periodic solutions for a class of non autonomous second order systems abstract the purpose of this paper is to study the existence of periodic solutions of non autonomous second order systems u t f t u t a e t 0 t u 0 u t u 0 u t 0 some new existence theorems are obtained by the least action principle

random perturbations of heteroclinic attractors estimates are derived for the mean recurrence time of orbits in the neighborhood of an attracting homoclinic orbit or heteroclinic cycle in an ordinary differential equation subject to small additive random noise the theory presented is illustrated with numerical simulations of several systems including ones invariant under symmetry groups for which such heteroclinic attractors are structurally stable the physical implications of the work presented are briefly discussed

how rare are multiple eigenvalues let a q be a differentiable family of self adjoint operators on a hilbert space h indexed by a parameter q that belongs to a separable banach manifold x assume that the spectrum of each operator a q is discrete of finite multiplicity and with no finite accumulation points we introduce a new concept of codimension in infinite dimensional space and then prove that under an appropriate transversality condition related to the strong arnold hypothesis the members of the family a q having multiple eigenvalues form a set of codimension at least 2 using this we show that a generic member of the family a q has a simple spectrum i e no repeated eigenvalues and that any two values q1 and q2 of the parameter can be connected by an analytic curve in x such that a q has a simple spectrum for all q in the interior of we then apply these results in two cases of physical interest to the laplace operator with the domain as parameter and to the schrodinger operator with a symmetric potential as parameter 1999 john wiley sons inc

exact controllability in projections of the bilinear schrodinger equation we consider the bilinear schrodinger equation with discrete spectrum drift we show for n n arbitrary exact controllability in projections on the first n given eigenstates the controllability result relies on a generic controllability hypothesis on some associated finite dimensional approximations the method is based on lie algebraic control techniques applied to the finite dimensional approximations coupled with classical topological arguments issuing from degree theory

low dimensional models of the wall region in a turbulent boundary layer new results abstract using an optimally convergent representation a low dimensional model is constructed which embodies in a streamwise invariant form the effects of streamwise structure results of stone show that the model is capable of mimicking the stability change due to favorable and unfavorable pressure gradients results of aubry et al suggest that polymer drag reduction is associated with stabilization of the secondary instabilities as has been speculated results of bloch and marsden indicate that drag can be reduced by feedback and that this is mathematically equivalent to polymer drag reduction the authors showed that dynamical systems based on the proper orthogonal decomposition have on the average the best short term tracking time the time that a model tracks the true system accurately essential for control for a given number of modes in recent work the authors have shown that several assumptions made on an intuitive basis in the work of aubry et al may be justified formally berkooz has made rigorous estimates using the proper orthogonal decomposition showing that a structured turbulent flow such as the wall layer has a phase space representation that remains within a thin slab centered on the most energetic modes for most of the time campbell and holmes have shown several results in connection with symmetry breaking in systems with structurally stable heteroclinic cycles this work is relevant to our models of interacting coherent structures in boundary layers with discrete spanwise symmetry such as that caused by riblets which are known to produce drag reduction

control of noisy heteroclinic cycles abstract we develop a control strategy for strongly nonlinear dynamical systems possessing heteroclinic cycles connecting saddle points and subject to small random perturbations in the absence of control attracting cycles lead to solutions which exhibit intermittent behavior consisting of quiescent periods interrupted by rapid bursts we show that suitable feedback control can retain solutions near the saddle points and hence reduce the typical bursting rate the model studied and the form of control are motivated by our interest in reducing turbulence production in the boundary layer

a low dimensional model of binocular rivalry using winnerless competition abstract we discuss a novel minimal model for binocular rivalry and more generally perceptual dominance effects the model has only three state variables but nonetheless exhibits a wide range of input and noise dependent switching the model has two reciprocally inhibiting input variables that represent perceptual processes active during the recognition of one of the two possible states and a third variable that represents the perceived output sensory inputs only affect the input variables we observe for rivalry inducing inputs the appearance of winnerless competition in the perceptual system this gives rise to a behaviour that conforms to well known principles describing binocular rivalry the levelt propositions in particular proposition iv monotonic response of residence time as a function of image contrast down to very low levels of stimulus intensity

optimal control in laser induced population transfer for two and three level quantum systems we apply the techniques of control theory and of sub riemannian geometry to laser induced population transfer in two and three level quantum systems the aim is to induce complete population transfer by one or two laser pulses minimizing the pulse fluences sub riemannian geometry and singular riemannian geometry provide a natural framework for this minimization where the optimal control is expressed in terms of geodesics we first show that in two level systems the well known technique of pulse transfer in the rotating wave approximation emerges naturally from this minimization in three level systems driven by two resonant fields we also find the counterpart of the pulse transfer this geometrical picture also allows one to analyze the population transfer by adiabatic passage

