two novel methods and multi mode periodic solutions for the fermi pasta ulam model we introduce two novel methods for studying periodic solutions of the fpu model both numerically and rigorously one is a variational approach based on the dual formulation of the problem and the other involves computer assisted proofs these methods are used e g to construct a new type of solutions whose energy is spread among several modes associated with closely spaced resonances

stability of low dimensional bushes of vibrational modes in the fermi pasta ulam chains bushes of normal modes represent the exact excitations in the nonlinear physical systems with discrete symmetries physica d 117 1998 43 the present paper is the continuation of our previous paper physica d 166 2002 208 where these dynamical objects of new type were discussed for the monoatomic nonlinear chains here we develop a simple crystallographic method for finding bushes in nonlinear chains and investigate stability of one and two dimensional vibrational bushes for both fpu and fpu models in particular of those revealed recently in physica d 175 2003 31

tangent bifurcation of band edge plane waves dynamical symmetry breaking and vibrational localization abstract we study tangent bifurcation of band edge plane waves in nonlinear hamiltonian lattices the lattice is translationally invariant we argue for the breaking of permutational symmetry by the new bifurcated periodic orbits the case of two coupled oscillators is considered as an example for the perturbation analysis where the symmetry breaking can be traced using poincare maps next we consider a lattice and derive the dependence of the bifurcation energy on the parameters of the hamiltonian function in the limit of large system sizes a necessary condition for the occurence of the bifurcation is the repelling of the band edge plane wave from the linear spectrum with increasing energy we conclude that the bifurcated orbits will consequently exponentially localize in the configurational space

interactions between normal modes in nonlinear dynamical systems with discrete symmetry exact results abstract we discuss in detail the concept of bush of modes introduced by us earlier for classical nonlinear systems with discrete point or space symmetry each bush comprises all modes singled out by the symmetry group of an initial excitation and may be considered as a geometrical and dynamical object we prove theorems that describe structure and some properties of bushes for hamiltonian and for a wide class of non hamiltonian systems theorems 2 a and b permit one to introduce new variables nonlinearly connected with normal modes which in a sense are independent of each other in case they are associated with different irreducible representations of the symmetry group of a system in equilibrium such independence provides a possibility of singling out specific dynamical regimes of an essentially lesser dimensionality since many different bushes are described by the same differential equations they may be classified by certain classes of universality possible physical applications of vibrational bushes are suggested

exact static solutions of a generalized discrete φ 4 model including short periodic solutions we carry out a comprehensive analysis of a generalized discrete 4 model of which virtually all 4 models discussed in the literature are particular cases for this model we construct the exact solutions in the form of the basic jacobi elliptic hyperbolic and sine functions and also give a list of short periodic and even aperiodic solutions some of those solutions coincide with the known ones others generalize the existing solutions and the rest of them are new we then discuss the relation between the models supporting exact static solutions and the two point maps in particular we show that some of the short periodic and sine solutions can be found from factorized difference equations and even from a set of two difference equations one of the first and another of the second order particular attention is paid to the discussion of the exceptional discrete ed models defined as models supporting the translationally invariant ti static solutions that can be placed arbitrarily with respect to the lattice we show that some of the derived short periodic solutions are ti ones while the others are not for the ti static solutions we demonstrate the existence of the translational goldstone mode for any location of the solution with respect to the lattice we then analyze numerically the stability and other properties of the ti kink solutions in conclusion we divide the ed models into two classes the ed i models support a two parameter set of ti static solutions while the ed ii models support only a one parameter set of such solutions

rectangular kronecker coefficients and plethysms in geometric complexity theory abstract we prove that in the geometric complexity theory program the vanishing of rectangular kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial moreover we prove the positivity of rectangular kronecker coefficients for a large class of partitions where the side lengths of the rectangle are at least quadratic in the length of the partition we also compare rectangular kronecker coefficients with their corresponding plethysm coefficients which leads to a new lower bound for rectangular kronecker coefficients moreover we prove that the saturation of the rectangular kronecker semigroup is trivial we show that the rectangular kronecker positivity stretching factor is 2 for a long first row and we completely classify the positivity of rectangular limit kronecker coefficients that were introduced by manivel in 2011

