blocks of small defect abstract let g be a finite solvable group let p be a prime such that p 5 and o p g 1 and we denote g p p n then g contains a block of defect less than or equal to 3 n 5 let g be a finite solvable group and let p a be the largest power of p dividing 1 for an irreducible character of g we show that g f g p p 3 a for p 5

dynamics of conservative bykov cycles tangencies generalized cocoon bifurcations and elliptic solutions abstract this paper presents a mechanism for the coexistence of hyperbolic and non hyperbolic dynamics arising in a neighbourhood of a conservative bykov cycle where trajectories turn in opposite directions near the two saddle foci we show that within the class of divergence free vector fields that preserve the cycle tangencies of the invariant manifolds of two hyperbolic saddle foci densely occur the global dynamics is persistently dominated by heteroclinic tangencies and by the existence of infinitely many elliptic points coexisting with non uniformly hyperbolic suspended horseshoes a generalized version of the cocoon bifurcations for conservative systems is obtained

using lin s method to solve bykov s problems abstract we consider nonwandering dynamics near heteroclinic cycles between two hyperbolic equilibria the constituting heteroclinic connections are assumed to be such that one of them is transverse and isolated such heteroclinic cycles are associated with the termination of a branch of homoclinic solutions and called t points in this context we study codimension two t points and their unfoldings in r n in our consideration we distinguish between cases with real and complex leading eigenvalues of the equilibria in doing so we establish lin s method as a unified approach to re gain and extend results of bykov s seminal studies and related works to a large extent our approach reduces the study to the discussion of intersections of lines and spirals in the plane

t points a codimension two heteroclinic bifurcation the local bifurcation structure of a heteroclinic bifurcation which has been observed in the lorenz equations is analyzed the existence of a particular heteroclinic loop at one point in a two dimensional parameter space a t point implies the existence of a line of heteroclinic loops and a logarithmic spiral of homoclinic orbits as well as countably many other topologically more complicatedt points in a small neighborhood in parameter space

abundance of strange attractors in 1976 he h non performed a numerical study of the family of diffeomorphisms of the plane ha b x y 1 ax2 y bx and detected for parameter values a l 4 b 0 3 what seemed to be a non trivial attractor with a highly intricate geometric structure this family has since then been the subject of intense research both numerical and theoretical but its dynamics is still far from being completely understood in particular one could not exclude the possibility that the attractor observed by h6non were just a periodic orbit with a very high period recently in a remarkable paper bc2 benedicks and carleson were able to show that this is not the case at least for a positive lebesgue measure set of parameter values near a 2 b o more precisely they showed that if b 0 is small enough then for a positive measure set of a values near a 2 the corresponding diffeomorphism ha b exhibits a strange attractor their argument is a very creative extension of the techniques they had previously developed in bull for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated nonhyperbolic dynamics when acquainted in 1985 with the work by benedicks and carleson then in progress palls suggested that one should in this context think of the h6non family as a particular although important model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies he proposed that the correct setting for benedicks carleson s results is within this more general framework of homoclinic bifurcations and stated the following

a survey of the theory of hypercube graphs abstract we present a comprehensive survey of the theory of hypercube graphs basic properties related to distance coloring domination and genus are reviewed the properties of the n cube defined by its subgraphs are considered next including thickness coarseness hamiltonian cycles and induced paths and cycles finally various embedding and packing problems are discussed including the determination of the cubical dimension of a given cubical graph

bifurcation phenomena near homoclinic systems a two parameter analysis the bifurcations of periodic orbits in a class of autonomous three variable nonlinear differential equation systems possessing a homoclinic orbit associated with a saddle focus with eigenvalues i where 1 sil nikov s condition are studied in a two parameter space the perturbed homoclinic systems undergo a countable set of tangent bifurcations followed by period doubling bifurcations leading to periodic orbits which may be attractors if 1 2 the accumulation rate of the critical parameter values at the homoclinic system is exp 2 a global mechanism for the onset of homoclinicity in strongly contractive flows is analyzed cusp bifurcations with bistability and hysteresis phenomena exist locally near the onset of homoclinicity a countable set of these cusp bifurcations with scaling properties related to the eigenvalues i of the stationary state are shown to occur in infinitely contractive flows in the two parameter space the periodic orbit attractor domain exhibits a spiral structure globally around the set of homoclinic systems in which all the different periodic orbits are continuously connected