noise induced chaos and phase space flux abstract we study the effect of additive noise on near integrable second order dynamical systems whose unperturbed flows have homoclinic or heteroclinic orbits the noise is represented by a type of shinozuka stochastic process capable of arbitrarily closely approximating gaussian noise with any specified spectrum we derive a formula for the flux factor applicable for any asymptotic mean stationary excitation this derivation shows that to first order the effect of the external excitation on the system is mediated by a linear filter associated with the system homoclinic or heteroclinic orbit it also shows that the stationary mean distribution of the filtered excitation determines the average phase space flux this is true for both random and nonrandom excitations and indicates that for the dynamical systems considered here these two classes of excitation play substantially equivalent roles in the promotion of chaos

on the conicity of eigenvalues intersections for parameter dependent self adjoint operators motivated by recent controllability results for the bilinear schrodinger equation based on the existence of conical intersections in this paper we identify two physically interesting families of parameter dependent hamiltonians that admit residual and prevalent subfamilies for which all double eigenvalues are conical in order to obtain such a result we exploit a characterization of conical intersections in terms of a transversality condition which allows to apply a suitable transversality theorem

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

on stable oscillations and equilibriums induced by small noise we consider the motion of a light particle when the force field is perturbed by a small nose if a certain relation between the mass of the particle and the noise intensity holds the motion of the particle will be close to periodic oscillations or to a stable equilibrium which do not exist without the noise we study various classes of random perturbations in particular we consider the question of computer simulation of these effects and calculate the correction term which appears when the gaussian perturbations are replaced by the simple random walk these are the stochastic resonance type effects and their mathematical description is based on the large deviation theory

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

destabilization by noise of transverse perturbations to heteroclinic cycles a simple model and an example from dynamo theory we show that transverse perturbations from structurally stable heteroclinic cycles can be destabilized by surprisingly small amounts of noise even when each individual fixed point of the cycle is stable to transverse modes a condition that favours this process is that the linearization of the dynamics in the transverse direction be characterized by a non normal matrix the phenomenon is illustrated by a simple two dimensional switching model and by a simulation of a convectively driven dynamo

analysis of excitability for the fitzhugh nagumo model via a stochastic sensitivity function technique we study excitability phenomena for the stochastically forced fitzhugh nagumo system modeling a neural activity noise induced changes in the dynamics of this model can be explained by the high stochastic sensitivity of its attractors computational methods based on the stochastic sensitivity functions technique are suggested for the analysis of these attractors our method allows us to construct confidence ellipses and estimate a threshold value of a noise intensity corresponding to the neuron excitement on the basis of the proposed technique a supersensitive limit cycle is found for the fitzhugh nagumo model

finite controllability of infinite dimensional quantum systems quantum phenomena of interest in connection with applications to computation and communication often involve generating specific transfers between eigenstates and their linear superpositions for some quantum systems such as spin systems the quantum evolution equation the schrodinger equation is finite dimensional and old results on controllability of systems defined on on lie groups and quotient spaces provide most of what is needed insofar as controllability of non dissipative systems is concerned however in an infinite dimensional setting controlling the evolution of quantum systems often presents difficulties both conceptual and technical in this paper we present a systematic approach to a class of such problems for which it is possible to avoid some of the technical issues in particular we analyze controllability for infinite dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre defined subspaces this result which we call the finite controllability theorem provides a set of sufficient conditions for controllability in an infinite dimensional setting we consider specific physical systems that are of interest for quantum computing and provide insights into the types of quantum operations gates that may be developed

mean field approximation of two coupled populations of excitable units the analysis on stability and bifurcations in the macroscopic dynamics exhibited by the system of two coupled large populations composed of n stochastic excitable units each is performed by studying an approximate system obtained by replacing each population with the corresponding mean field model in the exact system one has the units within an ensemble communicating via the time delayed linear couplings whereas the interensemble terms involve the nonlinear time delayed interaction mediated by the appropriate global variables the aim is to demonstrate that the bifurcations affecting the stability of the stationary state of the original system governed by a set of 4n stochastic delay differential equations for the microscopic dynamics can accurately be reproduced by a flow containing just four deterministic delay differential equations which describe the evolution of the mean field based variables in particular the considered issues include determining the parameter domains where the stationary state is stable the scenarios for the onset and the time delay induced suppression of the collective mode as well as the parameter domains admitting bistability between the equilibrium and the oscillatory state we show how analytically tractable bifurcations occurring in the approximate model can be used to identify the characteristic mechanisms by which the stationary state is destabilized under different system configurations like those with symmetrical or asymmetrical interpopulation couplings

on designing heteroclinic networks from graphs abstract robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems these networks may have a complicated structure determined to a large extent by the constraints and dimension of the system as these networks are of great interest as dynamical models of biological and cognitive processes it is useful to understand how particular directed graphs can be realised as attracting robust heteroclinic networks between states in phase space this paper presents two methods of realising arbitrarily complex directed graphs as robust heteroclinic networks for flows generated by odes we say the odes realise the graphs as heteroclinic networks between equilibria that represent the vertices suppose we have a directed graph on n v vertices with n e edges the simplex realisation embeds the graph as an invariant set of a flow on an n v 1 simplex this method realises the graph as long as it is one and two cycle free the cylinder realisation embeds a graph as an invariant set of a flow on a n e 1 dimensional space this method realises the graph as long as it is one cycle free in both cases we realise the graph as an invariant set within an attractor and discuss some illustrative examples including the influence of noise and parameters on the dynamics in particular we show that the resulting heteroclinic network may or may not display memory of the vertices visited

