scaling variables and asymptotic expansions in damped wave equations abstract we study the long time behavior of small solutions to the nonlinear damped wave equationeu u a u n u u u r 0 whereeis a positive not necessarily small parameter we assume that the diffusion coefficienta converges to positive limitsa as and that the nonlinearity n u u u vanishes sufficiently fast asu 0 introducing scaling variables and using various energy estimates we compute an asymptotic expansion of the solutionu in powers of 1 2as and we show that this expansion is entirely determined up to the second order by a linear parabolic equation which depends only on the limiting valuesa in particular this implies that the small solutions of the damped wave equation behave for large like those of the parabolic equation obtained by settinge 0

lp and bmo bounds of weighted hardy littlewood averages abstract this paper characterizes those nonnegative functions defined on 0 1 for which the weighted hardy littlewood average u f 0 1 f t t dt is bounded on either l p r n p 1 or bmo r n meanwhile the corresponding operator norms are worked out

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

noise induced instability an approach based on higher order moments noise induced transitions in the organization of systems far from equilibrium have been of vital interest although the effects of additive and multiplicative noise have been widely studied it is only the multiplicative noise that can be dealt with within the scope of a linear analysis of first moments of the spatiotemporal perturbations by the application of novikov s theorem for the case of additive noise the corresponding straightforward linear analysis of the first moment throws no light on the effect of the noise on stability conditions we propose here a simple approach based on higher order moments to show how additive noise can give rise to noise induced instability in spatially extended systems at times leading to pattern formation our theoretical analysis is corroborated by numerical simulations on two simple one component reaction diffusion systems in two dimensions

the norm inequalities for the weighted cesaro mean operators this paper gives some necessary and sufficient conditions for the weighted cesaro mean operators to be bounded on herz spaces the corresponding new operator norm inequalities are obtained

periodic solutions of abstract functional differential equations with infinite delay abstract we characterize the existence of periodic solutions for a class of abstract retarded functional differential equations with infinite delay we apply our results to the nonlinear equation x t a x t l x t n x t f t t r where a d a x x is a closed operator defined on a banach space x l is a bounded linear map n b x is a continuous function defined on an appropriate phase space b and f l p t x

classification and unfoldings of degenerate hopf bifurcations abstract this paper initiates the classification up to symmetry covariant contact equivalence of perturbations of local hopf bifurcation problems which do not satisfy the classical non degeneracy conditions the only remaining hypothesis is that i should be simple eigenvalues of the linearized right hand side at criticality then the lyapunov schmidt method allows a reduction to a scalar equation g x 0 where g x g x a definition is given of the codimension of g and a complete classification is obtained for all problems with codimension 3 together with the corresponding universal unfoldings the perturbed bifurcation diagrams are given for the cases with codimension 2 and for one case with codimension 3 for this last case one of the unfolding parameters is a modal parameter such that the topological codimension equals in fact 2 formulas are given for the calculation of the taylor coefficients needed for the application of the results and finally the results are applied to two simple problems a model of glycolytic oscillations and the fitzhugh nerve equations

fourier multipliers and periodic solutions of delay equations in banach spaces abstract in this paper we characterize the existence and uniqueness of periodic solutions of inhomogeneous abstract delay equations and establish maximal regularity results for strong solutions the conditions are obtained in terms of r boundedness of linear operators determined by the equations and l p fourier multipliers periodic mild solutions are also studied and characterized

finite dimensional dynamical system modeling thermal instabilities we describe a three dimensional dynamical system which is obtained as a pseudo spectral approximation to a free boundary problem modeling solid combustion and rapid solidification and is capable of generating its major dynamical patterns these patterns include a hopf bifurcation followed by a sequence of secondary period doubling and a transition to chaos reverse sequences and sequences followed by shilnikov type trajectories a computer assisted bifurcation analysis uncovers some novel mechanisms of stability exchange the most striking of them is an infinite period bifurcation which resembles the classical shilnikov bifurcation but instead of a funnel shaped spiral along which the period is continually increasing the continuation produced a series of isolas each isola is a closed branch of solutions of roughly the same period and with the same number of oscillations the isolas corresponding to consecutive numbers of low amplitude oscillations about the equilibrium are adjacent to each other and appear to accumulate on a saddle focus homoclinic connection of shilnikov type 2000 elsevier science b v all rights reserved

