishable from random chaotic motion lyapunov analysis distinguishes global instability from local fluctuations clustered motions have finite lifetime they have fractal geometric structure in the phase space as the orbits are trapped to ruins of kam tori and islands

special solutions from the variable separation approach the davey stewartson equation a variable separation procedure for the davey stewartson ds equation is proposed by using a prior ansatz to its bilinear form the reduced equations for two variable separated fields have the same trilinear form although they possess different independent variables the trilinear equation can be changed to a spacetime symmetric form and can be solved by means of a boussinesq type equation system whenever a pair of solutions of the reduced fields are obtained a corresponding solution of the ds equation can be obtained algebraically the single dromion solution and some kinds of positon solutions are obtained explicitly

some special types of solutions of a class of the n 1 dimensional nonlinear wave equations some special types of solutions with arbitrary functions for the n 1 dimensional sine cosine gordon equation the n 1 dimensional double sinh gordon equation and the n 1 dimensional sinh cosinh gordon equations are constructed by means of the separation transformation approach as an illustrative sample the properties of some solutions are shown by their figures

sinh gordon equation painleve property and backlund transformation abstract the painleve property for partial differential equation proposed by weiss et al is studied for the sinh gordon equation we performed a singular point analysis to show that the painleve property is a valuable analytical test for integrability and we found in a remarkably straightforward manner the backlund transformation of the sinh gordon equation

on characterizations of the class pspace poly abstract we present several characterizations of the class pspace poly these characterizations are given in terms of pspace complete problems certain algebraic closure operations vectorial straight line programs and kolmogorov complexity to end we characterize the problems having an exponential lower bound on their nonuniform complexity measured by vectorial straight line programs

on small generators abstract yap 1983 shows that each set having small generators is in the class np poly which was introduced by karp and lipton 1980 we show here that the converse is also true i e each set in np poly has small generators this settles a question left open by yap 1983 further an argument by meyer stated in berman and hartmanis 1977 is generalized to show that a set has small generators if and only if for some sparse set s a np s

exact travelling wave solutions for the n 1 dimensional double sine and sinh gordon equations for the n 1 dimensional double sine and sinh gordon equations by using the approach of dynamical systems to the travelling wave solutions in different regions of the parameter space all possible bounded solutions solitary wave solutions kink and anti kink wave solutions and periodic wave solutions et al are obtained explicit exact parametric representations are given

bifurcation studies on travelling wave solutions for nonlinear intensity klein gordon equation abstract in this paper we study the travelling waves for the n 1 dimensional klein gordon equation with nonlinear intensity by using the bifurcation theory and methods of planar dynamic systems with the aid of maple we obtain all bifurcations and phase portraits in the parametric space moreover we explain the reason for appearance of nonsmooth travelling waves and for persistence of smoothness of smooth travelling waves and show the existence of smooth and nonsmooth solitary and periodic waves under different regions of parametric space various sufficient conditions to guarantee the existence of the above solutions are given

logarithmic advice classes abstract karp and lipton 1980 introduced the notion of nonuniform complexity classes where a certain amount of additional information the advice is given for free the advice only depends on the length of the input karp and lipton initiated the study of classes with either logarithmic or polynomial advice however later yap 1983 schoning 1984 balcazar 1987 and ko and schoning 1985 concentrated on the study of classes of the form c poly where c is p np or pspace and poly denotes a polynomial size advice this paper considers classes of the form c log as a main result it is shown that in the context of an np log computation log bounded advice is equivalent to a sparse oracle in np in contrast it has been shown that a poly bounded advice to an arbitrary sparse oracle set furthermore a general theorem is presented that generalizes karp and lipton s round robin tournament method

scaling law for the maximum lyapunov characteristic exponent of infinite product of random matrices the authors present a simple derivation for the scaling behaviour of the maximum lyapunov characteristic exponent lambda of infinite product of symplectic random matrices the considered random matrices depend on a parameter epsilon and lambda 0 for epsilon 0 they obtain lambda varies as epsilon beta with either beta 1 2 or beta 2 3 depending on the probability distribution of the matrix elements the results are in agreement with a previous numerical simulation

statistical mechanical formulation of lyapunov exponents we show how the lyapunov exponents of a dynamic system can in general be expressed in terms of the free energy of a non hermitian quantum many body problem this puts their study as a problem of statistical mechanics whose intuitive concepts and techniques of approximation can hence be borrowed

traveling wave solutions to the n 1 dimensional sinh cosh gordon equation traveling wave solutions for a generalized sinh cosh gordon equation are studied the equation is transformed into an auxiliary partial differential equation without any hyperbolic functions by using the theory of planar dynamical system the existence of different kinds of traveling wave solutions of the auxiliary equation is obtained including smooth solitary wave periodic wave kink and antikink wave solutions some explicit expressions of the blow up solution kink like solution antikink like solution and periodic wave solution to the generalized sinh cosh gordon equation are given planar portraits of the solutions are shown