controllability of a quantum particle in a moving potential well we consider a nonrelativistic charged particle in a 1d moving potential well this quantum system is subject to a control which is the acceleration of the well it is represented by a wave function solution of a schrodinger equation the position of the well together with its velocity we prove the following controllability result for this bilinear control system given 0 close enough to an eigenstate and f close enough to another eigenstate the wave function can be moved exactly from 0 to f in finite time moreover we can control the position and the velocity of the well our proof uses moment theory a nash moser implicit function theorem the return method and expansion to the second order

two distinct mechanisms of coherence in randomly perturbed dynamical systems we carefully examine two mechanisms coherence resonance and self induced stochastic resonance by which small random perturbations of excitable systems with large time scale separation may lead to the emergence of new coherent behaviors in the form of limit cycles we analyze what controls the degree of coherence in these two mechanisms and classify their very different properties in particular we show that coherence resonance arises only at the onset of bifurcation and is rather insensitive against variations in the noise amplitude and the time scale separation ratio in contrast self induced stochastic resonance may arise away from bifurcations and the properties of the limit cycle it induces are controlled by both the noise amplitude and the time scale separation ratio

chaos from nonlinear markov processes why the whole is different from the sum of its parts abstract nonlinear markov processes have been frequently used to address bifurcations and multistability in equilibrium and non equilibrium many body systems however our understanding of the range of phenomena produced by nonlinear markov processes is still in its infancy we demonstrate that in addition to bifurcations and multistability nonlinear markov processes can exhibit another key phenomena well known in the realm of nonlinear physics chaos it is argued that chaotically evolving process probabilities are a generic feature of many body systems exhibiting nonlinear markov processes even if the isolated subsystems do not exhibit chaos that is when considering a nonlinear markov process as an entity of its own type then the nonlinear markov process in general is qualitatively different from its constituent subprocesses which reflects that the many body system as a whole is different from the sum of its parts

beating the curse of dimension with accurate statistics for the fokker planck equation in complex turbulent systems abstract solving the fokker planck equation for high dimensional complex dynamical systems is an important issue recently the authors developed efficient statistically accurate algorithms for solving the fokker planck equations associated with high dimensional nonlinear turbulent dynamical systems with conditional gaussian structures which contain many strong non gaussian features such as intermittency and fat tailed probability density functions pdfs the algorithms involve a hybrid strategy with a small number of samples l l where a conditional gaussian mixture in a high dimensional subspace via an extremely efficient parametric method is combined with a judicious gaussian kernel density estimation in the remaining low dimensional subspace in this article two effective strategies are developed and incorporated into these algorithms the first strategy involves a judicious block decomposition of the conditional covariance matrix such that the evolutions of different blocks have no interactions which allows an extremely efficient parallel computation due to the small size of each individual block the second strategy exploits statistical symmetry for a further reduction of l l the resulting algorithms can efficiently solve the fokker planck equation with strongly non gaussian pdfs in much higher dimensions even with orders in the millions and thus beat the curse of dimension the algorithms are applied to a 1 000 1 000 dimensional stochastic coupled fitzhugh nagumo model for excitable media an accurate recovery of both the transient and equilibrium non gaussian pdfs requires only l 1 l 1 samples in addition the block decomposition facilitates the algorithms to efficiently capture the distinct non gaussian features at different locations in a 240 240 dimensional two layer inhomogeneous lorenz 96 model using only l 500 l 500 samples

the palais smale condition versus coercivity possesses a convergent subsequence and cp is said to satisfy ps if ir satisfies ps for every c e ir recently among other results in critical point theory shujie i i showed that if a c functional p x k is bounded from below and satisfies the condition ps then p is coercive shujie s proof uses a gradient flow approach through the so called deformation theorem cf 3 io and for that he needs the notion of a pseudo gradient vector field tjassociated with the functional cp whose existence is guaranteed for c functionals by palais 8 in this paper we present some new results which relate the palais smale condition and the notion of coercivity and are based on the well known variational principle due to ekeland 5 61 in particular a new proof of the above mentioned result of shujie is given it should be pointed out that throughout the paper the given functional p could be assumed to be only gateaux differentiable rather than c and in addition to being conceptually simpler this approach could be used in more general situations where the functional is not even differentiable cf 4 the strong form of ekeland s variational principle to be repeatedly used is given in the following theorem

effects of non gaussian noise near supercritical hopf bifurcation abstract we have studied the effects of non gaussian colored noise in a chemical oscillation system the well known brusselator model in the parameter region close to the supercritical hopf bifurcation with the variation of the parameter q which quantifies the deviation from gaussian character the signal to noise ratio of noise induced oscillation exhibits a bell shaped change indicating the presence of resonant activity the cooperative effects of q and the correlation time on the performance of noise induced oscillation are also investigated interestingly resonance like behavior can be induced by either q or when the other parameter is properly fixed stochastic normal form theory is used to analyze these nontrivial effects and the simulation results are well reproduced this work provides us comprehensive understanding of how non gaussian noise influences the dynamics in chemical oscillation systems