time delay effects on coupled limit cycle oscillators at hopf bifurcation abstract we present a detailed study of the effect of time delay on the collective dynamics of coupled limit cycle oscillators at hopf bifurcation for a simple model consisting of just two oscillators with a time delayed coupling the bifurcation diagram obtained by numerical and analytical solutions shows significant changes in the stability boundaries of the amplitude death phase locked and incoherent regions a novel result is the occurrence of amplitude death even in the absence of a frequency mismatch between the two oscillators similar results are obtained for an array of n oscillators with a delayed mean field coupling and the regions of such amplitude death in the parameter space of the coupling strength and time delay are quantified some general analytic results for the n thermodynamic limit are also obtained and the implications of the time delay effects for physical applications are discussed

about riesz transforms on the heisenberg groups see also m1 subsequently several other proofs have been given of probabilistic b ba see also g v or analytic d r p nature also p a meyer has given in m2 a similar result in the gaussian setting the aim of this work is to prove analogous inequalities for the family of heisenberg groups h n instead of the family of euclidean spaces r n the euclidean proofs do not adapt to the non commutative situation of h however our method is reminiscent of some ideas of the proofs by duoandikoetxearubio de francia d r and pisier p a new feature is however the central r6te played by the heat kernel and our proof heavily relies on its explicit expression in h n given by gavean g and hulanicki h let us recall finally that apart from any question of uniformity the boundedness of the riesz transforms on every nilpotent lie group is a well known issue see l v

attractors in a sbil nikov hopf scenario and a related one dimensional map abstract we study a situation close to the sil nikov hopf bifurcation placing special emphasis on the attractors a model is proposed for the return map to a suitable surface which explains some numerical results concerning a particular physical system the model is quite general and can give reasonable explanations to many other examples later on we introduce a circle map as an additional simplification if the return map is strongly dissipative near the attractor this circle map has very interesting properties it plays a role similar to the logistic map in the case of maps on the interval however it doesn t have the disadvantage of escapes to infinity produced in the logistic map when the parameter is large enough

a period adding phenomenon a simple geometrical explanation is given for the period adding phenomenon observed in the 1927 experiment of van der pol and van der mark and refined recently by chua and kennedy the problem is reduced to a simple circle map in the second part of the paper some previous results are outlined to show how the same phenomenon arises in a class of van der pol type systems with periodic forcing and in conclusion some recent topological results are applied to such systems

phase locked patterns and amplitude death in a ring of delay coupled limit cycle oscillators we study the existence and stability of phase locked patterns and amplitude death states in a closed chain of delay coupled identical limit cycle oscillators that are near a supercritical hopf bifurcation the coupling is limited to nearest neighbors and is linear we analyze a model set of discrete dynamical equations using the method of plane waves the resultant dispersion relation which is valid for any arbitrary number of oscillators displays important differences from similar relations obtained from continuum models we discuss the general characteristics of the equilibrium states including their dependencies on various system parameters we next carry out a detailed linear stability investigation of these states in order to delineate their actual existence regions and to determine their parametric dependence on time delay time delay is found to expand the range of possible phase locked patterns and to contribute favorably toward their stability the amplitude death state is studied in the parameter space of time delay and coupling strength it is shown that death island regions can exist for any number of oscillators n in the presence of finite time delay a particularly interesting result is that the size of an island is independent of n when n is even but is a decreasing function of n when n is odd

multirhythmicity generated by slow variable diffusion in a ring of relaxation oscillators and noise induced abnormal interspike variability the deterministic and noise dependent dynamics of a ring of three ohmically coupled electronic relaxation oscillators are considered by means of numerical simulations each isolated oscillator is described by a set of two ordinary differential equations with very different characteristic times the emergence of the limit cycle via the hopf bifurcation results from the n shaped current versus voltage characteristic of the nonlinear resistor the phase diagram is calculated for a ring of three such oscillators in the presence of small detuning special attention is focused on two parameter areas one near a transition to the homogeneous and the other near the inhomogeneous stable steady state along with other nontrivial limit cycles essentially asymmetrical limit cycles termed dynamic traps may arise in these two areas a dynamic trap is a regime in which one or two oscillators do not perform full amplitude oscillations and correspondingly do not generate spikes the interspike interval isi distribution in the presence of noise is calculated as a function of the coupling strength in both areas of the parameter plane the distributions are extremely polymodal near the homogeneous steady state even if the in phase limit cycle is dominating the origins of this abnormal enhancement of isi variability are discussed in detail a similar analysis shows that nontrivial periodic attractors are observable in the vicinity of the inhomogeneous stable steady states only if the level of noise is relatively low in this case the dominance of the in phase limit cycle basin results in an almost unimodal distribution of interspike intervals

