neighborhood radius estimation for arnold s miniversal deformations of complex and p adic matrices abstract v i arnold 1971 constructed a simple normal form to which all complex matrices b in a neighborhood u of a given square matrix a can be reduced by similarity transformations that smoothly depend on the entries of b we calculate the radius of the neighborhood u a a mailybaev 1999 2001 constructed a reducing similarity transformation in the form of taylor series we construct this transformation by another method we extend arnold s normal form to matrices over the field q p of p adic numbers and the field f t of laurent series over a field f

miniversal deformations of pairs of skew symmetric matrices under congruence for each pair of complex symmetric matrices a b we provide a normal form with a minimal number of independent parameters to which all pairs of complex symmetric matrices a over tilde b ove

synchronized clusters in coupled map networks ii stability analysis we study self organized and driven synchronization in some simple coupled map networks namely globally coupled networks and complete bipartite networks using both linear stability analysis and lyapunov function approach and determine stability conditions for synchronization the phase diagrams for the networks studied here have features very similar to the different kinds of structurally similar networks studied in part i lyapunov function approach shows that when any two nodes are in driven synchronization all the coupling terms in the difference between the variables of these two nodes cancel out whereas when they are in self organized synchronization the direct coupling term between the two nodes adds an extra term while the other couplings cancel out we also discuss the conditions for the occurrence of a floating node and suggest that the fluctuations of the conditional lyapunov exponent about zero can be a criterion for its occurrence

invariant subset and set stability of boolean networks under arbitrary switching signals this study investigates the set stability of switched boolean networks bns with arbitrary switching signals based on invariant subsets set stability determines whether a bn converges to a given subset while arbitrary switching signals can be used to characterize uncertainties and disturbances an algorithm is proposed to calculate the largest invariant subset contained in any given set as this is the key step in establishing the set stability criterion based on this algorithm a necessary and sufficient condition for set stability is obtained the result is then used to solve the robust output synchronization problem and the robust node synchronization problem for bns in the presence of uncertainties and disturbances examples are presented to demonstrate the use of the proposed approach

stabilization of a class of slow fast control systems at non hyperbolic points in this document we deal with the local asymptotic stabilization problem of a class of slow fastsystems or singularly perturbed ordinary differential equations the systems studied here have the following properties 1 they have one fast and an arbitrary number of slow variables and 2 they have a non hyperbolic singularity at the origin of arbitrary degeneracy our goal is to stabilize such a point the presence of the aforementioned singularity complicates the analysis and the controller design in particular the classical theory of singular perturbations cannot be used we propose a novel design based on geometric desingularization which allows the stabilization of a non hyperbolic point of singularly perturbed control systems our results are exemplified on a didactic example and on an electric circuit

modeling intrinsic noise in random boolean networks abstract noise has crucial effects on biological processes chemical reactions and various chaotic systems this study explores the dynamical properties of noise in random boolean networks in which the stochasticity may cause significant deviations from deterministic descriptions such noise is intrinsic and results from the discrete dynamics of finite populations by using methods from statistical physics and nonlinear dynamics this study illustrates the dynamical characteristics of the inherent noise furthermore a modified mean field model is formulated to mimic the stochastic dynamics revealed by the discrete systems the proposed idea of modeling intrinsic noise can be potentially applied to other discrete systems belonging to the category of random boolean networks