crossing the quasi threshold manifold of a noise driven excitable system we consider the noise induced escapes in an excitable system possessing a quasi threshold manifold along which there exists a certain point of minimal quasi potential in the weak noise limit the optimal escaping path turns out to approach this particular point asymptotically making it analogous to an ordinary saddle numerical simulations are performed and an elaboration on the effect of small but finite noise is given which shows that the ridges where the prehistory probability distribution peaks are located mainly within the region where the quasi potential increases gently the cases allowing anisotropic noise are discussed and we found that varying the noise term in the slow variable would dramatically raise the whole level of quasi potentials leading to significant changes in both patterns of optimal paths and exit locations

mixed mode oscillations in a stochastic piecewise linear system abstract we analyse a piecewise linear fitzhugh nagumo model the system exhibits a canard near which both small amplitude and large amplitude periodic orbits exist the addition of small noise induces mixed mode oscillations mmos in the vicinity of the canard point we determine the effect of each model parameter on the stochastically driven mmos in particular we show that any parameter variation such as a modification of the piecewise linear function in the model that leaves the ratio of noise amplitude to time scale separation unchanged typically has little effect on the width of the interval of the primary bifurcation parameter over which mmos occur in that sense the mmos are robust furthermore we show that the piecewise linear model exhibits mmos more readily than the classical fitzhugh nagumo model for which a cubic polynomial is the only nonlinearity by studying a piecewise linear model we are able to explain results using analytical expressions and compare these with numerical investigations

robust steering of n level quantum systems robust open loop steering of a finite dimensional quantum system is a central problem in a growing number of applications of information engineering in this note we reformulate the problem in the classical control theoretic setting and provide a precise definition of robustness of the control strategy we then discuss and compare some significant problems from nuclear magnetic resonance in the light of the given definition we obtain quantitative results that are consistent with the qualitative ones available in the physics literature

a nontrivial scaling limit for multiscale markov chains we consider markov chains with fast and slow variables and show that in a suitable scaling limit the dynamics becomes deterministic yet is far away from the standard mean field approximation this new limit is an instance of self induced stochastic resonance which arises due to matching between a rare event timescale on the one hand and the natural timescale separation in the underlying problem on the other here it is illustrated on a model of a molecular motor where it is shown to explain the regularity of the motor gait observed in some experiments

approximate controllability exact controllability and conical eigenvalue intersections for quantum mechanical systems we study the controllability of a closed control affine quantum system driven by two or more external fields we provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the hamiltonian in dependence of the controls seen as parameters such spectral condition is structurally stable in the case of three controls or in the case of two controls when the hamiltonian is real the spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite dimensional case in the finite dimensional case it implies that the system is lie bracket generating when lifted to the group of unitary transformations and in particular that it is exactly controllable hence lie algebraic conditions are deduced from purely spectral properties

path connected yang mills moduli spaces on utilise des techniques de min max dans le calcul des variations pour demontrer que les espaces modules des connexions auto duales sur des fibres principaux su 2 ou su 3 sur s 4 sont chemin connexe

existence and multiplicity of nonzero periodic solution with saddle point character for some nonautonomous second order systems abstract the purpose of this paper is to study the existence and multiplicity of nonzero periodic solutions with saddle point character for the following nonautonomous second order systems u t f t u t a e t 0 t u 0 u t u 0 u t 0 some new existence and multiplicity theorems are obtained by using saddle point reduction method and a three critical point theorem proposed by brezis and nirenberg

hunting french ducks in a noisy environment we consider the effect of gaussian white noise on fast slow dynamical systems with one fast and two slow variables containing a folded node singularity in the absence of noise these systems are known to display mixed mode oscillations consisting of alternating large and small amplitude oscillations we quantify the effect of noise and obtain critical noise intensities above which the small amplitude oscillations become hidden by fluctuations furthermore we prove that the noise can cause sample paths to jump away from so called canard solutions with high probability before deterministic orbits do this early jump mechanism can drastically influence the local and global dynamics of the system by changing the mixed mode patterns

coherence resonance induced by non gaussian noise in a deterministic hodgkin huxley neuron in this paper we report a novel effect of a particular kind of non gaussian noise ngn especially its deviation from gaussian noise on the spiking activity in a deterministic hodgkin huxley hh neuron it is found via numerical simulations that the coefficient of variation cv characterizing the spiking regularity nonlinearly changes with varying deviation and passes through a minimum at an optimal deviation value representing the presence of the coherence resonance cr this phenomenon shows that when the ngn is optimal the neuron can exhibit the most regular firing behavior this result provides a constructive role of the ngn in the timing precision of information processing and signal transduction in neural systems