hausdorff operators on the power weighted hardy spaces abstract in this paper we study the high dimensional hausdorff operators on the power weighted hardy spaces we establish their sharp boundedness on power weighted hardy spaces and local hardy spaces particularly our results solve an open question posted by liflyand in 16

reaction cattaneo systems with fluctuating relaxation time a reaction cattaneo equation with fluctuating relaxation time of the diffusive flux has been explored it has been shown that depending on the strength of fluctuations the dynamical system exhibits new oscillatory solutions as a result of hopf and double hopf bifurcations leading to spatiotemporal patterns this analysis has been applied to two model nonlinear systems

complex mixed mode oscillations in a bonhoeffer van der pol oscillator under weak periodic perturbation abstract in this paper we elucidate the extremely complicated bifurcation structure of a weakly driven relaxation oscillator by focusing on chaos and notably on complex mixed mode oscillations mmos generated in a simple dynamical model our model uses the bonhoeffer van der pol bvp oscillator subjected to a weak periodic perturbation near a subcritical andronov hopf bifurcation ahb the mechanisms underlying the chaotic dynamics can be explained using an approximate one dimensional map the mmos that appear in this forced dynamical model may be more sophisticated than those appearing in three variable slow fast autonomous dynamics because the approximate one dimensional mapping of the dynamics used herein is a circle map whereas the one dimensional first return map that is derived from the three variable slow fast autonomous dynamics is usually a unimodal map in this study we generate novel bifurcations such as an mmo incrementing bifurcation mmoib and intermittently chaotic mmos mmoibs trigger an mmo sequence that upon varying a parameter is followed by another type of mmo sequence by constructing a two parameter bifurcation diagram we confirmed that mmoibs occur successively according to our numerical results mmoibs are often observed between two neighboring mmos numerically mmoibs may occur as many times as desired we also derive the universal constant of the associated successive mmoibs the existence of the universal constant suggests that mmoibs could occur infinitely many times furthermore intermittently chaotic mmos appear in this dynamical circuit the intermittently chaotic mmos relate to a type of intermittent chaos that resembles mmos at first glance but includes rare bursts over a long time interval complex intermittently chaotic mmos of various types are observed and we clarify that the intermittently chaotic mmos are generated by crisis induced intermittency

voltage interval mappings for activity transitions in neuron models for elliptic bursters abstract we performed a thorough bifurcation analysis of a mathematical elliptic bursting model using a computer assisted reduction to equationless one dimensional poincare mappings for a voltage interval using the interval mappings we were able to examine in detail the bifurcations that underlie the complex activity transitions between tonic spiking and bursting bursting and mixed mode oscillations and finally mixed mode oscillations and quiescence in the fitzhugh nagumo rinzel model we illustrate the wealth of information qualitative and quantitative that was derived from the poincare mappings for the neuronal models and for similar electro chemical systems

estimates of weighted hardy littlewood averages on the p adic vector space abstract in the p adic vector space q p n we characterize those non negative functions defined on z p w q p 0 w p 1 for which the weighted hardy littlewood average u f z p f t t d t is bounded on l r q p n 1 r and on bmo q p n also in each case we find the corresponding operator norm u

the slow passage through a hopf bifurcation delay memory effects and resonance this paper explores analytically and numerically in the context of the fitzhugh nagumo model of nerve membrane excitability an interesting phenomenon that has been described as a delay or memory effect it can occur when a parameter passes slowly through a hopf bifurcation point and the system s response changes from a slowly varying steady state to slowly varying oscillations on quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects the dynamic aspect of the parameter variation this delay and its dependence on the speed of the parameter variation are described the model involves several parameters and particular singular limits are investigated one in particular is the slow passage through a low frequency hopf bifurcation where the system s response changes from a slowly varying steady state to slowly varying relaxation oscillations we find in this case the onset o

