analytical search for homoclinic bifurcations in the shimizu morioka model the existence of a homoclinic butterfly to a saddle with zero saddle value is established analytically for the shimizu morioka model

pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction diffusion model we address the stability of solitary pulses as well as some other traveling structures near the onset of spatiotemporal chaos in a two species reaction diffusion model describing the oxidation of c

chaos in a low order model of magnetoconvection abstract in the limit of tall thin rolls weakly nonlinear convection in a vertical magnetic field is described by an asymptotically exact third order set of ordinary differential equations these equations are shown here to have three codimension two bifurcation points a takens bogdanov bifurcation at which a gluing bifurcation is created a point at which the gluing bifurcation is replaced by a pair of homoclinic explosions between which there are lorenz like chaotic trajectories and a new type of bifurcation point at which the first of a cascade of period doubling bifurcation lines originates the last two bifurcation points are analysed in terms of a one dimensional map the equations also have a t point at which there is a heteroclinic connection between a saddle and a pair of saddle foci emerging from this point is a line of sil nikov bifurcations involving homoclinic connections to a saddle focus

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

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

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

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

complex oscillations and waves of calcium in pancreatic acinar cells abstract we perform a bifurcation analysis of a model of ca2 wave propagation in the basal region of pancreatic acinar cells the model we consider was first presented in sneyd et al j sneyd k tsaneva atanasova j i e bruce s v straub d r giovannucci d i yule a model of calcium waves in pancreatic and parotid acinar cells biophys j 85 2003 1392 1405 where a partial bifurcation analysis was given of the model in the absence of diffusion we obtain more complete information about bifurcations of the diffusionless model via numerical studies then analyse the spatially extended model by numerical investigation of the travelling wave equations and direct numerical solution of the model equations we find solitary waves in the model equations arising from homoclinic bifurcations in the travelling wave equations the solitary waves exist and appear to be stable for a significant interval of the primary bifurcation parameter i e the concentration of inositol trisphosphate but are eventually replaced by irregular spatio temporal behaviour the homoclinic bifurcations are related to a number of complicated mathematical structures in the travelling wave equations including an anomalous homoclinic hopf bifurcation heteroclinic bifurcations between an equilibrium and a periodic orbit and homoclinic bifurcations of periodic orbits

the accumulation of eigenvalues in a stability problem abstract when two waves propagating in a one dimensional medium are locked together as a composite wave a natural question arises as to whether the new wave is stable an interesting and novel instability mechanism is exposed here in which a cascade of eigenvalues accumulates at a distinguished point in the unstable half plane the underlying assumption is that the transition between the two waves occurs at an unstable homogeneous steady state of the partial differential equations this causes the individual waves to have an unstable continuous spectrum but the instability of the full wave cannot be predicted from the configuration of these spectra alone

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

extra strong uncertainty principles in relation to phase derivative for signals in euclidean spaces abstract this paper devotes to studying uncertainty principles of heisenberg type for signals defined on r n taking values in a clifford algebra for real para vector valued signals possessing all first order partial derivatives we obtain two uncertainty principles of which both correspond to the strongest form of the heisenberg type uncertainty principles for the one dimensional space the lower bounds of the new uncertainty principles are in terms of a scalar valued phase derivative through hardy spaces decomposition we also obtain two forms of uncertainty principles for real valued signals of finite energy with the first order sobolev type smoothness

on the dimensions of the spectral measure of symmetric binary substitutions substitution rules defined on the binary alphabet and invariant under the interchange of both participating symbols are considered typically the resulting symbolic sequences are weakly disordered and their fourier spectra are singular continuous exact relations are derived which express the correlation dimension of the multifractal spectral measure in terms of the entries in the substitution pattern recurrence relations are obtained for the finite length spectral sums and for the autocorrelation function

an uncertainty principle for quaternion fourier transform we review the quaternionic fourier transform qft using the properties of the qft we establish an uncertainty principle for the right sided qft this uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion valued signals in the spatial and frequency domains it is shown that only a gaussian quaternion signal minimizes the uncertainty

neighbour balanced block designs for correlated errors summary the paper considers a simple block model and assumes that the errors within the blocks are correlated following a stationary first order autoregressive process gill shukla 1985 consider this model but they restrict the set of the competing designs to a small subset of the possible designs the binary and equireplicate designs in the present paper we show that the designs determined by gill shukla are optimal over all possible designs for positive correlations and that neighbour balanced designs in general are highly efficient

an uncertainty principle for real signals in the fractional fourier transform domain the fractional fourier transform frft can be thought of as a generalization of the fourier transform to rotate a signal representation by an arbitrary angle spl alpha in the time frequency plane a lower bound on the uncertainty product of signal representations in two frft domains for real signals is obtained and it is shown that a gaussian signal achieves the lower bound the effect of shifting and scaling the signal on the uncertainty relation is discussed an example is given in which the uncertainty relation for a real signal is obtained and it is shown that this relation matches with that given by the uncertainty relation derived

