sbil nikov hopf bifurcation abstract we derive a 2 parameter poincare map modeling a 3 dimensional vector flow near the codimension 2 point defined by a sil nikov homoclinic orbit to a fixed point undergoing a hopf bifurcation in one parameter regime the map reproduces glendinning and sparrow s results for the sil nikov bifurcation while in another regime it provides new results for a small hopf cycle with a global reinjection mechanism the map is analyzed finding primary homoclinic tangencies to the small hopf cycle parameter dependence of periodic orbits associated with these tangencies is described global behavior of the tube like unstable manifold of the hopf cycle is studied in depth analytically and numerically locating its simplest subsidiary tangencies

a distributed spatio temporal eeg meg inverse solver we propose a novel l1l2 norm inverse solver for estimating the sources of eeg meg signals based on the standard l1 norm inverse solvers this sparse distributed inverse solver integrates the l1 norm spatial model with a temporal model of the source signals in order to avoid unstable activation patterns and spiky reconstructed signals often produced by the currently used sparse solvers the joint spatio temporal model leads to a cost function with an l1l2 norm regularizer whose minimization can be reduced to a convex second order cone programming socp problem and efficiently solved using the interior point method the efficient computation of the socp problem allows us to implement permutation tests for estimating statistical significance of the inverse solution validation with simulated and human meg data shows that the proposed solver yields source time course estimates qualitatively similar to those obtained through dipole fitting but without the need to specify the number of dipole sources in advance furthermore the l1l2 norm solver achieves fewer false positives and a better representation of the source locations than the conventional l2 minimum norm estimates

structure ranks of matrices abstract the structure rank of a matrix i e the maximum order of a nonsingular submatrix all of whose entries are located in a given structure a subset of m n where m n are the row and column index sets respectively were previously studied for the off diagonal part here results on more general ranks e g off block diagonal strictly upper triangular and strictly upper block triangular are presented in particular it is shown that for a nonsingular matrix a in each of the last mentioned three cases the ranks of a and a 1 coincide

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

geometry and chaos near resonant equilibria of 3 dof hamiltonian systems in this paper we study the dynamics near resonant elliptic equilibria in three degree of freedom hamiltonian systems the resonances we consider have multiplicity two and the corresponding local normal form for the equilibrium is integrable at cubic order we prove the existence of families of 3 tori and whiskered 2 tori with nearby chaotic dynamics in the quartic normal form the whiskers of the 2 tori intersect in a non trivial way giving rise to multi pulse homoclinic and heteroclinic connections these connections survive in the full system as orbits homoclinic to invariant 3 spheres

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

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 existence of localized vegetation patterns in a systematically reduced model for dryland vegetation abstract in this paper we consider the 2 component reaction diffusion model that was recently obtained by a systematic reduction of the 3 component gilad et al model for dryland ecosystem dynamics gilad et al 2004 the nonlinear structure of this model is more involved than other more conceptual models such as the extended klausmeier model and the analysis a priori is more complicated however the present model has a strong advantage over these more conceptual models in that it can be more directly linked to ecological mechanisms and observations moreover we find that the model exhibits a richness of analytically tractable patterns that exceeds that of klausmeier type models our study focuses on the 4 dimensional dynamical system associated with the reaction diffusion model by considering traveling waves in 1 spatial dimension we use the methods of geometric singular perturbation theory to establish the existence of a multitude of heteroclinic homoclinic periodic orbits that jump between normally hyperbolic slow manifolds representing various kinds of localized vegetation patterns the basic 1 front invasion patterns and 2 front spot gap patterns that form the starting point of our analysis have a direct ecological interpretation and appear naturally in simulations of the model by exploiting the rich nonlinear structure of the model we construct many multi front patterns that are novel both from the ecological and the mathematical point of view in fact we argue that these orbits patterns are not specific for the model considered here but will also occur in a much more general singularly perturbed reaction diffusion setting we conclude with a discussion of the ecological and mathematical implications of our findings

generalized oscillatory matrices we define a new class of generalized oscillatory matrices shortly go matrices over a noncommutative ring with identity and a positive part similarly to the classical case this class consists of square generalized totally nonnegative matrices gtn matrices of which some power is generalized totally positive using the previously defined ordering of invertible gtn matrices we study in particular so called basic go matrices which form in a sense atoms of this ordering

traveling waves of calcium in a pancreatic acinar cells model construction and bifurcation analysis abstract we construct and study a model for intracellular calcium wave propagation with particular attention to pancreatic acinar cells the model is based on a model of the inositol trisphosphate ip3 receptor which assumes that calcium modulates the binding affinity of ip3 to the receptor two versions of the model one simpler than the other are studied numerically in both versions solitary waves in the excitable regime arise via homoclinic bifurcations in the traveling wave equations as the background concentration of ip3 is increased the wave speed increases and for some values of the ip3 concentration the initial pulse gives rise to secondary pulses that travel in both directions this can give rise to irregular spatio temporal behavior or to trains of pulses in the simpler model these secondary waves are related to the presence of a t point a heteroclinic cycle and an associated spiral of homoclinic orbits which terminate the branch of homoclinic orbits