synchronization and coherence in thermodynamic coupled map lattices with intermediate range coupling in spatially extended systems intermediate range interactions arise naturally in some physical contexts to study them we investigate a model of coupled map lattices cml s with intermediate range coupling and derive analytic conditions for its synchronization we find that in these cml s if the range of coupling is fixed the law of large numbers applies for the mean field the total normalized power in nonzero components of the power spectrum of the mean field goes to zero in the thermodynamic limit we also show that in the same limit the relevant parameter for synchronization and coherence is the fraction of sites coupled and not their number s1063 651x 99 07410 3 pacs number s 05 45 2a the past decade has seen a surge of interest in synchronization and coherence in coupled map lattices cml s 1 10 most work has focused on systems with finite degrees of freedom and short range global or random coupling in this paper we study the synchronization of one and two dimensional 1d and 2d cml s with intermediaterange couplings in the thermodynamic limit this study is related to systems with finite degrees of freedom studied by sinha et al 2 and kozma 10 in these works each site is coupled to a range of neighbors on either side rather than just its nearest neighbors this may be a valid approximation in systems where interaction strength decays slowly with distance just as cml s with nearest neighbor coupling could be considered to be a discretization of partial differential equation cml s with intermediate range coupling could be seen as a discretization of a partial integro differential equation in fact such integro differential equations have often been used in modeling certain physico chemical reactions 11 elimination of rapidly diffusing components in a system of diffusion coupling can also lead to effectively nonlocal coupling in resultant equations 12 cml s in which coupling between two sites separated by distance r decays as power law 1 r a with exponent a have also been considered as models for biological neural networks 6 in the present work we derive the analytic conditions for synchronization in our cml s we find that in the thermodynamic limit the relevant parameter for synchronization is the fraction of sites coupled and not their number we present numerical results which show that for the mean field in the cml s with a fixed range of coupling sites for each site the law of large numbers applies and the total normalized power in nonzero components of the power spectrum goes to zero in the thermodynamic limit this is in sharp contrast with the conjecture of sinha et al 2

canards in r3 abstract we give a geometric analysis of canard solutions in three dimensional singularly perturbed systems with a folded two dimensional critical manifold by analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold curve we classify these singularities called canard points as folded saddles folded nodes and folded saddle nodes we prove the existence of canard solutions in the case of the folded saddle we show the existence of canards in the folded node case provided a generic non resonance condition is satisfied and in a subcase of the folded saddle node the proof is based on the blow up method

kinetics of self induced aggregation of brownian particles non markovian and non gaussian features in this paper we have explored a model for self induced aggregation of brownian particles incorporating non markovian and non gaussian character of the associated random noise processes the time evolution of each individual is guided by an overdamped langevin equation of motion with a nonlocal drift arising out of the imbalance in the local distribution of the other individuals our simulation results show that colored noise enhances the tendency of cluster formation another observation is that the critical noise variance decreases at first with increase in noise correlation time followed by an increase after exhibiting a minimum furthermore in the long time limit the cluster number in the aggregation process exhibits depletion with time following a power law with an exponent which increases remarkably with non markovian character of the noise processes

empirical mode decomposition and synchrogram approach to cardiorespiratory synchronization we use the empirical mode decomposition method to decompose experimental respiratory signals into a set of intrinsic mode functions imfs and consider one of these imfs as a respiratory rhythm we then use the hilbert spectral analysis to calculate the instantaneous phase of the imf heartbeat data are finally incorporated to construct the cardiorespiratory synchrogram which is a visual tool for inspecting synchronization we perform analysis on 20 data sets collected by the harvard medical school from ten young 21 34 years old and ten elderly 68 81 years old rigorously screened healthy subjects our results support the existence of cardiorespiratory synchronization we also investigate the origin of the cardiorespiratory synchronization by addressing the problem of correlations between regularities of respiratory and cardiac signals our analysis shows that regularity of respiratory signals plays a dominant role in the cardiorespiratory synchronization

simplest miniversal deformations of matrices matrix pencils and contragredient matrix pencils abstract for a family of linear operators a u u over c that smoothly depend on parameters 1 k v i arnold obtained the simplest normal form of their matrices relative to a smoothly depending on change of a basis in u we solve the same problem for a family of linear operators a u u over r for a family of pairs of linear mappings a u v b u v over c and r and for a family of pairs of counter linear mappings a u v b v u over c and r