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

global bifurcations to strange attractors in a model for skew varicose instability in thermal convection abstract we present a global bifurcation study of a four dimensional system of differential equations proposed by f h busse and coworkers modeling instabilities of convection rolls in the rayleigh benard experiment the rayleigh and prandtl numbers are two natural parameters on which the system depends we focus on a detailed mathematical study combining numerical pathfollowing and bifurcation analysis of chaotic dynamics and transitions to chaotic dynamics numerical continuation makes clear how homoclinic and heteroclinic bifurcations organize the bifurcation diagram in the parameter plane combined with a theoretical bifurcation analysis this explains the development of patterns and the creation of chaotic spatio temporal dynamics in the model the organizing centers such as heteroclinic cycles with resonance conditions among eigenvalues and homoclinic loops with geometric degeneracies inclination flips are identified and their unfoldings are analyzed this ties the creation of strange attractors of various geometric structures to codimension two global bifurcations

numerical bifurcation and stability analysis of solitary pulses in an excitable reaction diffusion medium abstract we present a systematic computer assisted study of the bifurcations and instabilities of solitary pulses in an excitable medium capable of displaying both stable pulse propagation and spatiotemporally chaotic dynamics over intervals of parameter space the reaction diffusion model used is of the activator inhibitor type only the activator diffuses in this medium the control parameters are the ratio of time scales of the activator and inhibitor dynamics and the excitation threshold this study focuses on travelling pulses their domain of existence and the bifurcations that render them unstable these pulses are approximated as a homoclinic orbits in a travelling wave ode frame and b as solutions of the full partial differential equation pde with periodic boundary conditions in large domains a variety of bifurcations in the travelling wave ode frame are observed including heteroclinic loops so called t points a r champneys and y a kuznetsov numerical detection and continuation of codimension 2 homoclinic bifurcations int j bif chaos 4 1994 785 h kokobu homoclinic and heteroclinic bifurcations of vectorfields japan j appl math 5 1988 455 instabilities in the full pde frame include both hopf bifurcations to modulated travelling waves involving the discrete pulse spectrum as well as transitions involving the continuous spectrum such as the so called backfiring transition m bar m hildebrand m eiswirth m falcke h engel and m neufeld chemical turbulence and standing waves in a surface reaction model the influence of global coupling and wave instabilities chaos 4 1994 499 the stability of modulated pulses is computed through numerical floquet analysis and a cascade of period doubling bifurcations is observed as well as certain global bifurcations these results corroborated by observations from direct numerical integration provide a skeleton around which many features of the overall complex spatiotemporal dynamics of the pde are organized

some optimal designs for repeated measurements with autoregressive errors summary a number of individuals are observed at equally spaced time points different individuals behave independently but the observations on the same individual have the autocorrelation structure of a first order autoregressive process we are concerned with the optimal allocation of treatments assuming that at each time point a new selection must be made conditions for optimality are given for models without nuisance parameters and also with covariates and with block effects

contrasts under long range correlations the background of the paper is the empirical observation from a variety of subject areas that long range correlations appear to be much more frequent than has been previously assumed this includes high quality measurement series which are commonly treated as prototypes of i i d observations evidence is briefly cited in the paper it has already been shown elsewhere that long range dependence leads to results that can be qualitatively different from those obtained under short range dependence and in particular that long range dependence has drastic effects on the naive statistical treatment of absolute constants the natural question arising from this also of relevance for statistical practice is how the long range dependence affects the statistics for contrasts the main answer given in this paper is twofold