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

heteroclinic orbits for retarded functional differential equations abstract suppose is a heteroclinic orbit of a ck functional differential equation x t x i with limit set and limit set being either hyperbolic equilibrium points or periodic orbits necessary and sufficient conditions are given for the existence of an ftf close to in ck with the property that x t tf x t has a heteroclinic orbit gg close to the orbits gg are obtained from the zeros of a finite number of bifurcation functions g f r d r d 1 transversality of is characterized by the nondegeneracy of the derivative d g it is also shown that the tf which have heteroclinic orbits near are on a ck submanifold of finite codimension max 0 ind or on the closure of it where ind is the index of

two results on basic oscillatory matrices this paper continues the research of the authors on totally nonnegative and oscillatory matrices first the result of the second author stating that the hadamard product of oscillatory tridiagonal matrices of the same order is again an oscillatory tridiagonal matrix is extended to the class of basic oscillatory matrices introduced recently then it is shown that every oscillatory matrix contains a basic oscillatory matrix as a factor this explains the role of the class of basic oscillatory matrices within the class of oscillatory matrices

noninvasive reconstruction of three dimensional ventricular activation sequence from the inverse solution of distributed equivalent current density we propose a new electrocardiographic ecg inverse approach for imaging the three dimensional 3 d ventricular activation sequence based on the modeling and estimation of the equivalent current density throughout the entire volume of the ventricular myocardium the spatio temporal coherence of the ventricular excitation process has been utilized to derive the activation time from the estimated time course of the equivalent current density in the present study we explored four different linear inverse algorithms the minimum norm and weighted minimum norm estimates in combination with two regularization schemes the instant by instant regularization and the isotropy method to estimate the current density at each time instant during the ventricular depolarization the activation time at any given location within the ventricular myocardium was determined as the time point with the occurrence of the maximum local current density estimate computer simulations were performed to evaluate this approach using single and dual site pacing protocols in a physiologically realistic cellular automaton heart model the performance and stability of the proposed approach was evaluated with respect to the various levels of measurement noise 0 5 10 20 40 and 60 muv the various numbers of ecg electrodes and the modeling errors on the torso geometry and heart position the simulation results demonstrate that 1 the single site paced 3 d activation sequence can be well reconstructed from 200 channel body surface potential maps with additive gaussian white noise of 20 muv correlation coefficient 0 90 relative error 0 19 and localization error 5 49 mm 2 a higher imaging accuracy can be obtained when the activation is initiated from the left right ventricle lv rv compared to from the septum 3 the isotropy method gives rise to a better performance than the conventional instant by instant regularization 4 a decreased imaging accuracy results from a larger noise level a fewer number of electrodes or the volume conductor modeling errors however a reasonable imaging accuracy can still be obtained with a 60 muv noise level 64 electrodes or mild errors on both the torso geometry and heart position respectively 5 the dual site paced 3 d activation sequence can be imaged when the two sites are paced either simultaneously or with a time delay of 20 ms 6 two pacing sites can be resolved and localized in the imaged 3 d activation sequence when they are located at the contralateral sides of ventricles or at the ventricular lateral wall and the apex respectively

degenerate bifurcations of nontwisted heterodimensional cycles with codimension 3 abstract bifurcations of heterodimensional cycles with highly degenerate conditions are studied by establishing a suitable local coordinate system in three dimensional vector fields the existence coexistence and noncoexistence of the periodic orbit homoclinic loop heteroclinic loop and double periodic orbit are obtained under some generic hypotheses the bifurcation surfaces and the existence regions are located the number of the bifurcation surfaces exhibits variety and complexity of the bifurcation of degenerate heterodimensional cycles the corresponding bifurcation graph is also drawn

generalized totally nonnegative matrices we define a new class of generalized totally nonnegative matrices shortly gtn matrices over a noncommutative ring with identity and a positive part the class of gtn matrices is closed under multiplication every submatrix of a gtn matrix is a gtn matrix as well we also define a standard form of a gtn matrix we then study the invertible gtn matrices of a fixed order and classify them the classes form a partially ordered set can be noncommutatively multiplied and multiplication preserves ordering

absolute and convective instabilities of waves on unbounded and large bounded domains abstract instabilities of nonlinear waves on unbounded domains manifest themselves in different ways an absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain in contrast convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time perturbations decay locally at each given point in the unbounded domain wave packets are convected towards infinity in experiments as well as in numerical simulations bounded domains are often more relevant we are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the aforementioned in stability properties of a wave these effects depend upon the boundary conditions that are imposed on the bounded domain we compare the spectra of the linearized evolution operators on unbounded and bounded domains for two classes of boundary conditions it is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately spectra for separated boundary conditions behave in quite a different way firstly separated boundary conditions may generate additional isolated eigenvalues secondly the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain instead the so called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights we interpret the difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain in particular it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave but that convectively unstable waves can destabilize under domain truncation the theoretical predictions are compared with numerical computations