bifurcations of mixed mode oscillations in a stellate cell model experimental recordings of the membrane potential of stellate cells within the entorhinal cortex show a transition from subthreshold oscillations stos via mixed mode oscillations mmos to relaxation oscillations under increased injection of depolarizing current acker et al introduced a 7d conductance based model which reproduces many features of the oscillatory patterns observed in these experiments for the first time we present a comprehensive bifurcation analysis of this model by using the software package auto in particular we calculate the stable mmo branches within the bifurcation diagram of this model as well as other mmo patterns which are unstable we then use geometric singular perturbation theory to demonstrate how the bifurcations are governed by a 3d reduced model introduced by rotstein et al we extend their analysis to explain all observed mmo patterns within the bifurcation diagram a key role in this bifurcation analysis is played by a novel homoclinic bifurcation structure connecting to a saddle equilibrium on the unstable branch of the corresponding critical manifold this type of homoclinic connection is possible due to canards of folded node folded saddle node type

synchronization of weakly coupled canard oscillators synchronization has been studied extensively in the context of weakly coupled oscillators using the so called phase response curve prc which measures how a change of the phase of an oscillator is affected by a small perturbation this approach was based upon the work of malkin and it has been extended to relaxation oscillators namely synchronization conditions were established under the weak coupling assumption leading to a criterion for the existence of synchronous solutions of weakly coupled relaxation oscillators previous analysis relies on the fact that the slow nullcline does not intersect the fast nullcline near one of its fold points where canard solutions can arise in the present study we use numerical continuation techniques to solve the adjoint equations and we show that synchronization properties of canard cycles are different than those of classical relaxation cycles in particular we highlight a new special role of the maximal canard in separating two distinct synchronization regimes the hopf regime and the relaxation regime phase plane analysis of slow fast oscillators undergoing a canard explosion provides an explanation for this change of synchronization properties across the maximal canard

influence of noise on the synchronization of the stochastic kuramoto model we consider the kuramoto model of globally coupled phase oscillators subject to ornstein uhlenbeck and non gaussian colored noise and investigate the influence of noise on the order parameter of the synchronization process we use numerical methods to study the dependence of the threshold as well as the maximum degree of synchronization on the correlation time and the strength of the noise and find that the threshold of synchronization strongly depends on the nature of the noise it is found to be lower for both the ornstein uhlenbeck and non gaussian processes compared to the case of white noise a finite correlation time also favors the achievement of the full synchronization of the system in contract to the white noise process which does not allow that finally we discuss possible applications of the stochastic kuramoto model to oscillations taking place in biochemical systems

stokes efficiency and its stochastic properties abstract we study the stokes efficiency and its fluctuating properties in the case of a spatial asymmetric ratchet potential with a temporal asymmetric driving force from adiabatic to nonadiabatic regime our numerical investigations show that the average stokes efficiency and the average current decrease with increase in the frequency of driving for low frequency of driving i e in the case of an adiabatic regime we reproduced the analytical results supporting our numerical simulations by evaluating the probability distribution p s for stokes efficiency s we focus on the stochastic properties of stokes efficiency we find that in most of the parameter space fluctuations in s are comparable to or larger than the mean values in such a situation one has to study the nature of the full probability distribution of s we observe that with increase in frequency of driving the distribution becomes multipeaked at the same time the average stokes efficiency monotonically decreases with increase in frequency of the drive for high frequency of driving the distribution develops a peak at zero with further increase in frequency of the drive this peak gets sharper and finally at sufficiently high frequency we get a strong peak at zero indicating that there is no effective transport in this regime

miniversal deformations of matrices under congruence and reducing transformations arnold 1971 1 constructed a miniversal deformation of a square complex matrix under similarity that is a simple normal form to which not only a given square matrix a but all matrices b close to it can be reduced by similarity transformations that smoothly depend on the entries of b we give miniversal deformations of matrices of sesquilinear forms that is of square complex matrices under congruence and construct an analytic reducing transformation to a miniversal deformation analogous results for matrices under congruence were obtained by dmytryshyn futorny and sergeichuk 2012 11