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

efficient implementation of quaternion fourier transform convolution and correlation by 2 d complex fft the concepts of quaternion fourier transform qft quaternion convolution qcv and quaternion correlation which are based on quaternion algebra have been found to be useful for color image processing however the necessary computational algorithms and their complexity still need some attention we develop efficient algorithms for qft qcv and quaternion correlation the conventional complex two dimensional 2 d fourier transform ft is used to implement these quaternion operations very efficiently with these algorithms we only need two complex 2 d fts to implement a qft six complex 2 d fts to implement a one side qcv or a quaternion correlation and 12 complex 2 d fts to implement a two side qcv and the efficiency of these quaternion operations is much improved meanwhile we also discuss two additional topics the first one is about how to use qft and qcv for quaternion linear time invariant qlti system analysis this topic is important for quaternion filter design and color image processing besides we also develop the spectrum product qcv it is an improvement of the conventional form of qcv for any arbitrary input functions it always corresponds to the product operation in the frequency domain it is very useful for quaternion filter design

experimental unfolding of the nonlinear dynamics of a cable mass suspended system around a divergence hopf bifurcation systematic experimental investigation of the finite amplitude dynamics of a multiple internally resonant suspended cable mass subjected to anti phase support motion at primary resonance is accomplished upon getting hints from a basic system configuration assumed as reference setup about the multiple bifurcation event possibly governing transition to complex dynamics an improved experimental apparatus is used to make it technically accessible results obtained by varying three control parameters namely the frequency and amplitude of excitation and the temperature of a thermostatic chamber embedding the experimental system allow us to characterize in depth various occurring classes of motion in terms of time and spatial complexity to describe peculiar and or persistent features of transition to nonregular dynamics and to trace them back to a canonical scenario from bifurcation theory variable response paths are detected via bifurcation diagrams and spectra of singular values of measurement results and overall behaviour charts are built in the excitation parameter space considering the temperature as a controllable parameter shows to be fundamental for i indirectly setting cable material properties to values for which the conjectured codimension 2 bifurcation becomes apparent ii qualitatively referring the experimental unfolding of regular and nonregular cable dynamics to the theoretical unfolding of the divergence hopf bifurcation normal form and iii determining system response not only in the strict neighbourhood of the organizing divergence hopf bifurcation but also in the ensuing postcritical regions where the dependence of material damping on temperature affects secondary bifurcations to low dimensional homoclinic chaos

windowed fourier transform of two dimensional quaternionic signals in this paper we generalize the classical windowed fourier transform wft to quaternion valued signals called the quaternionic windowed fourier transform qwft using the spectral representation of the quaternionic fourier transform qft we derive several important properties such as reconstruction formula reproducing kernel isometry and orthogonality relation taking the gaussian function as window function we obtain quaternionic gabor filters which play the role of coefficient functions when decomposing the signal in the quaternionic gabor basis we apply the qwft properties and the right sided qft to establish a heisenberg type uncertainty principle for the qwft finally we briefly introduce an application of the qwft to a linear time varying system

wrinkled tori and bursts due to resonant temporal forcing the effect of resonant temporal forcing on a system undergoing a hopf bifurcation with d4 square symmetry is studied the forcing breaks the continuous normal form symmetry of the governing amplitude equations but the d4 symmetry is preserved for this system it is shown that bursts with very large dynamic range may occur the bursts are associated with visits near solutions at infinity and are related to those found previously for the hopf bifurcation with broken d4 symmetry moehlis and knobloch physica d 135 2000 263 the regime in which an attracting quasiperiodic solution that exists in the absence of forcing wrinkles into chaos as the amplitude of the forcing increases is investigated in detail the overall behavior is governed by the approach of the resulting attractor to solutions at infinity windows with stable periodic solutions are found and associated with the traversal in parameter space through arnol d tongues other aspects of the dynamics are related to the presence of a new type of gluing bifurcation which we call a supergluing bifurcation 2001 elsevier science b v all rights reserved

on bifurcations of the lorenz attractor in the shimizu morioka model abstract bifurcations of lorenz like attractors are studied in the shimizu morioka model two scenarios for their destruction are suggested the first is connected with the appearance of a lacuna the second with a transition to a quasi attractor the effects of codimension two bifurcations are considered

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

homoclinic phenomena in laser models abstract a description of the principal bifurcations which lead to the appearance of the lorenz attractor is given for the 3d normal form for codimension 3 bifurcations of equilibria and periodic orbits in systems with symmetry we pay special attention to two bifurcation points corresponding to the formation of a homoclinic butterfly of a saddle with unit saddle index and to a homoclinic butterfly with zero separatrix value