entropy complexity analysis in some globally coupled systems globally coupled n body systems are well known to possess an intricate dynamics when n is large collective effects may drastically lower the effective dimension of the dynamics breaking the conditions on ergodicity necessary for the applicability of statistical mechanics these problems are here illustrated and discussed through an entropy complexity analysis of the repulsive hamiltonian mean field model using a poincare section of the mean field time series provides a natural sampling time in the entropy complexity treatment this approach is shown to single out the out of equilibrium dynamical features and to uncover a transition of the system dynamics from low energy non boltzmann quasi stationary states to high energy stochastic like behavior

on similarity and duality of computation i this paper describes what is the parallel time for sequential models what is the sequential time for parallel models and proves that the well known computational models ram vector machines turing machines uniform circuits uniform aggregates storage modification machines hardware modification machines are all similar in the sense that their parallel time their sequential time and their space complexities are polynomially related simultaneously

n tree approximation for the largest lyapunov exponent of a coupled map lattice the n tree approximation scheme introduced in the context of random directed polymers is applied here to the computation of the maximum lyapunov exponent in a coupled map lattice we discuss both an exact implementation for small tree depth n and a numerical implementation for larger n we find that the phase transition predicted by the mean field approach shifts towards larger values of the coupling parameter when the depth n is increased we conjecture that the transition eventually disappears

characterizing dynamics with covariant lyapunov vectors a general method to determine covariant lyapunov vectors in both discrete and continuous time dynamical systems is introduced this allows us to address fundamental questions such as the degree of hyperbolicity which can be quantified in terms of the transversality of these intrinsic vectors for spatially extended systems the covariant lyapunov vectors have localization properties and spatial fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the lyapunov exponents

the computational power of discrete hopfield nets with hidden units we prove that polynomial size discrete hopfield networks with hidden units compute exactly the class of boolean functions pspace poly i e the same functions as are computed by polynomial space bounded nonuniform turing machines as a corollary to the construction we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions p poly i e the class computed by polynomial time bounded nonuniform turing machines

the variable separated ode and the tanh methods for solving the combined and the double combined sinh cosh gordon equations abstract exact travelling wave solutions for the combined sinh cosh gordon and the double combined sinh cosh gordon equation are formally derived the painleve property v eu will be used to back up the tanh method to derive a first set of solutions the newly developed variable separated ode method is also used to derive a second set of solutions each method provides travelling wave solutions of distinct physical structures

hyperbolic decoupling of tangent space and effective dimension of dissipative systems we show using covariant lyapunov vectors that the tangent space of spatially extended dissipative systems is split into two hyperbolically decoupled subspaces one comprising a finite number of frequently entangled physical modes which carry the physically relevant information of the trajectory and a residual set of strongly decaying spurious modes the decoupling of the physical and spurious subspaces is defined by the absence of tangencies between them and found to take place generally we find evidence in partial differential equations in one and two spatial dimensions and even in lattices of coupled maps or oscillators we conjecture that the physical modes may constitute a local linear description of the inertial manifold at any point in the global attractor

the tanh method exact solutions of the sine gordon and the sinh gordon equations in this work we use the tanh method for traveling wave solutions of the sine gordon and the sinh gordon equations several exact solutions of distinct physical structures are obtained the method is powerful with minimal algebra work and is demonstrated for four models

on the numerical solution of the sine gordon equation in this the first of two papers on the numerical solution of the sine gordon equation we investigate the numerical behavior of a double discrete completely integrable discretization of the sine gordon equation for certain initial values in the vicinity of homoclinic manifolds this discretization admits an instability in the form of grid scale oscillations we clarify the nature of the instability through an analytical investigation supported by numerical experiments in particular a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold this paves the way for the second paper where we use the non linear spectrum as a basis for comparing different numerical schemes

discrete singular convolution for the sine gordon equation abstract this paper explores the utility of a discrete singular convolution dsc algorithm for the integration of the sine gordon equation the initial values are chosen close to a homoclinic manifold for which previous methods have encountered significant numerical difficulties such as numerically induced spatial and temporal chaos a number of new initial values are considered including a case where the initial value is exactly on the homoclinic orbit the present algorithm performs extremely well in terms of accuracy efficiency simplicity stability and reliability

chaos and relaxation to equilibrium in systems with long range interactions in the thermodynamic limit systems with long range interactions do not relax to equilibrium but become trapped in nonequilibrium stationary states for a finite number of particles a nonequilibrium state has a finite lifetime so that eventually a system will relax to thermodynamic equilibrium the time that a system remains trapped in a quasistationary state qss scales with the number of particles as n with 0 and diverges in the thermodynamic limit in this paper we will explore the role of chaotic dynamics on the time that a system remains trapped in a qss we discover that chaos measured by the lyapunov exponents favors faster relaxation to equilibrium surprisingly weak chaos favors faster relaxation than strong chaos

scaling law for the maximal lyapunov exponent the authors study the scaling law for epsilon to 0 of the maximal lyapunov exponent for coupled chaotic map lattices and for products of random jacobi matrices to this purpose they develop approximate analytical treatments of the random matrix problem inspired by the theory of directed polymers in a random medium a type of mean field method and a tree approximation which introduces correlations the theoretical results suggest a leading mod log epsilon mod 1 increase in the maximal lyapunov exponent near epsilon 0 which is confirmed by numerical simulations also for coupled map lattices a dynamical mechanism responsible for this behaviour is investigated for a 2 2 random matrix model the theory also predicts a phase transition at a critical value of the coupling epsilon e which is not observed in numerical simulations and might be an artifact of the approximation