three dimensional chaotic flows with discrete symmetries there are a number of well known three dimensional flows with quadratic nonlinearities which demonstrate a chaotic behavior the most popular among them are lorenz and rossler systems using an exhaustive computer search j sprott found 19 examples of chaotic flows with either five terms and two quadratic nonlinearities or six terms and one nonlinearity phys rev e 50 r647 1994 in contrast to this approach we use symmetry related considerations to construct types of chaotic flows with an arbitrary dimension the discussion is based on our previous work devoted to nonlinear dynamics of the physical systems with discrete symmetries see physica d 117 43 1998 etc here we present all possible chaotic flows with quadratic nonlinearities which are invariant under the action of 32 point groups of crystallographic symmetry these systems demonstrate a typical chaotic behavior as well as general dynamical properties of nonlinear systems with discrete symmetries in particular we found a dynamical system with the point symmetry group d 2 which seems to be more simple and more elegant than those by lorenz and rossler

bushes of vibrational modes for fermi pasta ulam chains abstract some exact solutions and multimode invariant submanifolds were found for the fermi pasta ulam fpu model by poggi and ruffo physica d 103 1997 251 in the present paper we demonstrate how results of such a type can be obtained for an arbitrary n particle chain with periodic boundary conditions with the aid of our group theoretical approach physica d 117 1998 43 based on the concept of bushes of normal modes in mechanical systems with discrete symmetry the integro differential equation describing the fpu dynamics in the modal space is derived the loss of stability of the bushes of modes for the fpu model in particular for the limiting case n for the dynamical regime with displacement pattern having period twice the lattice spacing mode is studied our results for the fpu chain are compared with those by poggi and ruffo for the fpu chain

large deviations from classical paths hamiltonian flows as classical limits of quantum flows we prove that in the limit 0 the probability for the paths of the stochastic jump process associated to the quantum time evolution to be in a tublet around the classical trajectory is of order 1 exp a we give some applications of this result to the study of the classical limit of wigner functions

symmetric invariant manifolds in the fermi pasta ulam lattice abstract the fermi pasta ulam fpu lattice with periodic boundary conditions and n particles admits a large group of discrete symmetries the fixed point sets of these symmetries naturally form invariant symplectic manifolds that are investigated in this short note for each k dividing n we find k degree of freedom invariant manifolds they represent short wavelength solutions composed of k fourier modes and can be interpreted as embedded lattices with periodic boundary conditions and only k particles inside these invariant manifolds other invariant structures and exact solutions are found which represent for instance periodic and quasi periodic solutions and standing and travelling waves some of these results have been found previously by other authors via a study of mode coupling coefficients and recently also by investigating bushes of normal modes the method of this paper is similar to the latter method and much more systematic than the former we arrive at previously unknown results without any difficult computations it is shown moreover that similar invariant manifolds exist also in the klein gordon kg lattice and in the thermodynamic and continuum limits

computers and group theoretical methods for studying structural phase transitions abstract the present work is a review of a complex of programs developed at rostov university for a group theoretical analysis of phase transitions in crystals a comparative analysis of various approaches and results from different authors is also presented

q breathers in fermi pasta ulam chains existence localization and stability the fermi pasta ulam fpu problem consists of the nonequipartition of energy among normal modes of a weakly anharmonic atomic chain model in the harmonic limit each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q we continue normal modes from the harmonic limit into the fpu parameter regime and obtain persistence of these periodic orbits termed here q breathers qb they are characterized by time periodicity exponential localization in the q space of normal modes and linear stability up to a size dependent threshold amplitude trajectories computed in the original fpu setting are perturbations around these exact qb solutions the qb concept is applicable to other nonlinear lattices as well