the 1 2 mode interaction and heteroclinic networks in boussinesq convection abstract methods of equivariant bifurcation theory are applied to boussinesq convection in a plane layer with stress free horizontal boundaries and an imposed square lattice periodicity in the horizontal directions we consider the problem near the onset of instability of the uniform conducting state where spatial roll patterns with two different wavelengths in the ratio 1 2 become simultaneously unstable at a mode interaction center manifold reduction yields a normal form on c 4 with very rich dynamical behavior for a fixed prandtl number p the mode interaction occurs at an isolated point in the parameter plane r l where r is the rayleigh number and l the length of the horizontal periodicities and acts as an organizing center for many nearby bifurcations the normal form predicts the appearance of many steady states and travelling waves which are classified by their symmetries it also predicts the appearance of robust heteroclinic networks involving steady states with several different symmetries and robust attractors of generalized heteroclinic type that include connections from equilibria to subcycles this is the first example of a heteroclinic network in a fluid dynamical system that has depth greater than one the normal form dynamics is in good correspondence both quantitatively and qualitatively with direct numerical simulations of the full convection equations

noise induced intermittency in a model of a turbulent boundary layer abstract the presence of attracting heteroclinic cycles in a set of truncated ordinary differential equations for the modal amplitudes in a proper orthogonal decomposition of the navier stokes equations leads to burst like intermittent phenomenon in the reconstructed velocity field the time between bursts or the cycle time is effectively randomized by the addition of small random perturbations in this case taking the form of irregular pressure fluctuations from the outer flow this suggests that interaction of the inner and outer flow is important in the generation and triggering of these burst events a scaling law is developed for the time between burst events as well as an expression for the probability distribution of these time intervals

pattern selection in three dimensional magnetoconvection abstract weakly nonlinear boussinesq convection in an imposed vertical magnetic field is studied stress free boundary conditions are used at top and bottom of the layer where the magnetic field is assumed to remain vertical both steady and oscillatory convection are considered among all the spatially periodic patterns steady rolls are the preferred pattern when the onset of convection is steady state in the overstable regime a variety of spatially and temporally periodic patterns may be stable depending on the parameters these include in particular a pattern called alternating rolls the location of stable alternating rolls appears to be consistent with recent numerical simulations of compressible magnetoconvection where such patterns have been found in certain cases the conduction solution loses stability directly to spatially periodic patterns with nonperiodic time dependence the analysis of the possible patterns and their stability takes advantage of existing results from equivariant bifurcation theory and uses the critical wavenumber selected by the linear stability calculation as a function of the imposed magnetic field

the turnpike property in finite dimensional nonlinear optimal control turnpike properties have been established long time ago in finite dimensional optimal control problems arising in econometry they refer to the fact that under quite general assumptions the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces the first and the last of which being transient short time arcs and the middle piece being a long time arc staying exponentially close to the optimal steady state solution of an associated static optimal control problem we provide in this paper a general version of a turnpike theorem valuable for nonlinear dynamics without any specific assumption and for very general terminal conditions not only the optimal trajectory is shown to remain exponentially close to a steady state but also the corresponding adjoint vector of the pontryagin maximum principle the exponential closedness is quantified with the use of appropriate normal forms of riccati equations we show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method or a shooting method in particular we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector not at the initial time but at the middle of the trajectory

stochastic resonance in an autonomous system with a nonuniform limit cycle in a recent numerical study gang et al phys rev lett 71 807 1993 presented the first example of stochastic resonance in an autonomous system they considered a two variable model in which a limit cycle is born as some parameter is varied their numerical experiments revealed various noise induced effects including a noise induced shift in the frequency of the limit cycle oscillations and noise induced oscillations in the absence of a deterministic limit cycle we show that both of these effects are simple consequences of the nonuniformity of the motion along the limit cycle

adiabatic control of the schrodinger equation via conical intersections of the eigenvalues in this paper we present a constructive method to control the bilinear schrodinger equation via two controls the method is based on adiabatic techniques and works if the spectrum of the hamiltonian admits eigenvalue intersections and if the latter are conical as it happens generically in this framework we are able to spread on several levels connected by conical intersections a state initially concentrated in a single energy level we provide sharp estimates on the dependence of the error with respect to the controllability time moreover we identify some special curves in the space of controls that improve the precision of the adiabatic approximation when passing through conical intersections with respect to classical adiabatic theory

chaos and the effect of noise in a model of three wave mode coupling abstract the equations governing the nonlinear interaction of three resonant waves are studied in the limit that one wave is slightly unstable and the other two modes more heavily damped it is shown that the dynamics can be described in terms of a bimodal one dimensional map which allows bifurcation sequences etc to be simply constructed the effect of small additive noise can be studied in the same framework it is shown that even at very low levels the effect of noise can be extremely important in determining the period and amplitude of the oscillations

noise and slow fast dynamics in a three wave resonance problem recent research on the dynamics of certain fluid dynamical instabilities shows that when there is a slow invariant manifold subject to fast timescale instability the dynamics are extremely sensitive to noise the behaviour of such systems can be described in terms of a one dimensional map and previous work has shown how the effect of noise can be modelled by a simple adjustment to the map here we undertake an in depth investigation of a particular set of equations using the methods of stochastic integration we confirm the prediction of the earlier studies that the noise becomes important when mu log epsilon o 1 where mu is the small timescale ratio and epsilon is the noise level in addition we present detailed information about the statistics of the solution when the noise is a dominant effect the analytical results show excellent agreement with numerical simulations