death of period doublings locating the homoclinic doubling cascade abstract this paper studies a natural mechanism called a homoclinic doubling cascade for the disappearance of period doubling cascades in vector fields simply put an entire period doubling cascade collides with a saddle type equilibrium homoclinic doubling cascades are known to have self similar structure in contrast to the well known feigenbaum constant the scaling constants for homoclinic doubling depend on the eigenvalues of the saddle equilibrium specifically we present here for the first time a detailed study of homoclinic doubling cascades in a smooth vector field namely a three dimensional polynomial model proposed by sandstede a numerical algorithm is presented for computing homoclinic doubling cascades in general vector fields which makes use of the program auto homcont this allows us to compute two types of homoclinic doubling cascades one where the primary homoclinic orbit undergoes an inclination flip bifurcation and one where it undergoes an orbit flip bifurcation our results bring out the scaling constants in good agreement with analytical estimates obtained from one dimensional maps

the non transverse shil nikov hopf bifurcation uncoupling of homoclinic orbits and homoclinic tangencies abstract it is known that in a neighbourhood of a codimension two shil nikov hopf bifurcation primary periodic orbits lie on a single wiggly curve in period parameter space with accumulation points at parameter values of a pair of homoclinic tangencies to a periodic orbit in contrast it has recently been shown by hirschberg and laing that primary periodic orbits lie on an infinity of isolas in a neighbourhood of certain degenerate homoclinic tangency to a periodic orbit this paper analyses the codimension three bifurcation caused by a non transverse i e degenerately parametrically unfolded shil nikov hopf bifurcation which contains nearby dynamics akin to both degeneracies two cases are classified as being downward pointing or upward pointing depending on whether the variation of a third parameter causes either the annihilation of a locus of saddle focus homoclinic orbits to equilibria or the uncoupling of this locus from the locus of hopf bifurcations we undertake a heuristic analysis of the unfolding showing that in both cases it contains codimension two non transversal homoclinic orbits to equilibria and non transversal homoclinic tangencies to periodic orbits unfolding the former non transverse orbit is shown to cause two wiggly curves to coalesce and leave finitely many isolas of periodic orbits unfolding the latter causes two wiggly curves to coalesce into first infinitely many and then finitely many isolas asymptotic expressions are given for the accumulation of two types of isola forming bifurcations the implications of z 2 equivariance on the unfolding is discussed finally numerical evidence is presented for both upward and downward pointing non transverse shil nikov hopf bifurcations occuring in a model of an autonomous non linear electronic circuit both with respect to a z 2 equivariant and a z 2 non equivariant equilibrium numerical computation of curves of periodic orbits homoclinic orbits and homoclinic tangencies to periodic orbits are shown to agree broadly with the theory but to uncover extra complications

analytical search for bifurcation surfaces in parameter space the method of resultants can be used to compute hopf bifurcations in ode systems we discuss this method from the applicant s point of view furthermore we show by theory and examples that the method may be extended to cover other bifurcation situations as well among them are the real hopf situation which plays an important role in the transition to shil nikov chaos as well as some higher codimension bifurcations such as takens bogdanov gavrilov guckenheimer and double hopf bifurcations the method yields an analytical test function that can be solved analytically or by computer algebra systems in contrast to common analytical techniques based on eigenvalue computation which can only be applied to systems of size n 4 the method is applicable for systems of intermediate size n 10 we illustrate the power of the method by discussing examples from different disciplines of science a lorenz like oscillator two coupled oscillators and a five species food chain

bounds for the chaotic region in the lorenz model abstract in a previous paper the authors made an extensive numerical study of the lorenz model changing all three parameters of the system we conjectured that the region of parameters where the lorenz model is chaotic is bounded for fixed r in this paper we give a theoretical proof of the conjecture by obtaining theoretical bounds for the chaotic region and by using fenichel theory the theoretical bounds are complemented with numerical studies performed using the maximum lyapunov exponent and ofli2 techniques and a comparison of both sets of results is shown finally we provide a complete three dimensional model of the chaotic regime depending on the three parameters

two mechanisms of the transition to chaos in finite dimensional models of convection abstract the onset of chaos in systems of ordinary differential equations possessing a stationary solution with a one dimensional unstable manifold is studied both numerically and qualitatively with the help of an auxiliary piecewise monotomic discontinuous recursion relation a connection is established between the route to chaos and the ratio of two leading eigenvalues of the vector field linearized near the fixed point this connection is confirmed by numerical data obtained from the investigation of differential equations originating from a hydrodynamical problem two routes are considered a well known mechanism suggested by lorenz and another one which is due to the accumulation of bifurcations corresponding to the emergence of homoclinic orbits of a saddle point the asymptotical properties of the latter route prove to be entirely determined by the above ratio

a new type of period doubling bifurcations in one dimensional transformations with two extrema abstract we consider the iteration of a one dimensional transformation x f x symmetrical with the origin and with two symmetrical extrema there exists a sequence of period doubling bifurcations where period 2k cycles are created at parameter values k and which is not of the feigenbaum type we show that the k s sequence converge toward a limit value as a double exponential function of k