stability of nonlinear modes and chaotic properties of 1d fermi pasta ulam lattices abstract the stability properties of certain simple periodic solutions nonlinear modes of the equations of motion of one dimensional n particle fpu lattices are obtained analytically by uncoupling the n linear variational equations the energy per particle e c n at which these modes first become unstable is calculated and its asymptotic behavior as n is determined we find that for lattices which experience strong energy sharing e c n 0 as n while for a lattice where little energy sharing is observed e c n const 0 as n increases certain possible connections between our local stability results and some global chaotic properties of fpu lattices are discussed

a decomposition rule for certain tensor product representations of the symmetric groups abstract in this paper we give a combinatorial rule to calculate the decomposition of the tensor product kronecker product of two irreducible complex representations of the symmetric group s n when one of the representations corresponds to a hook n m 1 m

strange attractors and asymptotic measures of discrete time dissipative systems we investigate asymptotic properties of certain discrete time dynamical systems in two and three dimensions with solenoidal attractor it is proved that the asymptotic measures relevant for the generalized version of the ergodic theorem all derive from one haar measure

relation between multidimensional representations of the fedorov groups and the groups of color symmetry abstract relations have been traced between multidimensional irreducible representations ir of the fedorov groups and the groups of color symmetry using as an example the phase transitions in crystals unlike the case of the one dimensional real ir s for which the indenbom niggli theorem is valid for the multidimensional ir s a one to one relationship between them and the color symmetry groups is disturbed namely each multidimensional ir of the fedorov group is associated not with one but rather with a whole set of the color symmetry groups and conversely each group of the color symmetry is associated not with one ir but with a certain set of these complete correspondence between the color symmetry groups and the ir s of the fedorov group can be established on the basis of an analysis of the complete condensate of order parameters it has been shown that on the set of different domains of a low symmetry phase macroscopic level there are acting certain p symmetry groups when however the structure of one domain of such a phase microscopic level is described the w p symmetry groups have to be used the distinguishing feature of the w p symmetry as compared to the p symmetry is this each element of the classical group is assigned not one color permutation but different permutations depending on which point of the space is subjected to the action of this element it has also been shown that to the w p symmetry groups describing the structure of one domain there corresponds generally speaking nonrectangular young diagrams

isochronism and tangent bifurcation of band edge modes in hamiltonian lattices abstract in physica d s flach physica d 91 1996 223 results were obtained regarding the tangent bifurcation of the band edge modes q 0 of nonlinear hamiltonian lattices made of n coupled identical oscillators introducing the concept of partial isochronism which characterises the way the frequency of a mode depends on its energy e we generalize these results and show how the bifurcation energies of these modes are intimately connected to their degree of isochronism in particular we prove that in a lattice of coupled purely isochronous oscillators oscillators with an energy independent frequency the in phase mode q 0 never undergoes a tangent bifurcation whereas the out of phase mode q does provided the strength of the nonlinearity in the coupling is sufficient we derive a discrete nonlinear schrodinger equation governing the slow modulations of small amplitude band edge modes and show that its nonlinear exponent is proportional to the degree of isochronism of the corresponding orbits this equation may be seen as a link between the tangent bifurcation of band edge modes and the possible emergence of localized modes such as discrete breathers

classical waves on nonlinear lattices ii qn exact analytical solutions are found for classical vibrations of nonlinear oscillators with higher powers of spring constants such as q6 and q8 the solutions are based on a new type of elliptic function that has been created to solve these problems

quantisation of the kaplan yorke model a classically chaotic model with a strange attractor introduced by kaplan and yorke 1979 is quantised and analysed in the heisenberg picture the equivalence is shown to quantisation in the schrodinger picture given earlier by one of the authors as an advantage of the approach presented here the expectation values of observables can be calculated it is shown that in expectation values the limits to 0 and t to infinity commute