stochastic resonance in the brusselator model using the brusselator model we show that in a simple dynamical system small noise can be converted into stochastic spikewise oscillations of huge amplitude bursting noises in the vicinity of a hopf bifurcation small periodic signals with amplitude several times less than the noise intensity transform these stochastic oscillations into quasiperiodic large amplitude spikewise oscillations or small amplitude quasiharmonic oscillations depending on the signal form

saddle point reduction method for some non autonomous second order systems abstract the purpose of this paper is to study the existence of solutions with saddle point character for non autonomous second order systems u t f t u t a e t 0 t u 0 u t u 0 u t 0 some new solvability conditions are obtained

non uniform decay of predictability and return of skill in stochastic oscillatory models we examine the dynamical mechanisms that lead to the loss of predictability in low dimensional stochastic models that exhibit three main types of oscillatory behavior damped self sustained and heteroclinic we show that the information that an initial ensemble provides about the state of the system decays non uniformly with time long intervals during which the forecast provided by the ensemble does not loose any of its power are typical in all the three cases moreover the information that the forecast provides about the individual variables in the model may increase despite the fact that information about the entire system always decreases we analyze the fully solvable case of the linear oscillator and use it to provide a general heuristic explanation for the phenomenon we also show that during the intervals during which the forecast loses little of its power there is a flow of information between the marginal and conditional distributions

multi input schrodinger equation controllability tracking and application to the quantum angular momentum abstract we present a sufficient condition for approximate controllability of the bilinear discrete spectrum schrodinger equation in the multi input case the controllability result extends to simultaneous controllability approximate controllability in h s and tracking in modulus the sufficient condition is more general than those present in the literature even in the single input case and allows the spectrum of the uncontrolled operator to be very degenerate e g to have multiple eigenvalues or equal gaps among different pairs of eigenvalues we apply the general result to a rotating polar linear molecule driven by three orthogonal external fields a remarkable property of this model is the presence of infinitely many degeneracies and resonances in the spectrum

product dynamics for homoclinic attractors heteroclinic cycles may occur as structurally stable asymptotically stable attractors if there are invariant subspaces or symmetries of a dynamical system even for cycles between equilibria it may be difficult to obtain results on the generic behaviour of trajectories converging to the cycle for more complicated cycles between chaotic sets the non trivial dynamics of the nodes can interact with that of the connections this paper focuses on some of the simplest problems for such dynamics where there are direct products of an attracting homoclinic cycle with various types of dynamics using a precise analytic description of a general planar homoclinic attractor we are able to obtain a number of results for direct product systems we show that for flows that are a product of a homoclinic attractor and a periodic orbit or a mixing hyperbolic attractor the product of the attractors is a minimal milnor attractor for the product on the other hand we present evidence to show that for the product of two homoclinic attractors typically only a small subset of the product of the attractors is an attractor for the product system

a mountain pass theorem 143 in other words the mountain separating 0 and e may even be assumed of zero altitude and the conclusion is still true moreover as we shall show if c a the critical point can be chosen with llx ll r it is interesting to ask whether this extension of the ambrosetti rabinowitz theorem remains true in the infinite dimensional case we prove here that if 1 is strengthened a little to the form 1 there exist real numbers u r r such that 0 a for every x e a x e x y a moreover if c a the critical point can be chosen with r i x0 i r roughly speaking in brief the mountain pass theorem continues to hold for a mountain of zero altitude provided it also has non zero thickness in addition if c a the pass itself occurs precisely on the mountain i e satisfying r llroll r there are two interesting and immediate corollaries the first one says that a c function which has iwo local minimum points also has a third critical point the second states that a u periodic c function with a local minimum e has a critical point x e ku k 0 1 2 the precise statements of the above results will be given in the next sec tions the proof depends upon a lemma of clark 3 formulated here in a version suitable to our purpose in section 3 some applications are presen ted for the forced pendulum equation 2

coherence collective rhythm and phase difference distribution in populations of stochastic genetic oscillators with cellular communication an ensemble of stochastic genetic relaxation oscillators via phase attractive or repulsive cell to cell communication are investigated in the phase attractive coupling case it is found that cellular communication can enhance self induced stochastic resonance as well as collective rhythms and that different intensities of noise resulting from the fluctuation of intrinsic chemical reactions or the extrinsic environment can induce stochastic limit cycles with different amplitudes for a large cell density in contrast in the phase repulsive coupling case the distribution of phase differences among the stochastic oscillators can display such characteristic as unimodality bimodality or polymodality depending on both noise intensity and cell number but the modality of phase difference distribution almost keeps invariant for an arbitrary noise intensity as the cell number is beyond a threshold