boundedness of multidimensional hausdorff operators on h1 rn abstract in this paper we consider the h 1 boundedness of multidimensional hausdorff operators defined by h a f x r n u f a u x d u where l l o c 1 r n a u a i j u i j 1 n is an n n matrix and each a i j u is a measurable function of u let b i j 1 n b i j for the matrix b b i j u i j 1 n we prove that h a is bounded from the hardy space h 1 to itself if r n u det a 1 u ln 1 a 1 u n det a 1 u d u our result improves known results in addition we show that the above condition is optimal in the size condition

bifurcations of mixed mode oscillations in a three variable autonomous van der pol duffing model with a cross shaped phase diagram abstract the bifurcation structure of a three variable van der pol duffing type model is studied in some detail with special attention to the mixed mode solutions a type of complex periodic behavior frequently encountered in oscillating chemical reactions the mixed mode oscillations in the model occur close to two hopf bifurcations which are arranged with the saddle node bifurcations in a so called cross shaped phase diagram a bifurcation diagram also typical for chemical reactions the mixed mode oscillations are shown to lie on isolated bifurcation curves which are all born in a single codimension two bifurcation known as the neutrally twisted homoclinic orbit or inclination switch with the introduction of an additional slow time scale the same model can exhibit more complex mixed mode oscillations and torus bifurcations

multimodal regimes in a compartmental model of the dopamine neuron we study chains of strongly electrically coupled relaxation oscillators modeling dopamine neurons when individual oscillators are in the regime close to an andronov hopf bifurcation ahb the coupled system exhibits a variety of oscillatory behavior we show that the proximity of individual oscillators to the ahb has a significant impact on the system dynamics in a wide range of parameters it manifests itself through a family of stable multimodal periodic solutions that are composed out of large amplitude relaxation oscillations and small amplitude oscillations this family of solutions has a rich bifurcation structure the waveform and the period vary greatly across the family the structure and bifurcations of the stable periodic solutions of the coupled system are investigated using numerical and analytic techniques

the p adic weighted hardy cesaro operator and an application to discrete hardy inequalities abstract this paper aims to investigate the boundedness of the p adic analog of the weighted hardy cesaro operator u s f z p f s t t d t on weighted lebesgue spaces and weighted bmo spaces in each case we obtain the corresponding operator norms u s in particular these results have a surprising relevance to discrete hardy inequalities on the real field we prove a reverse bmo hardy inequality and give a necessary condition on so that the commutator of u s is bounded on l r q p n with symbols in bmo q p n

multivariate hausdorff operators on the spaces lp rn abstract the operators h indicated in the title are characterized by a finite borel measure on r n a borel measurable function c on r n and an n n a e nonsingular matrix a whose entries are also borel measurable functions on r n and h f is defined by means of a lebesgue stieltjes integral with respect to we give simple sufficient conditions in order that these operators be bounded on the lebesgue spaces l p r n for some 1 p these sufficient conditions are exact even in the well known special cases of the cesaro and copson operators we also determine the hausdorff operator h which is adjoint to h in a certain sense we reveal interrelations among these operators and the fourier transform of a function f in l 1 r n on closing we present further special hausdorff operators

time delay autosynchronization of the spatiotemporal dynamics in resonant tunneling diodes the double barrier resonant tunneling diode exhibits complex spatiotemporal patterns including low dimensional chaos when operated in an active external circuit we demonstrate how autosynchronization by time delayed feedback control can be used to select and stabilize specific current density patterns in a noninvasive way we compare the efficiency of different control schemes involving feedback in either local spatial or global degrees of freedom the numerically obtained floquet exponents are explained by analytical results from linear stability analysis

exact treatment of mode locking for a piecewise linear map a piecewise linear map with one discontinuity is studied by analytic means in the two dimensional parameter space when the slope of the map is less than unity periodic orbits are present and we give the precise symbolic dynamic classification of these the localization of the periodic domains in parameter space is given by closed expressions the winding number forms a devil s terrace a two dimensional function whose cross sections are complete devils s staircases in such a cross section the complementary set to the periodic intervals is a cantor set with dimensiond 0