stability analysis of dynamical regimes in nonlinear systems with discrete symmetries we present a theorem that allows one to simplify the linear stability analysis of periodic and quasiperiodic nonlinear regimes in n particle mechanical systems with different kinds of discrete symmetry this theorem suggests a decomposition of the linearized system arising in the standard stability analysis into a number of subsystems whose dimensions can be considerably less than the dimension of the full system as an example of such a simplification we discuss the stability of bushes of modes invariant manifolds for the fermi pasta ulam chains and prove another theorem about the maximal dimension of the above mentioned subsystems

conditions on the existence of localized excitations in nonlinear discrete systems we use recent results that localized excitations in nonlinear hamiltonian lattices can be viewed and described as multiple frequency excitations their dynamics in phase space takes place on tori of corresponding dimension for a one dimensional hamiltonian lattice with nearest neighbour interaction we transform the problem of solving the coupled differential equations of motion into a certain mapping m l 1 f m l m l 1 where m l for every l lattice site is a function defined on an infinite discrete space of the same dimension as the torus we consider this mapping in the tails of the localized excitation i e for l rightarrow pm infty for a generic hamiltonian lattice the thus linearized mapping is analyzed we find conditions of existence of periodic one frequency localized excitations as well as of multiple frequency excitations the symmetries of the solutions are obtained as a result we find that the existence of localized excitations can be a generic property of nonlinear hamiltonian lattices in contrast to nonlinear hamiltonian fields

discrete breathers on symmetry determined invariant manifolds for scalar models on the plane square lattice a group theoretical approach for studying localized periodic and quasiperiodic vibrations in two and three dimensional lattice dynamical models is developed this approach is demonstrated for the scalar models on the plane square lattice the symmetry determined invariant manifolds admitting existence of localized vibrations are found and some types of discrete breathers are constructed on these manifolds a general method using the apparatus of matrix representations of symmetry groups to simplify the standard linear stability analysis is discussed this method allows one to decompose the corresponding system of linear differential equations with time dependent coefficients into a number of independent subsystems whose dimensions are less than the full dimension of the considered system

stability of simple periodic orbits and chaos in a fermi pasta ulam lattice we investigate the connection between local and global dynamics in the fermi pasta ulam fpu beta model from the point of view of stability of its simplest periodic orbits spo s in particular we show that there is a relatively high q mode q 2 n 1 3 of the linear lattice having one particle fixed every two oppositely moving ones called spo2 here which can be exactly continued to the nonlinear case for n 5 3m m 0 1 2 and whose first destabilization e 2u as the energy or beta increases for any fixed n practically coincides with the onset of a weak form of chaos preceding the breakdown of fpu recurrences as predicted recently in a similar study of the continuation of a very low q 3 mode of the corresponding linear chain this energy threshold per particle behaves like e 2u n alpha n 2 we also follow exactly the properties of another spo with q n 1 2 in which fixed and moving particles are interchanged called spo1 here and which destabilizes at higher energies than spo2 since e 1u n alpha n 1 we find that immediately after their first destabilization these spo s have different positive lyapunov spectra in their vicinity however as the energy increases further at fixed n these spectra converge to the same exponentially decreasing function thus providing strong evidence that the chaotic regions around spo1 and spo2 have merged and large scale chaos has spread throughout the lattice since these results hold for n arbitrarily large they suggest a direct approach by which one can use local stability analysis of spo s to estimate the energy threshold at which a transition to ergodicity occurs and thermodynamic properties such as kolmogorov sinai entropies per particle can be computed for similar one dimensional lattices

the zone boundary mode in periodic nonlinear electrical lattices abstract we study the existence and linear stability of the zone boundary mode in a nonlinear electrical lattice consisting of n inductors and n voltage dependent capacitors with periodic boundary conditions the inductances are allowed to alternate while the capacitors are identical and each have a quadratic dependence on voltage by block diagonalizing a 2 n 2 n floquet problem we reduce the question of the stability of the mode to a single hill s equation that is analyzed using methods of perturbation theory and averaging we show that periodicity of the lattice inductances degrades stability and also show that the instability threshold is proportional to n 2 numerical computations validate the perturbative results