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

suppression of bursting abstract we investigate the possibility of using a single small amplitude control input and feedback to stabilize equilibrium sets in a class of highly nonlinear o 2 symmetric dynamical systems possessing structurally stable heteroclinic cycles the leading order behavior near the equilibria is bilinear and homogeneous in the state variables while nonlinearities representing the symmetry breaking effect of the controller are crucial after a series of simplifying transformations we use ideas from optimal control theory to construct a stabilizing controller this study is motivated by the desire to stabilize the burst sweep cycle in low dimensional models of a turbulent boundary layer in the last two sections we apply the techniques to the 10 dimensional system of aubry et al 1988 kw bilinear systems dynamical systems heteroclinic cycles normal forms optimal control symmetry

metastable switching in a planar limit cycle system with additive noise abstract additive noise is known to produce counter intuitive behaviors in nonlinear dynamical systems previously it was shown that systems with a deterministic limit cycle can display switching between metastable states in the presence of asymmetric additive white noise here we systematically analyze the dynamics of this metastable behavior and show how the vector field away from the limit cycle influences the rate and directionality of the metastable switching using stochastic phase reduction methods as well as asymptotic approximations we identify mechanisms underlying different rates of switching and predict when the system will rotate in the opposite direction of the deterministic limit cycle thus this work presents an alternative mechanism for generating a range of metastable switch behaviors that have been observed in a number of physical systems

noise stabilized random attractor a two dimensional flow model is introduced with deterministic behavior consisting of bursts that become successively larger with longer interburst time intervals between them the system is symmetric in one variable and there are bursts on either side of separated by the presence of an invariant manifold at in the presence of arbitrarily small additive noise in the direction the successive bursts have bounded amplitudes and interburst intervals this system with noise is proposed as a model for edge localized modes in tokamaks with noise the bursts can switch from positive to negative and vice versa the probability distribution of burst heights and interburst periods is studied as is the dependence of the statistics on the noise variance the modification of this behavior as the symmetry in is broken is studied showing qualitatively similar behavior if the symmetry breaking is small enough experimental observations of a nonlinear circuit governed by the same equations are presented showing good agreement

finding the quasipotential for nongradient sdes abstract the quasipotential with respect to a given attractor of a system evolving according to a stochastic differential equation sde with a small noise is a function that allows us to find the expected exit time from its basin of attraction and to estimate the equilibrium probability density in its neighborhood this work is devoted to the study of the quasipotential of nongradient sdes with isotropic diffusion the goals are i to understand the theoretical behavior of the quasipotential and ii to develop tools for its numerical study this work is focused on 2d equations with asymptotically stable equilibria and limit cycles in the theoretical part we establish a number of properties of the quasipotential for 2d linear equations we derive an exact formula for the quasipotential for 2d nonlinear equations we establish how the quasipotential grows near the stable equilibria and the stable limit cycles in the numerical part of this work we adjust the oum for computing the quasipotential on a regular mesh we show that the quasipotential satisfies a hamilton jacobi bellman equation with an anisotropic and unbounded speed function the unbounded speed function may lead to an additional consistency error in specific areas of the phase space however we conduct an error analysis and show that even if this error appears it decays quadratically as we refine the mesh we apply the proposed numerical algorithm to a number of examples linear systems the maier stein model a system with stochastic resonance and a system with two stable limit cycles whenever it is appropriate we compute the minimum action paths by shooting characteristics the minimum action paths found in this way are necessarily the global minimizers of the freidlin wentzell action

controllability for distributed bilinear systems this paper studies controllability of systems of the form dw dt mathcal a w p t mathcal b w where mathcal a is the infinitesimal generator of a c 0 semigroup of bounded linear operators e mathcal a t on a banach space x mathcal b x to x is a c 1 map and p in l 1 0 t mathbb r is a control the paper i gives conditions for elements of x to be accessible from a given initial state w 0 and ii shows that controllability to a full neighborhood in x of w 0 is impossible for dim x infty examples of hyperbolic partial differential equations are provided

deterministic noise amplifiers although we are accustomed to considering only overall stability properties of dynamical systems local stability properties can also have a dramatic physical effect local instabilities in overall stable motion can cause otherwise imperceptible ambient noise to be amplified to macroscopic proportions the result can be easily mistaken for deterministic chaos we give measures for local instability and noise amplification in terms of the covariance matrix and local divergence rates and analyze several examples an experimental test can be made by varying the external noise level for sufficiently small amplitudes the noisy response due to local instabilities scales linearly with the noise level whereas noisy behavior due to deterministic chaos is largely unaffected

propagation through conical crossings an asymptotic semigroup we consider the standard model problem for a conical intersection of electronic surfaces in molecular dynamics our main result is the construction of a semi group in order to approximate the wigner function associated with the solution of the schrodinger equation at leading order in the semiclassical parameter the semigroup stems from an underlying markov process that combines deterministic transport along classical trajectories within the electronic surfaces and random jumps between the surfaces near the crossing our semigroup can be viewed as a rigorous mathematical counterpart of so called trajectory surface hopping algorithms which are of major importance in molecular simulations in chemical physics the key point of our analysis the incorporation of the nonadiabatic transitions is based on the landau zener type formula of fermanian kammerer and gerard10 for the propagation of two scale wigner measures through conical crossings 2005 wiley periodicals inc