shil nikov configurations in any generic unfolding of the nilpotent singularity of codimension three on r3 abstract we are interested in the dynamics arising in generic unfoldings of the nilpotent singularity of codimension three since shil nikov type homoclinic orbits make up the simplest scenario where very complicated dynamics for three dimensional vector fields can be shown the aim of this paper is to prove analytically their existence in any of such unfoldings

dense heteroclinic tangencies near a bykov cycle abstract this article presents a mechanism for the coexistence of hyperbolic and non hyperbolic dynamics arising in a neighbourhood of a bykov cycle where trajectories turn in opposite directions near the two nodes we say that the nodes have different chirality we show that in the set of vector fields defined on a three dimensional manifold there is a class where tangencies of the invariant manifolds of two hyperbolic saddle foci occur densely the class is defined by the presence of the bykov cycle and by a condition on the parameters that determine the linear part of the vector field at the equilibria this has important consequences the global dynamics is persistently dominated by heteroclinic tangencies and by newhouse phenomena coexisting with hyperbolic dynamics arising from transversality the coexistence gives rise to linked suspensions of cantor sets with hyperbolic and non hyperbolic dynamics in contrast with the case where the nodes have the same chirality we illustrate our theory with an explicit example where tangencies arise in the unfolding of a symmetric vector field on the three dimensional sphere

accumulations of t points in a model for solitary pulses in an excitable reaction diffusion medium we consider a family of differential equations that describes traveling waves in a reaction diffusion equation modeling oxidation of carbon oxide on a platinum surface near the onset of spatio temporal chaos the organizing bifurcation for the bifurcation structure with small carbon oxide pressures turns out to be a codimension 3 bifurcation involving a homoclinic orbit to an equilibrium undergoing a transcritical bifurcation we show how infinitely many t point bifurcations of multi loop heteroclinic cycles occur in the unfolding

heteroclinic bifurcations near hopf zero bifurcation in reversible vector fields in r 3 abstract we study the dynamics near a symmetric hopf zero also known as saddle node hopf or fold hopf bifurcation in a reversible vector field in r 3 with involutory an reversing symmetry whose fixed point subspace is one dimensional we focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddle foci we study local perturbations of this degenerate family of heteroclinics within the class of reversible vector fields and establish the generic existence of hyperbolic basic sets horseshoes independent of the eigenvalues of the saddle foci as well as cascades of bifurcations of periodic heteroclinic and homoclinic orbits finally we discuss the application of our results to the michelson system describing stationary states and travelling waves of the kuramoto sivashinsky pde

evaluation of effective resistances in pseudo distance regular resistor networks the effective resistance or two point resistance between two nodes of a resistor network is the potential difference that appears across them when a unit current source is applied between the nodes as terminals this concept arises in problems which deal with graphs as electrical networks including random walks distributed detection and estimation sensor networks distributed clock synchronization collaborative filtering clustering algorithms and etc in the previous paper jafarizadeh et al in j math phys 50 023302 2009 a recursive formula for evaluation of effective resistances on the so called distance regular networks was given based on the christoffel darboux identity in this paper we consider more general networks called pseudo distance regular networks or qd type networks where we use the stratification of these networks and show that the effective resistances between a given node say and all of the nodes belonging to the same stratum with respect to are the same then based on the spectral techniques for those s which satisfy l 1 alpha alpha l 1 beta beta l 1 is the pseudo inverse of the laplacian of the network an analytical formula for effective resistances r alpha beta m the equivalent resistance between terminals and so that belongs to the m th stratum with respect to is given in terms of the first and second orthogonal polynomials associated with the network from the fact that in distance regular networks l 1 alpha alpha l 1 beta beta is satisfied for all nodes of the network the effective resistances r alpha beta m for m 1 2 d d is diameter of the network which is the same as the number of strata are calculated directly by using the given formula

