smoothness of transition maps in singular perturbation problems with one fast variable abstract this paper deals with the smoothness of the transition map between two sections transverse to the fast flow of a singularly perturbed vector field one fast multiple slow directions orbits connecting both sections are canard orbits i e they first move rapidly towards the attracting part of a critical surface then travel a distance near this critical surface even beyond the point where the orbit enters the repelling part of the critical surface and finally repel away from the surface we prove that the transition map is smooth in a transcritical situation however where orbits from an attracting part of one critical manifold follow the repelling part of another critical manifold the smoothness of the transition map may be limited due to resonance phenomena that are revealed by blowing up the turning point we present a polynomial example in r 3

relaxation oscillations in a class of predator prey systems abstract we consider a class of three dimensional singularly perturbed predator prey systems having two predators competing exploitatively for the same prey in a constant environment by using dynamical systems techniques and the geometric singular perturbation theory we give precise conditions which guarantee the existence of stable relaxation oscillations for systems within the class such result shows the coexistence of the predators and the prey with quite diversified time response which typically happens when the prey population grows much faster than those of predators as an application a well known model will be discussed in detail by showing the existence of stable relaxation oscillations for a wide range of parameters values of the model

the constant variation formulae for singular fractional differential systems with delay abstract this paper considers the caputo singular fractional differential systems with delay and the riemann liouville singular fractional differential systems with delay a new function is defined by the d inverse matrix and function two fundamental solutions are given the constant variation formulae for singular fractional differential systems with delay are obtained

analytical solution of the linear fractional system of commensurate order a useful representation of fractional order systems is the state space representation for the linear fractional systems of commensurate order the state space representation is defined as for regular integer state space representation with the state vector differentiated to a real order this paper presents a solution of the linear fractional order systems of commensurate order in the state space the solution is obtained using a technique based on functions of square matrices and the cayley hamilton theorem the technique developed for linear systems of integer order is extended to derive analytical solutions of linear fractional systems of commensurate order the basic ideas and the derived formulations of the technique are presented both homogeneous and inhomogeneous cases with usual input functions are solved the solution is calculated in the form of a linear combination of suitable fundamental functions the presented results are illustrated by analyzing some examples to demonstrate the effectiveness of the presented analytical approach

the general vector solutions of such linear coupled matrix fractional differential equations by using kronecker structures in this paper we demonstrate how kronecker products and vector operator can be combined beautifully to solve and fit the non homogenous linear matrix fractional differential equations and such coupled linear matrix fractional differential equations the way exists which transform the given coupled matrix fractional differential equations by using kronecker structures into forms for which solutions may be readily computed some important and interesting special cases of the general system of non homogeneous linear matrix fractional differential equations of fractional order are also considered which includes the non homogenous linear fractional dynamical system with delays in control finally a brief comparison between the fractional and integer order vector solutions and some examples are given to illustrate our new approaches

exchange lemmas for singular perturbation problems with certain turning points abstract in this work singular perturbation problems with a certain type of turning point are studied from a geometric point of view we first describe the delay phenomenon initially studied by pontryagin of dynamics near the slow manifold possessing the turning points based on the local properties of such turning points and the center manifold theory for general invariant manifolds developed by chow liu and yi we extend the well known exchange lemma first formulated by jones and kopell for problems with normally hyperbolic slow manifolds to problems with this type of turning point applications to singular boundary value problems with turning points are discussed

fisher s information for discretely sampled levy processes this paper studies the asymptotic behavior of the fisher information for a levy process discretely sampled at an increasing frequency we show that it is possible to distinguish not only the continuous part of the process from its jumps part but also different types of jumps and derive the rates of convergence of efficient estimators

heteroclinic bifurcation and singularly perturbed boundary value problems abstract we study a singularly perturbed boundary value problem in r m n x f x y e e y g x y e b 1 x 0 y 0 e 0 b 2 x 0 y 0 e 0 given a candidate for the 0th order approximation which exhibits both boundary layers and interior layers we present a complete procedure to compute higher order expansions and a procedure to compute the real solution near a truncated asymptotic expansion assuming the hyperbolicity of the regular layers and some generic assumptions similar results concerning the existence of periodic solutions relaxation oscillations are also presented several ideas from dynamical systems theory are employed e g exponential dichotomies fredholm alternatives and heteroclinic bifurcations