periodic solution and almost periodic solution for a nonautonomous lotka volterra dispersal system with infinite delay abstract this paper studies a nonautonomous lotka volterra dispersal systems with infinite time delay which models the diffusion of a single species into n patches by discrete dispersal our results show that the system is uniformly persistent under an appropriate condition the sufficient condition for the global asymptotical stability of the system is also given by using mawhin continuation theorem of coincidence degree we prove that the periodic system has at least one positive periodic solution further obtain the uniqueness and globally asymptotical stability for periodic system by using functional hull theory and directly analyzing the right functional of almost periodic system we show that the almost periodic system has a unique globally asymptotical stable positive almost periodic solution we also show that the delays have very important effects on the dynamic behaviors of the system

quasiconformal mappings on the heisenberg group on etudie les applications conformes et quasi conformes sur le groupe de heisenberg la metrique invariante a gauche naturelle sur cet espace est non isotrope sous des dilatations et ne peut se decrire par une metrique de riemann ceci engendre de nouveaux phenomenes

multimodal oscillations in systems with strong contraction one and two parameter families of flows in r 3 near an andronov hopf bifurcation ahb are investigated in this work we identify conditions on the global vector field which yield a rich family of multimodal orbits passing close to a weakly unstable saddle focus and perform a detailed asymptotic analysis of the trajectories in the vicinity of the saddle focus our analysis covers both cases of sub and supercritical ahb for the supercritical case we find that the periodic orbits born from the ahb are bimodal when viewed in the frame of coordinates generated by the linearization about the bifurcating equilibrium if the ahb is subcritical it is accompanied by the appearance of multimodal orbits which consist of long series of nearly harmonic oscillations separated by large amplitude spikes we analyze the dependence of the interspike intervals which can be extremely long on the control parameters in particular we show that the interspike intervals grow logarithmically as the boundary between regions of sub and supercritical ahb is approached in the parameter space we also identify a window of complex and possibly chaotic oscillations near the boundary between the regions of sub and supercritical ahb and explain the mechanism generating these oscillations this work is motivated by the numerical results for a finite dimensional approximation of a free boundary problem modeling solid fuel combustion c 2007 elsevier b v all rights reserved

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

synchronization in light controlled oscillators we have built light controlled oscillators lcos that mimic gregarious fireflies in the sense that their interactions are episodic and almost pulse like the ability of the lcos to synchronize their light emission constitutes a good experimental setup to validate different models of populations of integrate and fire oscillators and to analyze the role of the interactions and the spatial distribution of the lcos experimental measurements on two and three interacting lcos enable us to find synchronization ranges despite intrinsic differences among the oscillators we develop a mathematical model that we have used to solve analytically the simplified case of two identical oscillators in which we have found synchronous and anti synchronous states we have constructed the phase response curve for an lco finally we solved the model numerically finding that it reproduces our different experimental results with two lcos master slave interaction and mutual interaction and three interacting lcos in a linear configuration

delayed feedback control of chaos bifurcation analysis we study the effect of time delayed feedback control in the form proposed by pyragas on deterministic chaos in the rossler system we reveal the general bifurcation diagram in the parameter plane of time delay and feedback strength k which allows one to explain the phenomena that have been discovered in some previous works we show that the bifurcation diagram has essentially a multi leaf structure that constitutes multistability the larger the time delay the larger the number of attractors that can coexist in the phase space feedback induces a large variety of regimes non existent in the original system among them tori and chaotic attractors born from them finally we estimate how the parameters of delayed feedback influence the periods of limit cycles in the system

amplitude response of coupled oscillators abstract we investigate the interaction of a pair of weakly nonlinear oscillators eg each near a hopf bifurcation when the coupling strength is comparable to the attraction of the limit cycles changes in amplitude cannot then be ignored and there are new phenomena we show that even for simple forms of these equations there are parameter regimes in which the interaction causes the system to stop oscillating and the rest state at zero stabilized by the interaction is the only stable solution when the uncoupled oscillators have a local frequency that is independent of amplitude zero shear and the coupling is scalar we give a complete description of the behavior we then analyze more complicated equations to show the effects of amplitude dependent frequency in the uncoupled equations nonzero shear and nonscalar coupling for example when there is shear there can be bistability between a phase locked solution and an unlocked drift solution in which the oscillators go at different average frequencies there can also be bistability between a phase locked solution and an equilibrium nonoscillatory solution the equations for the zero shear case have a pair of degenerate bifurcation points and the new phenomena in the nonzero shear case arise from a partial unfolding of these singularities when the coupling is nonscalar there are a variety of new behaviors including bistability of an in phase and an anti phase solution for two identical oscillators parameters for which only the anti phase solution is stable and phase trapping i e nonlocked solutions in which the oscillators still have the same average frequency it is shown that whether the coupling is diffusive or direct has many effects on the behavior of the system