sector delayed hopf type mixed mode oscillations in a prototypical three time scale model we consider a three dimensional three time scale system that was first proposed by krupa et al 2008 under the additional assumption that two singular perturbation parameters are present in the equations while the presence of three scales was shown to give rise to canard induced periodic mixed mode oscillations mmos desroches et al 2012 in the parameter regime studied by krupa et al 2008 we additionally observe mixed mode patterns that display delayed hopf type behaviour ne shtadt 1987 we present analytical and numerical evidence for the occurrence of stable periodic dynamics that realises both mechanisms and we discuss the transition between them to the best of our knowledge the resulting mixed sector delayed hopf type mmo trajectories represent a novel class of mixed mode dynamics in singularly perturbed systems of ordinary differential equations

periodic orbits and chaos in fast slow systems with bogdanov takens type fold points abstract the existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast slow type having bogdanov takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow up method in particular the blow up method is effectively used for analyzing the flow near the bogdanov takens type fold point in order to show that a slow manifold near the fold point is extended along the boutroux s tritronquee solution of the first painleve equation in the blow up space

collective synchronization in populations of globally coupled phase oscillators with drifting frequencies we generalize the kuramoto model for coupled phase oscillators by allowing the frequencies to drift in time according to ornstein uhlenbeck dynamics such drifting frequencies were recently measured in cellular populations of circadian oscillator and inspired our work linear stability analysis of the fokker planck equation for an infinite population is amenable to exact solution and we show that the incoherent state is unstable past a critical coupling strength k c gamma sigma f where gamma is the inverse characteristic drifting time and sigma f the asymptotic frequency dispersion expectedly k c agrees with the noisy kuramoto model in the large gamma schmolukowski limit but increases slower as gamma decreases asymptotic expansion of the solution for gamma 0 shows that the noiseless kuramoto model with gaussian frequency distribution is recovered in that limit thus varying a single parameter allows us to interpolate smoothly between two regimes one dominated by the frequency dispersion and the other by phase diffusion

analysis of a slow fast system near a cusp singularity this paper studies a slow fast system whose principal characteristic is that the slow manifold is given by the critical set of the cusp catastrophe our analysis consists of two main parts first we recall a formal normal form suitable for systems as the one studied here afterwards taking advantage of this normal form we investigate the transition near the cusp singularity by means of the blow up technique our contribution relies heavily in the usage of normal form theory allowing us to refine previous results

local analysis near a folded saddle node singularity abstract folded saddle nodes occur generically in one parameter families of singularly perturbed systems with two slow variables we show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems canards and delayed hopf bifurcations we combine techniques from geometric singular perturbation theory the blow up technique and from delayed hopf bifurcation theory complex time path analysis to analyze the flow near such folded saddle nodes in particular we show the existence of canards as intersections of stable and unstable slow manifolds to derive these canard results we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths this enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost furthermore we define a way in way out function describing the maximal delay expected for generic solutions passing through a folded saddle node singularity branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical slow manifold make our analysis significantly different from the classical delayed hopf bifurcation analysis where these eigenvalues are complex only

effective markovian approximation for non gaussian noises a path integral approach abstract we have analyzed diffusion in a double well potential driven by a colored non gaussian noise using a path integral approach we have obtained a consistent markovian approximation to the initially non markovian problem such an approximation allows us to get analytical expressions for the mean first passage time that has been tested against extensive numerical simulations