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

bursts in oscillatory systems with broken d 4 symmetry a new mechanism responsible for generating regular and irregular bursts of large dynamic range near onset of an oscillatory instability is identified the bursts are present in systems with nearly square symmetry and are the result of heteroclinic cycles involving infinite amplitude states created when the square symmetry is broken all possible cycles of this type are identified and the resulting bursts described global connections involving finite amplitude states are also present the intricate sequence of bifurcations that result is described in several cases 2000 elsevier science b v all rights reserved

computing absolute and essential spectra using continuation a continuation approach to the computation of essential and absolute spectra of dierential operators on the real line is presented the advantage of this approach compared with direct eigenvalue computations for the discretized operator are the ecient and accurate computation of selected parts of the spectrum typically those near the imaginary axis and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability we also discuss the implementation and usage of this approach with the software package auto and provide example computations for the fitzhugh nagumo and the complex ginzburg landau equation

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

noninvasive transmural electrophysiological imaging based on minimization of total variation functional while tomographic imaging of cardiac structure and kinetics has improved substantially electrophysiological mapping of the heart is still restricted to the surface with little or no depth information beneath the progress in reconstructing 3 d action potential from surface voltage data has been hindered by the intrinsic ill posedness of the problem and the lack of a unique solution in the absence of prior assumptions in this work we propose a novel adaption of the total variation tv prior to exploit the unique spatial property of transmural action potential of being piecewise smooth with a steep boundary gradient separating depolarized and repolarized regions we present a variational tv prior instead of a common discrete tv prior for improved robustness to mesh resolution and solve the tv minimization by a sequence of weighted first order l2 norm minimization in a large set of phantom experiments the proposed method is shown to outperform existing quadratic methods in preserving the steep gradient of action potential along the border of infarcts as well as in capturing the disruption to the normal path of electrical wavefronts real data experiments also further demonstrate the potential of the proposed method in revealing the location and shape of infarcts when quadratic methods fail to do so

examining the impact of prior models in transmural electrophysiological imaging a hierarchical multiple model bayesian approach noninvasive cardiac electrophysiological ep imaging aims to mathematically reconstruct the spatiotemporal dynamics of cardiac sources from body surface electrocardiographic ecg data this ill posed problem is often regularized by a fixed constraining model however a fixed model approach enforces the source distribution to follow a pre assumed structure that does not always match the varying spatiotemporal distribution of actual sources to understand the model data relation and examine the impact of prior models we present a multiple model approach for volumetric cardiac ep imaging where multiple prior models are included and automatically picked by the available ecg data multiple models are incorporated as an lp norm prior for sources where p is an unknown hyperparameter with a prior uniform distribution to examine how different combinations of models may be favored by different measurement data the posterior distribution of cardiac sources and hyperparameter p is calculated using a markov chain monte carlo mcmc technique the importance of multiple model prior was assessed in two sets of synthetic and real data experiments compared to fixed model priors using laplace and gaussian priors the results showed that the posterior combination of models the posterior distribution of p as determined by the ecg data differed substantially when reconstructing sources with different sizes and structures while the use of fixed models is best suited in situations where the prior assumption fits the actual source structures the use of an automatically adaptive set of models may have the ability to better address model data mismatch and to provide consistent performance in reconstructing sources with different properties

heteroclinic cycles in rings of coupled cells abstract symmetry is used to investigate the existence and stability of heteroclinic cycles involving steady state and periodic solutions in coupled cell systems with d n symmetry using the lattice of isotropy subgroups we study the normal form equations restricted to invariant fixed point subspaces and prove that it is possible for the normal form equations to have robust asymptotically stable heteroclinic cycles connecting periodic solutions with steady states and periodic solutions with periodic solutions a center manifold reduction from the ring of cells to the normal form equations is then performed using this reduction we find parameter values of the cell system where asymptotically stable cycles exist simulations of the cycles show trajectories visiting steady states and periodic solutions and reveal interesting spatio temporal patterns in the dynamics of individual cells we discuss how these patterns are forced by normal form symmetries

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

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