delay induced multistable synchronization of biological oscillators we analyze the dynamics of pulse coupled oscillators depending on strength and delay of the interaction for two oscillators we derive return maps for subsequent phase differences and construct phase diagrams for a broad range of parameters in phase synchronization proves stable for inhibitory coupling and unstable for excitatory coupling if the delay is not zero if the coupling strength is high additional regimes with marginally stable synchronization are found simulations with n 2 oscillators reveal a complex dynamics including spontaneous synchronization and desynchronization with excitatory coupling and multistable phase clustering with inhibitory coupling we simulate a continuous description of the system for n oscillators and demonstrate that these phenomena are independent of the size of the system phase clustering is shown to relate to stability and basins of attraction of fixed points in the return map of two oscillators our findings are generic in the sense that they qualitatively are robust with respect to modeling details we demonstrate this using also pulses of finite rise time and the more realistic model by hodgkin and huxley which exhibits multistable synchronization as predicted from our analysis as well s1063 651x 98 09902 4

analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies a population of oscillatory hodgkin huxley hh model neurons is shown numerically to exhibit a behavior in which the introduction of excitatory synaptic coupling synchronizes and dramatically slows firing this effect contrasts with the standard theory that recurrent synaptic excitation promotes states of rapid sustained activity independent of intrinsic neuronal dynamics the observed behavior is not due to simple depolarization block nor to standard elliptic bursting although it is related to these phenomena we analyze this effect using a reduced model for a single self coupled hh oscillator the mechanism explained here involves an extreme form of delayed bifurcation in which the development of a vortex structure through interaction of fast and slow subsystems pins trajectories near a surface that consists of unstable equilibria of a certain reduced system in a canard like manner using this vortex structure a new passage time calculation is used to approximate the interspike time interval we

hardy space estimates of hausdorff operators on the heisenberg group abstract this is a continuation of a series of works about hausdorff operator on hardy spaces in the setting of the heisenberg group in this paper we obtain the boundedness from lebesgue spaces to hardy spaces for fractional hausdorff operators and their compositions with riesz transforms we also establish the boundedness for two kinds of special hausdorff operators the hausdorff poisson operator and the hausdorff gauss operator on hardy spaces

total and partial amplitude death in networks of diffusively coupled oscillators networks of weakly nonlinear oscillators are considered with diffusive and time delayed coupling averaging theory is used to determine parameter ranges for which the network experiences amplitude death whereby oscillations are quenched and the equilibrium solution has a large domain of attraction the amplitude death is shown to be a common phenomenon which can be observed regardless of the precise nature of the nonlinearities and under very general coupling conditions in addition when the network consists of dissimilar oscillators there exist parameter values for which only parts of the network are suppressed sufficient conditions are derived for total and partial amplitude death in arbitrary network topologies with general nonlinearities coupling coefficients and connection delays

strongly singular convolution operators on the heisenberg group we consider the l2 mapping properties of a model class of strongly singular integral operators on the heisenberg group hn these are convolution operators on hn whose kernels are too singular at the origin to be of calderon zygmund type this strong singularity is compensated for by introducing a suitably large oscillation our results are obtained by utilizing the group fourier transform and uniform asymptotic forms for laguerre functions due to erdelyi

bursting oscillations induced by small noise we consider a model of a square wave bursting neuron residing in the regime of tonic spiking upon introduction of small stochastic forcing the model generates irregular bursting the statistical properties of the emergent bursting patterns are studied in the present work in particular we identify two principal statistical regimes associated with the noise induced bursting in the first case type i bursting oscillations are created mainly due to the fluctuations in the fast subsystem in the alternative scenario type ii bursting the random perturbations in the slow dynamics play a dominant role we propose two classes of randomly perturbed slow fast systems that realize type i and type ii scenarios for these models we derive the poincare maps the analysis of the linearized poincare maps of the randomly perturbed systems explains the distributions of the number of spikes within one burst and reveals their dependence on the small and control parameters present in the models the mathematical analy