fisher information and criticality in the kuramoto model of nonidentical oscillators we use the fisher information to provide a lens on the transition to synchronization of the kuramoto model of nonidentical frequencies on a variety of undirected graphs we numerically solve the equations of motion for a n 400 complete graph and n 1000 small world scale free uniform random and random regular graphs for large but finite graphs of small average diameter the fisher information f as a function of coupling shows a peak closely coinciding with the critical point as determined by kuramoto s order parameter or synchronization measure r however for graphs of larger average diameter the position of the peak in f differs from the critical point determined by estimates of r on the one hand this is a finite size effect even at n 1000 however we show across a range of topologies that the fisher information peak points to a transition for smaller graphs that indicates structural changes in the numbers of locally phase synchronized clusters often directly from metastable to stable frequency synchronization solving explicitly for a two cluster ansatz subject to gaussian noise shows that the fisher infomation peaks at such a transition we discuss the implications for fisher information as an indicator for edge of chaos phenomena in finite coupled oscillator systems

stability measures in metastable states with gaussian colored noise we present a study of the escape time from a metastable state of an overdamped brownian particle in the presence of colored noise generated by ornstein uhlenbeck process we analyze the role of the correlation time on the enhancement of the mean first passage time through a potential barrier and on the behavior of the mean growth rate coefficient as a function of the noise intensity we observe the noise enhanced stability effect for all the initial unstable states used and for all values of the correlation time tau c investigated we can distinguish two dynamical regimes characterized by weak and strong correlated noises depending on the value of tau c with respect to the relaxation time of the system

perturbation theory of matrix pencils through miniversal deformations abstract we give new proofs of several known results about perturbations of matrix pencils in particular we give a direct and constructive proof of andrzej pokrzywa s theorem 1983 in which the closure of the orbit of each kronecker canonical matrix pencil is described in terms of inequalities for invariants of matrix pencils a more abstract description is given by klaus bongartz 1996 by methods of representation theory we formulate and prove pokrzywa s theorem in terms of successive replacements of direct summands in a kronecker canonical pencil first we show that it is sufficient to prove pokrzywa s theorem in two cases for matrices under similarity and for each matrix pencil p q that is a direct sum of two indecomposable pencils then we calculate the kronecker canonical form of pencils that are close to p q in fact the kronecker canonical form is calculated for only those pencils that belong to a miniversal deformation of p q this is sufficient since all pencils in a neighborhood of p q are reduced to them by a smooth strict equivalence transformation

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

spike adding in parabolic bursters the role of folded saddle canards abstract the present work develops a new approach to studying parabolic bursting and also proposes a novel four dimensional canonical and polynomial based parabolic burster in addition to this new polynomial system we also consider the conductance based model of the aplysia r15 neuron known as the plant model and a reduction of this prototypical biophysical parabolic burster to three variables including one phase variable namely the baer rinzel carillo brc phase model revisiting these models from the perspective of slow fast dynamics reveals that the number of spikes per burst may vary upon parameter changes however the spike adding process occurs in an explosive fashion that involves special solutions called canards this spike adding canard explosion phenomenon is analysed by using tools from geometric singular perturbation theory in tandem with numerical bifurcation techniques we find that the bifurcation structure persists across all considered systems that is spikes within the burst are incremented via the crossing of an excitability threshold given by a particular type of canard orbit namely the true canard of a folded saddle singularity however there can be a difference in the spike adding transitions in parameter space from one case to another according to whether the process is continuous or discontinuous which depends upon the geometry of the folded saddle canard using these findings we construct a new polynomial approximation of the plant model which retains all the key elements for parabolic bursting including the spike adding transitions mediated by folded saddle canards finally we briefly investigate the presence of spike adding via canards in planar phase models of parabolic bursting namely the theta model by ermentrout and kopell

canonical matrices for linear matrix problems abstract we consider a large class of matrix problems which includes the problem of classifying arbitrary systems of linear mappings for every matrix problem from this class we construct belitskii s algorithm for reducing a matrix to a canonical form which is the generalization of the jordan normal form and study the set c mn of indecomposable canonical m n matrices considering c mn as a subset in the affine space of m n matrices we prove that either c mn consists of a finite number of points and straight lines for every m n or c mn contains a 2 dimensional plane for a certain m n