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

the bifurcations of separatrix contours and chaos abstract bifurcations of systems close to systems having contours composed of separatrices of a pair of saddle points are studied and cases of the appearance of nontrivial hyperbolic sets are pointed out

some uncertainty principles for time frequency transforms of the cohen class the question of how the time frequency concentration of transforms of the cohen class is affected by fundamental properties is addressed the investigated transform properties are the marginal property the property of energy conservation and moyal s relation we show that each of these transform properties imply restrictions on the time frequency concentration of the cohen class and is thus related to an uncertainty principle different notions of time frequency concentration adapted to the property under consideration are applied

including homoclinic connections and t point heteroclinic cycles in the same global problem for a reversible family of piecewise linear systems global behavior is analyzed in a family of reversible piecewise linear systems the aim is to prove the existence of certain homoclinic orbits and t point cycles a problem including these two different objects as particular cases is constructed this problem leads to a common theorem of existence and local uniqueness the analytical proof of this result is given apart from being a complex task even in piecewise linear systems the proof of the existence of homoclinic connections and t point heteroclinic cycles must be usually carried out in separate ways because they are obviously different dynamical objects despite this some features of the system may narrow the disparities between such global bifurcations and help us to look for alternative methods to analyze them in this work taking advantage of the reversibility and some geometrical features of a piecewise linear version of the michelson system we construct by adding a suitable parameter a global problem that includes homoclinic connections and t point heteroclinic cycles as particular cases moreover this problem leads to a common result for the existence and local uniqueness of these global bifurcations whose proof is also given

asymptotic behaviour of the quaternion linear canonical transform and the bochner minlos theorem there have been numerous proposals in the literature to generalize the classical fourier transform by making use of the hamiltonian quaternion algebra the present paper reviews the quaternion linear canonical transform qlct which is a generalization of the quaternion fourier transform and it studies a number of its properties in the first part of this paper we establish a generalized riemann lebesgue lemma for the right sided qlct this lemma prescribes the asymptotic behaviour of the qlct extending and refining the classical riemann lebesgue lemma for the fourier transform of 2d quaternion signals we then introduce the qlct of a probability measure and we study some of its basic properties such as linearity reconstruction formula continuity boundedness and positivity finally we extend the classical bochner minlos theorem to the qlct setting showing the applicability of our approach

hypercomplex fourier transforms of color images fourier transforms are a fundamental tool in signal and image processing yet until recently there was no definition of a fourier transform applicable to color images in a holistic manner in this paper hypercomplex numbers specifically quaternions are used to define a fourier transform applicable to color images the properties of the transform are developed and it is shown that the transform may be computed using two standard complex fast fourier transforms the resulting spectrum is explained in terms of familiar phase and modulus concepts and a new concept of hypercomplex axis a method for visualizing the spectrum using color graphics is also presented finally a convolution operational formula in the spectral domain is discussed

variety of strange pseudohyperbolic attractors in three dimensional generalized henon maps abstract in the present paper we focus on the problem of the existence of strange pseudohyperbolic attractors for three dimensional diffeomorphisms such attractors are genuine strange attractors in that sense that each orbit in the attractor has positive maximal lyapunov exponent and this property is robust i e it holds for all close systems we restrict attention to the study of pseudohyperbolic attractors that contain only one fixed point then we show that three dimensional maps may have only 5 different types of such attractors which we call the discrete lorenz figure 8 double figure 8 super figure 8 and super lorenz attractors we find the first four types of attractors in three dimensional generalized henon maps of form x y y z z b x a z c y g y z where a b and c are parameters b is the jacobian and g 0 0 g 0 0 0

global structure of periodicity hubs in lyapunov phase diagrams of dissipative flows infinite cascades of periodicity hubs were predicted and very recently observed experimentally to organize stable oscillations of some dissipative flows here we describe the global mechanism underlying the genesis and organization of networks of periodicity hubs in control parameter space of a simple prototypical flow namely a r ossler s oscillator we show that spirals associated with periodicity hubs emerge and accumulate at the folding of certain fractal like sheaves of shilnikov homoclinic bifurcations of a common saddle focus equilibrium the specific organization of hub networks is found to depend strongly on the interaction between the homoclinic orbits and the global structure of the underlying attractor