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

generalized poincare embeddings and weighted hardy operator on qpα q spaces abstract the poincare embeddings w 1 n r n bmo r n and the john nirenberg inequality in bmo r n are classical results in modern analysis and partial differential equations this paper studies the generalized poincare embeddings and the john nirenberg inequality in q p q r n spaces furthermore for the weighted hardy operator u and its adjoint v this paper gives sufficient and necessary conditions on weight function to ensure the boundedness of them on q p q r n spaces

hopf bifurcation to repetitive activity in nerve various nerve axons and other excitable systems exhibit repetitive activity e g trains of propagated impulses in response to a spatially localized time independent stimulus if the stimulus is too weak or in some cases too strong one finds rather a spatially nonuniform steady response which attenuates with distance from the input site we consider these stimulus response properties for a qualitative model of nerve conduction the fitzhugh nagumo parabolic partial differential equations for each stimulus amplitude there is a unique steady state solution at critical stimulus values this steady solution loses stability and a branch of time periodic spatially nonuniform solutions appears via hopf bifurcation we associate this with the onset of repetitive activity we derive analytic bifurcation formulae and evaluate these for the case of a cubic nonlinearity to determine regions in parameter space where the steady solution is unstable we interpret physiologically the effect of stimulus form e

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

localized and asynchronous patterns via canards in coupled calcium oscillators in this paper we consider the mechanism for localized behavior in coupled calcium oscillators described by the canonical two pool model localization occurs when the individual cells oscillate with amplitudes of different orders of magnitude our analysis and computations show that a combination of diffusive coupling heterogeneity and the underlying canard structure of the oscillators all contribute to the localized behavior two key quantities characterize the different states of the system by representing the effects of both the autonomous and the non autonomous terms which are due to the coupling by highlighting the influence of the canard phenomenon these quantities identify stabilizing and destabilizing effects of the coupling on the localized behavior we compare our analysis with computations describing multi mode states and asynchronized large oscillations in addition to the localized states

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

synchronization and transient dynamics in the chains of electrically coupled fitzhugh nagumo oscillators chains of n fitzhugh nagumo oscillators with a gradient in natural frequencies and strong diffusive coupling are analyzed in this paper we study the system s dynamics in the limit of infinitely large coupling and then treat the case when the coupling is large but finite as a perturbation of the former case in the large coupling limit the 2n dimensional phase space has an unexpected structure there is an n 1 dimensional cylinder foliated by periodic orbits with an integral that is constant on each orbit when the coupling is large but finite this cylinder becomes an analog of an inertial manifold the phase trajectories approach the cylinder on the fast time scale and then slowly drift along it toward a unique limit cycle we analyze these dynamics using geometric theory for singularly perturbed dynamical systems asymptotic expansions of solutions rigorously justified and lyapunov s method

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

boundedness of hausdorff operators on real hardy spaces h1 over locally compact groups abstract results of liflyand and collaborators on the boundedness of hausdorff operators on the hardy space h 1 over finite dimensional real space are generalized to the case of locally compact groups that are spaces of homogeneous type special cases and examples of compact lie groups homogeneous groups in particular the heisenberg group and finite dimensional spaces over division rings are considered in conclusion we solve for the space l 2 an open question on compactness of a hausdorff operator posed by liflyand

reduction of a model of an excitable cell to a one dimensional map we use qualitative methods for singularly perturbed systems of differential equations and the principle of averaging to compute the first return map for the dynamics of a slow variable calcium concentration in the model of an excitable cell the bifurcation structure of the system with continuous time endows the map with distinct features it is a unimodal map with a boundary layer corresponding to the homoclinic bifurcation in the original model this structure accounts for different periodic and aperiodic regimes and transitions between them all parameters in the discrete system have biophysical meaning which allows for precise interpretation of various dynamical patterns our results provide analytical explanation for the numerical studies reported previously