relaxation oscillations in r3 abstract the existence of periodic relaxation oscillations in singularly perturbed systems with two slow and one fast variable is analyzed geometrically it is shown that near a singular periodic orbit a return map can be defined which has a one dimensional slow manifold with a stable invariant foliation under a natural hyperbolicity assumption on the singular periodic orbit this allows to prove the existence of a periodic relaxation orbit for small values of the perturbation parameter additionally the existence of an invariant torus is proved for the periodically forced van der pol oscillator the analysis is based on methods from geometric singular perturbation theory the blow up method is used to analyze the dynamics near the fold curves

canards heteroclinic and homoclinic orbits for a slow fast predator prey model of generalized holling type iii abstract for a classical ratio dependent predator prey model with the generalized holling type iii functional response it was previously investigated in 20 by hsu and huang for global stability of an equilibrium and in 21 by huang ruan and song for subcritical hopf and bogdanov takens bifurcations here in this model when prey reproduces much faster than predator by using geometric singular perturbation theory we achieve much richer new dynamical phenomena than the existing ones such as the existence of canard cycles canard explosion and relaxation oscillations heteroclinic and homoclinic orbits cyclicity of slow fast cycles and the coexistence of the hopf cycle and the relaxation oscillation on global stability of the equilibrium we also provide less restricted conditions than the existing ones

synchronized clusters in coupled map networks i numerical studies we study the synchronization of coupled maps on a variety of networks including regular one and two dimensional networks scale free networks small world networks tree networks and random networks for small coupling strengths nodes show turbulent behavior but form phase synchronized clusters as coupling increases when nodes show synchronized behavior we observe two interesting phenomena first there are some nodes of the floating type that show intermittent behavior between getting attached to some clusters and evolving independently second we identify two different ways of cluster formation namely self organized clusters which have mostly intracluster couplings and driven clusters which have mostly intercluster couplings the synchronized clusters may be of dominant self organized type dominant driven type or mixed type depending on the type of network and the parameters of the dynamics we define different states of the coupled dynamics by considering the number and type of synchronized clusters for the local dynamics governed by the logistic map we study the phase diagram in the plane of the coupling constant and the logistic map parameter ensuremath mu for large coupling strengths and nonlinear coupling we find that the scale free networks and the caley tree networks lead to better cluster formation than the other types of networks with the same average connectivity for most of our study we use the number of connections of the order of the number of nodes as the number of connections increases the number of nodes forming clusters and the size of the clusters in general increase

escape through an unstable limit cycle driven by multiplicative colored non gaussian and additive white gaussian noises in a previous paper bag and hu phys rev e 73 061107 2006 we studied the mean lifetime mlt for the escape of a brownian particle through an unstable limit cycle driven by multiplicative colored gaussian and additive gaussian white noises and found resonant activation ra behavior in the present paper we switch from gaussian to non gaussian multiplicative colored noise we find that in the ra phenomenon the minimum appears at a smaller noise correlation time tau for non gaussian noises compared to gaussian noises in the plot of mlt vs tau for a fixed noise variance the same plot for a given noise strength increases linearly and the increasing rate is smaller for non gaussian noises than for the gaussian noises the plot of logarithm of inverse of mlt vs inverse of the strength of additive noise is arrhenius like for gaussian colored noise and it becomes similar to the quantum kramers rate if the multiplicative noise is non gaussian

transformation to versal deformations of matrices abstract in the paper versal deformations of matrices are considered the versal deformation is a matrix family inducing an arbitrary multi parameter deformation of a given matrix by an appropriate smooth change of parameters and basis given a deformation of a matrix it is suggested to find transformation functions the change of parameters and the change of basis dependent on parameters in the form of taylor series the general method of construction of recurrent procedures for calculation of coefficients in the taylor expansions is developed and used for the cases of real and complex matrices elements of classical lie and jordan algebras and infinitesimally reversible matrices several examples are given and studied in detail applications of the suggested approach to problems of stability singularity and perturbation theories are discussed