new travelling wave solutions to some nonlinear equations via a combined method abstract based on the sine gordon and the sinh gordon equations transformations are introduced in a variable separated ode method to construct entirely new doubly periodic jacobian elliptic function solutions of some nonlinear differential equations which are powerful and simple distinct sets of exact doubly periodic solutions that possess distinct physical structures are obtained for the mkdv sine gordon the mkdv sinh gordon equations and born infeld equation when the module m 1 or 0 the corresponding solitary wave solutions are also found this approach can be also applied to solve other nonlinear differential equations

new exact solutions to sinh cosh gordon equation by using techniques based on projective riccati equations in this paper we present some methods for solving nonlinear partial differential equations which are based on the idea of the projective riccati equations we show how to derive well known methods such as conte s projective riccati equation method tanh coth method he s exp function method and a new method we called sn ns method we illustrate the effectiveness of the sn ns method for the problem of finding new exact solutions to the combined sinh cosh gordon equation with the aid of maple 14 and mathematica 7

travelling wave solutions for the mkdv sine gordon and the mkdv sinh gordon equations by using a variable separated ode method in this work we formally derive exact travelling wave solutions for the modified kdv sine gordon mkdv sine gordon equation and the modified kdv sinh gordon equation mkdv sinh gordon the proposed analysis depends mainly on a variable separated ode method two distinct sets of exact solitary wave solutions that possess distinct physical structures are formally derived for each equation the derived solutions include others results and introduce entirely new solutions

dynamical survey of a generalized zakharov equation and its exact travelling wave solutions we investigate the dynamical behavior of a generalized zakharov equation for the complex envelope of the high frequency wave and the real low frequency field by analyzing its phase portraits following a dynamical system approach in different parameter regions we depict phase portraits of a travelling wave system through discussing the bifurcation of phase portraits we unwrap explicit miscellaneous travelling waves including localized and periodic ones

exact solutions to the double sinh gordon equation by the tanh method and a variable separated ode method new exact travelling wave solutions for the double sinh gordon equation and its generalized form are formally derived by using the tanh method and the variable separated ode method the painleve property v e u is employed to support the tanh method in deriving exact solutions the work emphasizes the power of the methods in providing distinct solutions of different physical structures

scaling laws for the largest lyapunov exponent in long range systems a random matrix approach we investigate the laws that rule the behavior of the largest lyapunov exponent lle in many particle systems with long range interactions we consider as a representative system the so called hamiltonian ensuremath alpha ensuremath mathrm xy model where the adjustable parameter ensuremath alpha controls the range of the interactions of n ferromagnetic spins in a lattice of dimension d in previous work the dependence of the lle with the system size n for sufficiently high energies was established through numerical simulations in the thermodynamic limit the lle becomes constant for ensuremath alpha gd whereas it decays as an inverse power law of n for ensuremath alpha ld a recent theoretical calculation based on a geometrization of the dynamics is consistent with these numerical results here we show that the scaling behavior can also be explained by a random matrix approach in which the tangent mappings that define the lyapunov exponents are modeled by random simplectic matrices drawn from a suitable ensemble

on the notion of infinite pseudorandom sequences abstract three definitions of infinite pseudorandom sequences with respect to polynomial time and space complexity are introduced and compared with each other it is shown that the first two definitions based on martin lof s notion of sequential tests and levin and schnorr s notion of monotonic operator complexity are equivalent with respect to polynomial space complexity while both are strictly stronger than the third definition which is derived from von mises s notion of collectives

analytic estimation of the lyapunov exponent in a mean field model undergoing a phase transition the parametric instability contribution to the largest lyapunov exponent lle is derived for a mean field hamiltonian model with attractive long range interactions this uses a recent riemannian approach to describe hamiltonian chaos with a large number n of degrees of freedom through microcanonical estimates of suitable geometrical observables the mean field behavior of the lle is analytically computed and related to the second order phase transition undergone by the system it predicts that chaoticity drops to zero at the critical temperature and remains vanishing above it with the lle scaling as n 1 3 to the leading order in n

bifurcations of travelling wave solutions for the generalized double sinh gordon equation in this paper the generalized double sinh gordon equation is studied the existence of periodic wave solitary wave kink and anti kink wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given some exact explicit parametric representations of the above waves are determined

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations

a type of bounded traveling wave solutions for the fornberg whitham equation abstract in this paper by using bifurcation method we successfully find the fornberg whitham equation u t u x x t u x u u x x x u u x 3 u x u x x has a type of traveling wave solutions called kink like wave solutions and antikink like wave solutions they are defined on some semifinal bounded domains and possess properties of kink waves and anti kink waves their implicit expressions are obtained for some concrete data the graphs of the implicit functions are displayed and the numerical simulation is made the results show that our theoretical analysis agrees with the numerical simulation