global bifurcations close to symmetry abstract heteroclinic cycles involving two saddle foci where the saddle foci share both invariant manifolds occur persistently in some symmetric differential equations on the 3 dimensional sphere we analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1 dimensional connection the saddle foci have the same chirality when part of the symmetry is broken the 2 dimensional invariant manifolds intersect transversely creating a heteroclinic network of bykov cycles we show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics there are n pulse heteroclinic tangencies trajectories that follow the original cycle n times around before they arrive at the other node each n pulse heteroclinic tangency is accumulated by a sequence of n 1 pulse ones this coexists with the suspension of horseshoes defined on an infinite set of disjoint strips where the first return map is hyperbolic we also show how as the system approaches full symmetry the suspended horseshoes are destroyed creating regions with infinitely many attracting periodic solutions

period doubling cascades of attractors a prerequisite for horseshoes this paper shows that if a horseshoe is created in a natural manner as a parameter is varied then the process of creation involves the appearance of attracting periodic orbits of all periods furthermore each of these orbits will period double repeatedly with those periods going to infinity

some two vertex resistances of the three towers hanoi graph formed by a fractal graph the resistance distance between two vertices of a simple connected graph g is equal to the resistance between two equivalent points on an electrical network constructed to correspond to g with each edge being replaced by a unit resistor in this study we apply the principle of substitution to three towers hanoi graph to procure an equivalent network that enables us to only use the series and parallel principles and the delta wye transformation to acquire the some two vertex resistance distances of three towers hanoi graph

on selecting models for nonlinear time series abstract constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models or to choose between competing classes this paper discusses a method of building nonlinear models of possibly chaotic systems from data while maintaining good robustness against noise the models that are built are close to the simplest possible according to a description length criterion the method will deliver a linear model if that has shorter description length than a nonlinear model we show how our models can be used for prediction smoothing and interpolation in the usual way we also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series

global generic dynamics close to symmetry abstract our object of study is the dynamics that arises in generic perturbations of an asymptotically stable heteroclinic cycle in s 3 the cycle involves two saddle foci of different type and is structurally stable within the class of z 2 z 2 symmetric vector fields the cycle contains a two dimensional connection that persists as a transverse intersection of invariant surfaces under symmetry breaking perturbations gradually breaking the symmetry in a two parameter family we get a wide range of dynamical behaviour an attracting periodic trajectory other heteroclinic trajectories homoclinic orbits n pulses suspended horseshoes and cascades of bifurcations of periodic trajectories near an unstable homoclinic cycle of shilnikov type we also show that generically the coexistence of linked homoclinic orbits at the two saddle foci has codimension 2 and takes place arbitrarily close to the symmetric cycle

spiralling dynamics near heteroclinic networks abstract there are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically we construct explicitly a two parameter family of vector fields on the three dimensional sphere s 3 whose flow has a spiralling attractor containing the following two hyperbolic equilibria heteroclinic trajectories connecting them transversely and a non trivial hyperbolic invariant and transitive set the spiralling set unfolds a heteroclinic network between two symmetric saddle foci and contains a sequence of topological horseshoes semiconjugate to full shifts over an alphabet with more and more symbols coexisting with newhouse phenomena the vector field is the restriction to s 3 of a polynomial vector field in r 4 in this article we also identify global bifurcations that induce chaotic dynamics of different types