analysis of the t point hopf bifurcation abstract in a parameterized three dimensional system of autonomous differential equations a t point is a point of the parameter space where a special kind of codimension 2 heteroclinic cycle occurs if the parameter space is three dimensional such a bifurcation is located generically on a curve a more degenerate scenario appears when this curve reaches a surface of hopf bifurcations of one of the equilibria involved in the heteroclinic cycle we are interested in the analysis of this codimension 3 bifurcation which we call t point hopf in this work we propose a model based on the construction of a poincare map that describes the global behavior close to a t point hopf bifurcation the existence of certain kinds of homoclinic and heteroclinic connections between equilibria and or periodic orbits is proved the predictions deduced from this model strongly agree with the numerical results obtained in a modified van der pol duffing electronic oscillator

optimal repeated measurements designs for correlated observations and analysis by weighted least squares summary we consider a model for repeated measurements designs where the measurements on the same unit are assumed to be correlated with known correlation coefficient it is shown that latin squares with an additional balancing property are e optimal for the weighted least squares estimate

on models with non rough poincare homoclinic curves abstract the possibility of an a priori complete description of finite parameter models including systems with structurally unstable poincare homoclinic curves is studied the main result reported here is that systems having a countable set of moduli of equivalence and systems having infinitely many degenerate periodic and homoclinic orbits are dense in the newhouse regions of non stability we discuss the question of correctly setting a problem for the analysis of models of such type

a sharp clifford wavelet heisenberg type uncertainty principle in the present work we are concerned with the development of a new uncertainty principle based on the wavelet transform in the clifford analysis algebra framework we precisely derive a sharp heisenberg type uncertainty principle for the continuous clifford wavelet transform

two dimensional quaternion wavelet transform abstract in this paper we introduce the continuous quaternion wavelet transform cqwt we express the admissibility condition in terms of the right sided quaternion fourier transform we show that its fundamental properties such as inner product norm relation and inversion formula can be established whenever the quaternion wavelets satisfy a particular admissibility condition we present several examples of the cqwt as an application we derive a heisenberg type uncertainty principle for these extended wavelets

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

pitt s inequality and the uncertainty principle associated with the quaternion fourier transform abstract the quaternion fourier transform a generalized form of the classical fourier transform has been shown to be a powerful analyzing tool in image and signal processing this paper investigates pitt s inequality and uncertainty principle associated with the two sided quaternion fourier transform it is shown that by applying the symmetric form f f 1 i f 2 f 3 j i f 4 j of quaternion from hitzer and the novel module or l p norm of the quaternion fourier transform f then any nonzero quaternion signal and its quaternion fourier transform cannot both be highly concentrated two part results are provided one part is heisenberg weyl s uncertainty principle associated with the quaternion fourier transform it is formulated by using logarithmic estimates which may be obtained from a sharp of pitt s inequality the other part is the uncertainty principle of donoho and stark associated with the quaternion fourier transform

lyapunov schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies abstract this article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits in addition tangencies in the intersection of un stable manifolds are allowed the main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time and their parameter values a leading order expansion is given in terms of the time spent near vertices and if applicable the location on the non trivial tangent directions the only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit the essential assumptions are hyperbolicity of the vertices and transversality of parameters using the result conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven finally equilibrium to periodic orbit heteroclinic cycles of various types are considered

amplitude equations for systems with competing instabilities we describe how to derive ordinary differential equations to predict the time dependence of a system governed by partial differential equations when that system is near to the polycritical condition for the onset of several instabilities the method is illustrated for the general case of two competing instabilities and then a specific physical example of this case is worked out in detail the basic idea is to extend the krylov bogolyubov mitropolsky method of nonlinear mechanics to bifurcation problems

homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit we analyze homoclinic orbits near codimension 1 and 2 heteroclinic cycles between an equilibrium and a periodic orbit for ordinary differential equations in three or higher dimensions the main motivation for this study is a self organized periodic replication process of travelling pulses which has been observed in reaction diffusion equations we establish conditions for existence and uniqueness of countably infinite families of curve segments of 1 homoclinic orbits which accumulate at codimension 1 or 2 heteroclinic cycles the main result shows the bifurcation of a number of curves of 1 homoclinic orbits from such codimension 2 heteroclinic cycles which depends on a winding number of the transverse set of heteroclinic points in addition a leading order expansion of the associated curves in parameter space is derived its coefficients are periodic with one frequency from the imaginary part of the leading stable floquet exponents of the periodic orbit and one from the winding number