existence and uniqueness of holomorphic solutions for fractional cauchy problem by employing majorant functions the existence and uniqueness of holomorphic solutions to nonlinear fractional partial differential equations the cauchy problems are introduced furthermore the analytic continuation of solutions is studied

the controllability of fractional control systems with control delay this paper discusses the controllability of fractional control systems with control delay we firstly give the solution expression for fractional control systems with control delay then give the necessary and sufficient conditions for the controllability of fractional control systems with control delay

tracking invariant manifolds with differential forms in singularly perturbed systems abstract a technique is developed for finding homoclinic orbits in singularly perturbed systems the desired orbit lies near a singular solution that has both fast and slow time scales the orbit is to be constructed as the transverse intersection of a center stable and center unstable manifold which requires tracking the invariant manifolds near the singular solution the main technical advance is the exchange lemma which allows one to convert information about transversality of manifolds associated with the singular limit equations to information about behavior of the invariant manifold as it leaves a neighborhood of a slow segment of the singular solution the computations for this lemma are done with the use of differential forms and the lemma is applied to prove the existence of a homoclinic orbit in a class of singularly perturbed equations

numerical simulation of the fractional bloch equations in physics and chemistry specifically in nmr nuclear magnetic resonance or mri magnetic resonance imaging or esr electron spin resonance the bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization m m x m y m z as a function of time when relaxation times t 1 and t 2 are present recently some fractional models have been proposed for the bloch equations however effective numerical methods and supporting error analyses for the fractional bloch equation fbe are still limited in this paper the time fractional bloch equations tfbe and the anomalous fractional bloch equations afbe are considered firstly we derive an analytical solution for the tfbe with an initial condition secondly we propose an effective predictor corrector method pcm for the tfbe and the error analysis for pcm is investigated furthermore we derive an effective implicit numerical method inm for the anomalous fractional bloch equations afbe and the stability and convergence of the inm are investigated we prove that the implicit numerical method for the afbe is unconditionally stable and convergent finally we present some numerical results that support our theoretical analysis

numerical algorithm to solve system of nonlinear fractional differential equations based on wavelets method and the error analysis in this paper a system of nonlinear fractional differential equations fdes are considered they have been solved by legendre wavelets method combining with its operational matrix however there are no articles about solving this system using wavelets method the main purpose of this technique is to transform the initial equations into a nonlinear system of algebraic equations which can be solved easily the convergence and error analysis are presented to show the correctness and feasibility of method proposed for solving the above mentioned problem finally the applicability and efficiency of the mentioned approach are demonstrated by three numerical examples

on bifurcation delay an alternative approach using geometric singular perturbation theory abstract to explain the phenomenon of bifurcation delay which occurs in planar systems of the form x f x z z g x z z where f x 0 0 0 and g x 0 0 changes sign at least once on the x axis we use the exchange lemma in geometric singular perturbation theory to track the limiting behavior of the solutions using the trick of extending dimension to overcome the degeneracy at the turning point we show that the limiting attracting and repulsion points are given by the well known entry exit function and the minimum of z on the trajectory is of order exp 1 also we prove smoothness of the return map up to arbitrary finite order in

on a riemann liouville generalized taylor s formula in this paper a generalized taylor s formula of the kindfx j 0najx a j 1 1 tnx whereaj r x a 0 1 is established such expression is precisely the classical taylor s formula in the particular case 1 in addition detailed expressions fortn x andaj involving the riemann liouville fractional derivative and some applications are also given

center stable manifold for planar fractional damped equations in this paper we discuss the existence of a center stable manifold for planar fractional damped equations by constructing a suitable lyapunovperron operator via giving asymptotic behavior of mittagleffler function we obtain an interesting center stable manifold theorem finally an example is provided to illustrate the result