heteroclinic cycles and periodic orbits for the o 2 equivariant 0 1 2 mode interaction we study the quadratic normal form describing the generic interaction of fourier modes of wavenumbers 0 1 and 2 under the symmetry group o 2 of rotations and reflections in the case that the homogeneous quadratic terms preserve energy the sum of the squares of absolute values of the complex variables this system is a generalization of the 1 2 mode interaction studied by dangelmayr g dangelmayr steady state mode interactions in the presence of o 2 symmetry dyn stab syst 1 2 1986 159 185 armbruster et al d armbruster j guckenheimer p holmes heteroclinic cycles and modulated travelling waves in systems with o 2 symmetry physica d 29 1988 257 282 and others and its restriction to the 1 2 subspace is a degenerate case of that system it displays all the classes of fixed points periodic orbits standing and travelling waves invariant tori modulated travelling waves and heteroclinic cycles found in the 1 2 interaction as well as new heteroclinic cycles connecting pure and mixed modes chaotic cycles and strange periodic orbits we describe the key dynamical features show that the degenerate 1 2 case possesses a second organizing center at which bifurcation curves coalesce provide representative bifurcation sets and diagrams for the 1 2 and 0 1 2 systems and use a conservative limit to understand the periodic orbits in the latter system 2005 elsevier b v all rights reserved

possible new strange attractors with spiral structure we define a class of three dimensional differential equations which might present strange attractors of a new kind this is illustrated by numerical observations on an explicit example

bifurcations in a model of magnetoconvection abstract bifurcations from oscillatory solutions are studied in a truncated model of two dimensional boussinesq magnetoconvection the fifth order system of nonlinear differential equations is integrated numerically and in certain parameter regimes there is a bifurcation from symmetrical to asymmetrical oscillations followed by a period doubling cascade after the accumulation point there is a semiperiodic cascade leading to chaotic behaviour then the semiperiodic cascade is repeated in reverse followed by a period halving cascade and a bifurcation back to symmetry finally the branch of oscillatory solutions terminates with a symmetrical heteroclinic orbit that connects two saddle foci the interval with aperiodic solutions contains many pairs of narrow windows with asymmetrical or symmetrical periodic solutions each with its own cascade this pattern of behaviour is likely to occur when a periodic orbit approaches a symmetrical pair of saddle foci with eigenvalues that satisfy shil nikov s inequality

heteroclinic dynamics in the nonlocal parametrically driven nonlinear schrodinger equation faraday waves are described under appropriate conditions by a damped nonlocal parametrically driven nonlinear schrodinger equation as the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics the origin of these transitions is explained using a careful study of a two mode galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves these connections are associated with cascades of gluing and symmetry switching bifurcations such bifurcations are located in the partial differential equations as well

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

imperfect homoclinic bifurcations experimental observations of an almost symmetric electronic circuit show complicated sequences of bifurcations these results are discussed in the light of a theory of imperfect global bifurcations it is shown that much of the dynamics observed in the circuit can be understood by reference to imperfect homoclinic bifurcations without constructing an explicit mathematical model of the system

on the resistance diameter of hypercubes abstract the resistance distance among two vertices in some resistor networks has been widely considered by many physicists and mathematicians the resistance diameter of a graph is defined by the maximum resistance distance among all pairs of vertices in the graph the study of resistance diameter was initiated here it seems to be possible that the resistance diameter of a graph could play a significant role in analyzing the efficiency of the interconnection networks for wave or fluid like communication in order to improve the efficiency of wave or fluid like communication we need to minimize the resistance diameter of graphs the class of hypercubes is the most common multipurpose and efficient topological structure class of interconnection networks hypercube networks satisfy most requirements for the basic principles of network design the main result of this paper is to give a combinatorial formula for the resistance diameter of hypercubes in addition the minimum resistance distance among all pairs of vertices in hypercubes is also obtained these two results are deduced via the electrical network approach and the hitting time of random walks respectively