singular hopf bifurcations and mixed mode oscillations in a two cell inhibitory neural network abstract recent studies of a firing rate model for neural competition as observed in binocular rivalry and central pattern generators r curtu a shpiro n rubin j rinzel mechanisms for frequency control in neuronal competition models siam j appl dyn syst 7 2 2008 609 649 showed that the variation of the stimulus strength parameter can lead to rich and interesting dynamics several types of behavior were identified such as fusion equivalent to a steady state of identical activity levels for both neural units oscillations due to either an escape or a release mechanism and a winner take all state of bistability the model consists of two neural populations interacting through reciprocal inhibition each endowed with a slow negative feedback process in the form of spike frequency adaptation in this paper we report the occurrence of another complex oscillatory pattern the mixed mode oscillations mmos they exist in the model at the transition between the relaxation oscillator dynamical regime and the winner take all regime the system distinguishes itself from other neuronal models where mmos were found by the following interesting feature there is no autocatalysis involved as in the examples of voltage gated persistent inward currents and or intrapopulation recurrent excitation and therefore the two cells in the network are not intrinsic oscillators the oscillations are instead a combined result of the mutual inhibition and the adaptation we prove that the mmos are due to a singular hopf bifurcation point situated in close distance to the transition point to the winner take all case we also show that in the vicinity of the singular hopf other types of bifurcations exist and we construct numerically the corresponding diagrams

miniversal deformations of matrices of bilinear forms arnold v i arnold on matrices depending on parameters russian math surveys 26 2 1971 29 43 constructed miniversal deformations of square complex matrices under similarity that is a simp

paths to globally generalized synchronization in scale free networks we apply the auxiliary system approach to study paths to globally generalized synchronization in scale free networks of identical chaotic oscillators including h enon maps logistic maps and lorentz oscillators as the coupling strength ensuremath epsilon between nodes of the network is increased transitions from partially to globally generalized synchronization and intermittent behaviors near the synchronization thresholds are found the generalized synchronization starts from the hubs of the network and then spreads throughout the whole network with the increase of ensuremath epsilon our result is useful for understanding the synchronization process in complex networks

asymptotic analysis of subcritical hopf homoclinic bifurcation abstract this paper discusses the mathematical analysis of a codimension two bifurcation determined by the coincidence of a subcritical hopf bifurcation with a homoclinic orbit of the hopf equilibrium our work is motivated by our previous analysis of a hodgkin huxley neuron model which possesses a subcritical hopf bifurcation j guckenheimer r harris warrick j peck a willms j comput neurosci 4 1997 257 277 in this model the hopf bifurcation has the additional feature that trajectories beginning near the unstable manifold of the equilibrium point return to pass through a small neighborhood of the equilibrium that is the hopf bifurcation appears to be close to a homoclinic bifurcation as well this model of the lateral pyloric lp cell of the lobster stomatogastric ganglion was analyzed for its ability to explain the phenomenon of spike frequency adaptation in which the time intervals between successive spikes grow longer until the cell eventually becomes quiescent the presence of a subcritical hopf bifurcation in this model was the one identified mechanism for oscillatory trajectories to increase their period and finally collapse to a non oscillatory solution the analysis presented here explains the apparent proximity of homoclinic and hopf bifurcations we also develop an asymptotic theory for the scaling properties of the interspike intervals in a singularly perturbed system undergoing subcritical hopf bifurcation that may be close to a codimension two subcritical hopf homoclinic bifurcation