dynamics near a periodically perturbed robust heteroclinic cycle abstract robust heteroclinic cycles rhcs arise naturally in collections of symmetric differential equations derived as dynamical models in many fields including fluid mechanics game theory and population dynamics in this paper we present a careful study of the complicated dynamics generated by small amplitude periodic perturbations of a stable robust heteroclinic cycle rhc we give a detailed derivation of the poincare map for trajectories near the rhc asymptotically correct in the limit of small amplitude perturbations this reduces the nonautonomous system in r 3 to a 2d map we identify three distinct dynamical regimes the distinctions between these regimes depend subtly on different distinguished limits of the two small parameters in the problem the first regime corresponds to the rhc being only weakly attracting here we show that the system is equivalent to a damped nonlinear pendulum with a constant torque in the second regime the periodically perturbed rhc is more strongly attracting and the system dynamics corresponds to that of a non invertible or invertible circle map in the third regime of yet stronger attraction the dynamics of the return map is chaotic and no longer reducible to a one dimensional map this third regime has been noted previously our analysis in this paper focusses on providing quantitative results in the first two regimes

on the controllability of quantum transport in an electronic nanostructure we investigate the controllability of quantum electrons trapped in a two dimensional device typically a metal oxide semiconductor mos field effect transistor the problem is modeled by the schrodinger equation in a bounded domain coupled to the poisson equation for the electrical potential the controller acts on the system through the boundary condition on the potential on a part of the boundary modeling the gate we prove that generically with respect to the shape of the domain and boundary conditions on the gate the device is controllable we also consider control properties of a more realistic nonlinear version of the device taking into account the self consistent electrostatic poisson potential

nonlinear dynamics of neuronal excitability oscillations and coincidence detection we review some widely studied models and firing dynamics for neuronal systems both at the single cell and network level and dynamical systems techniques to study them in particular we focus on two topics in mathematical neuroscience that have attracted the attention of mathematicians for decades single cell excitability and bursting we review the mathematical framework for three types of excitability and onset of repetitive firing behavior in single neuron models and their relation with hodgkin s classification in 1948 of repetitive firing properties we discuss the mathematical dissection of bursting oscillations using fast slow analysis and demonstrate the approach using single cell and mean field network models finally we illustrate the properties of type iii excitability in which case repetitive firing for constant or slow inputs is absent rather firing is in response only to rapid enough changes in the stimulus our case study involves neuronal computations for sound localization for which neurons in the auditory brain stem perform extraordinarily precise coincidence detection with submillisecond temporal resolution

on off intermittency power spectrum and fractal properties of time series abstract some dynamical systems possess invariant submanifolds such that the dynamics restricted to the invariant submanifold is chaotic this situation arises in systems with a spatial symmetry or in the synchronization of chaotic oscillators the invariant submanifold could become unstable to perturbations in the transverse directions when a parameter of the system is changed through a critical blow out value this could result in an extreme form of temporally intermittent bursting called on off intermittency we propose a model that incorporates the universal features of systems that display on off intermittency we study this model both with and without additive noise and we derive scaling results for the power spectral density of the on off intermittent process and for the box counting dimension for the set of time intervals when the process takes on values above a given threshold we then present numerical simulations realizing these results

low dimensional chaotic dynamics versus intrinsic stochastic noise a paradigm model several prototype models are introduced here which are designed to elucidate the interaction between heteroclinic low dimensional chaos in the projected nonlinear dynamics and intrinsic stochasticity induced by energy exchange with a bath of fast variables these models are built by coupling a four dimensional ode with known analytical properties including heteroclinic cycles with a suitable deterministic bath of fast variables a systematic strategy for stochastic mode reduction is applied to these models with 104 degrees of freedom to derive four dimensional reduced stochastic equations for the slow variables due to the internal chaotic dynamics of the slow variables the stochastic mode reduction strategy is very robust in this case and yields reduced models which accurately capture the statistical behavior of the original deterministic system furthermore it is also shown here that even in the regime of a weak coupling between the slow variables and the fast heat bath the detailed structure of the stochastic terms derived through the mode elimination procedure is essential for reproducing the statistical behavior of the slow dynamics

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

power spectrum of a noisy system close to a heteroclinic orbit we consider a two dimensional dynamical system that possesses a heteroclinic orbit connecting four saddle points this system is not able to show self sustained oscillations on its own if endowed with white gaussian noise it displays stochastic oscillations the frequency and quality factor of which are controlled by the noise intensity this stochastic oscillation of a nonlinear system with noise is conveniently characterized by the power spectrum of suitable observables in this paper we explore different analytical and semianalytical ways to compute such power spectra besides a number of explicit expressions for the power spectrum we find scaling relations for the frequency spectral width and quality factor of the stochastic heteroclinic oscillator in the limit of weak noise in particular the quality factor shows a slow logarithmic increase with decreasing noise of the form q sim ln 1 d 2 our results are compared to numerical simulations of the respective langevin equations