chaos in a three dimensional single loop feedback system with a piecewice linear feedbackk function abstract this paper describes chaotic behaviour in a single loop feedback system of low order the system is particularly simple and because it is piecewise linear and hence can be undersdtood globally a description of the flow much more complete than usual can be given an orbit structure is suggested for the system after a minor approximation is considered the approximation gives two equivalent discrete one dimensional maps of the interval to itself and these maps are discussed in the light of recent work the validity of the approximation is confirmed by computer simulation

computation of resistance distance and kirchhoff index of the two classes of silicate networks abstract the resistance distance between two vertices of a simple connected graph g is equal to the resistance between two equivalent points on an electrical network constructed so as to correspond to g with each edge being replaced by a unit resistor the kirchhoff index of g is the sum of resistance distances between all pairs of vertices in g in this paper the resistance distance between any two arbitrary vertices of a chain silicate network and a cyclic silicate network was procured by utilizing techniques from the theory of electrical networks i e the series and parallel principles the principle of elimination the star triangle transformation and the delta wye transformation two closed formulae for the kirchhoff index of the chain silicate network and the cyclic silicate network were obtained respectively

existence and stability of multiple impulse solutions of a nerve equation a piecewise linear partial differential equation which models the propagation of a nerve impulse along an axon is studied for the equation parameters used there are two solutions which represent solitary impulses traveling with fixed form one with a slower velocity c s1 than the velocity c f1 of the other we show that near these velocities there exist multiple impulse solutions and exhibit four such solutions each of which has two pulses the local stability of these solutions is analyzed the two solutions whose velocities are near c s1 demonstrate instabilities which indicate growth or decay of the separate pulses for one of the multiple impulse solutions with velocity near c s1 and for one of those with velocity near c f1 the second pulse deomonstrates an instability which indicates a change in the spacing between the two pulses the other solution with velocity near c f1 is stable

the unicyclic graphs with maximum degree resistance distance for a connected graph with order n and size m the cyclomatic number number of independent cycles is equal to m n 1 the graphs with 1 are referred to the unicyclic graphs in this paper we characterized completely the unicyclic graphs with n vertices having maximum degree resistance distance

homoclinic and heteroclinic orbits in a modified lorenz system this paper presents a mathematically rigorous proof for the existence of chaos in a modified lorenz system using the theory of shil nikov bifurcations of homoclinic and heteroclinic orbits together with its dynamical behaviors which have been extensively studied the chaotic dynamics of the modified lorenz system are now much better understood providing a rigorous theoretic foundation to support studies and applications of this important class of chaotic systems

oscillators with chaotic behavior an illustration of a theorem by shil nikov using an explicit one parameter family of differential equations describing oscillators with feedback effects we prove the existence of values of the parameters such that there exist infinitely many unstable periodic orbits of saddle type the proof relies on a theorem by shil nikov which we propose as an explanation for the origin and structure of the chaotic behavior displayed by many well known third order differential systems

what can we learn from homoclinic orbits in chaotic dynamics state diagrams of two model systems involving three variables are constructed the parameter dependence of different forms of complex nonperiodic behavior and particularly of homoclinic orbits is analyzed it is shown that the onset of homoclinicity is reflected by deep changes in the qualitative behavior of the system

single and multiple pulse waves for the fitzhugh nagumo the fitzhugh nagumo equations describe the propagation of impulses along a nerve axon we study traveling wave solutions which satisfy a set of ordinary differential equations with a cubic nonlinearity f v v 1 v v a our emphasis is on solutions which tend to zero at pm infty an improved proof of a previous result that there are at least two such solutions is presented further by allowing the middle root v a of the nonlinearity to be a small parameter better estimates of the parameter range where such solutions exist are given in view of recent work of evans fenichel feroe and sil nikov our results imply that there are a great many more bounded traveling waves than was known previously in particular there is an infinite family of periodic waves all traveling with the same speed a reasonable transversality hypothesis introduced by evans fenichel and feroe implies also that there are multiple pulse waves at speeds close to that of a single pulse