gaussian noise and the two network frustrated kuramoto model abstract we examine analytically and numerically a variant of the stochastic kuramoto model for phase oscillators coupled on a general network two populations of phased oscillators are considered labelled blue and red each with their respective networks internal and external couplings natural frequencies and frustration parameters in the dynamical interactions of the phases we disentangle the different ways that additive gaussian noise may influence the dynamics by applying it separately on zero modes or normal modes corresponding to a laplacian decomposition for the sub graphs for blue and red under the linearisation ansatz that the oscillators of each respective network remain relatively phase synchronised centroids or clusters we are able to obtain simple closed form expressions using the fokker planck approach for the dynamics of the average angle of the two centroids in some cases this leads to subtle effects of metastability that we may analytically describe using the theory of ratchet potentials these considerations are extended to a regime where one of the populations has fragmented in two the analytic expressions we derive largely predict the dynamics of the non linear system seen in numerical simulation in particular we find that noise acting on a more tightly coupled population allows for improved synchronisation of the other population where deterministically it is fragmented

remarks on lipschitz properties of matrix groups actions it is proved that a large class of matrix group actions including joint similarity and congruence like actions as well as actions of the type of matrix equivalence have local lipschitz property under additional hypotheses global lipschitz property is proved these results are specialized and applied to obtain local lipschitz property of canonical bases of matrices that are selfadjoint in an indefinite inner product real complex and quaternionic matrices are considered

mixed mode oscillations with multiple time scales mixed mode oscillations mmos are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes mmos have been observed and studied for over thirty years in chemical physical and biological systems few attempts have been made thus far to classify different patterns of mmos in contrast to the classification of the related phenomena of bursting oscillations this paper gives a survey of different types of mmos concentrating its analysis on mmos whose small amplitude oscillations are produced by a local multiple time scale mechanism recent work gives substantially improved insight into the mathematical properties of these mechanisms in this survey we unify diverse observations about mmos and establish a systematic framework for studying their properties numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper

synchronisation of networked kuramoto oscillators under stable levy noise abstract we study the kuramoto model on several classes of network topologies examining the dynamics under the influence of levy noise such noise exhibits heavier tails than gaussian and allows us to understand how shocks influence the individual oscillator and collective system behaviour skewed stable levy noise equivalent to fractional diffusion perturbations are considered we perform numerical simulations for erd s renyi er and barabasi albert ba scale free networks of size n 1000 while varying the levy index for the noise we find that synchrony now assumes a surprising variety of forms not seen for gaussian type noise and changing with a noise generated drift a smooth dependence of the point of cross over of er and ba networks in the degree of synchronisation and a severe loss of synchronisation at low values of we also show that this robustness of the ba network across most values of can also be understood as a consequence of the laplacian of the graph working within the fractional fokker planck equation of the linearised system close to synchrony with both eigenvalues and eigenvectors alternately contributing in different regimes of

sungular hopf bifurcation to relaxation oscillations this paper is a continuation of our study of singular hopf bifurcations to relaxation oscillations siam journal on applied mathematics 46 1986 pp 721 739 in our previous analysis a system of two first order nonlinear equatins is investigated and it is shown how the bifurcation problem can be reduced to a weakly perturbed conservative system of two first order equations in this paper a method based on analyzing nearly conservative quantities was developed to find the bifurcation diagram of the periodic solutions emerging from singular hopf bifurcation points specifically the subcritical hopf bifurcation of the fitzhugh nagumo equations is analyzed the extension from the earlier bifurcation analysis is achieved by performing a perturbation analysis from the first integral of the leading order nonlinear oscillator at a critical value of a parameter the first integral exists up to an additional order in the perturbed equation this condition permits the resolution of features determined by h

escape through an unstable limit cycle resonant activation we consider a brownian particle acted on by a linear conservative force a nonlinear frictional force and multiplicative colored and additive white noises the frictional force can be negative when the external energy supply is large enough we numerically calculate the mean first passage time mfpt for the particle to escape from an unstable limit cycle and find resonant activation i e the mfpt first decreases followed by a rise after passing through a minimum with increasing noise correlation time tau for a fixed noise variance for fixed noise strength of the multiplicative noise the mfpt increases linearly with tau this is in sharp contrast to the case of fluctuations of nonlinear potentials in which the mfpt first increases nonlinearly before reaching a limiting value our model could be useful for understanding some biological processes