viscous singular shock profiles for a system of conservation laws modeling two phase flow abstract this paper is concerned with singular shocks for a system of conservation laws via the dafermos regularization u t f u x t u x x for a system modeling incompressible two phase fluid flow the existence of viscous profiles is proved using geometric singular perturbation theory the weak convergence and the growth rate of the viscous solution are also derived the weak limit is the sum of a piecewise constant function and a measure supported on a shock line and the maximum value of the viscous solution is of order exp 1

turning points and relaxation oscillation cycles in simple epidemic models we study the interplay between effects of disease burden on the host population and the effects of population growth on the disease incidence in an epidemic model of sir type with demography and disease caused death we revisit the classical problem of periodicity in incidences of certain autonomous diseases under the assumption that the host population has a small intrinsic growth rate using singular perturbation techniques and the phenomenon of the delay of stability loss due to turning points we prove that large amplitude relaxation oscillation cycles exist for an open set of model parameters simulations are provided to support our theoretical results our results offer new insight into the classical periodicity problem in epidemiology our approach relies on analysis far away from the endemic equilibrium and contrasts sharply with the method of hopf bifurcations

generalized fractional controller for singular systems of differential equations abstract in this article we consider a class of singular linear systems of first order and introduce a generalized fractional order feedback controller of caputo type the closed loop system in question is a singular system of differential equations having both first and fractional order derivatives we provide a comprehensive theory for the existence and uniqueness of solutions as well as for the stability of the system with inclusion of the fractional order controller an example of a singular system with a fractional order proportional integral controller as well as an example on a 3 bus power system with inclusion of a fractional order damping controller is given to illustrate our theory

viscous standing asymptotic states of isentropic compressible flows through a nozzle in this work we consider a viscous regularization of a well known one dimensional model for isentropic viscous compressible flows through a nozzle for the existence and multiplicity of standing asymptotic states for a certain type of ducts a complete analysis in a framework of dynamical systems is provided as an application of the geometric singular perturbation theory we show that all standing asymptotic states admit viscous profiles

invariant manifolds in singular perturbation problems for ordinary differential equations based on fenichel s geometric idea invariant manifold theory is applied to singular perturbation problems this approach clarifies the nature of outer and inner solutions a specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds a method to approximate the connections is also presented

on cauchy problem for nonlinear fractional differential equation with random discrete data abstract this paper is concerned with finding the solution u x t of the cauchy problem for nonlinear fractional elliptic equation with perturbed input data this study shows that our forward problem is severely ill posed in sense of hadamard for this ill posed problem the trigonometric of non parametric regression associated with the truncation method is applied to construct a regularized solution under prior assumptions for the exact solution the convergence rate is obtained in both l2 and h q for q 0 norm moreover the numerical example is also investigated to justify our results

controllability and observability of impulsive fractional linear time invariant system in this paper we deal with the controllability and observability of impulsive fractional linear time invariant if lti for short system our main purpose is to built some necessary and sufficient conditions of controllability and observability for the if lti system at the same time we establish some conclusions of controllability and observability for a continuous fractional lti system which is a special case of the if lti system examples are given to illustrate our results

the numerical solution of a challenging class of turning point problems a curious class of challenging singularly perturbed turning point problems is considered and properties of the solutions to corresponding initial value problems are studied the solutions are exponentially small near the turning point and become unstable after passing it various state of the art codes available in matlab as well as one step and multistep methods on a uniform mesh are tested by examining a number of examples one finds that the usual error control strategies may not work when the solution near the turning point is small while one step and multistep methods on a uniform mesh work only for a moderately small perturbation parameter a scale amplification transformation however seems to give the correct solution when the solution is extremely small and or zero at the turning point extensions to problems with more equilibria are also briefly considered

shadowing lemma and singularly perturbed boundary value problems a complete procedure is given to determine the outer and inner expansions of a singularly perturbed boundary value problem in mathbb r n the validity of such expansions is deduced from a gene

a complete analysis of a classical poisson nernst planck model for ionic flow abstract in this work we examine the stationary one dimensional classical poisson nernst planck cpnp model for ionic flow a singularly perturbed boundary value problem bvp for the case of zero permanent charge we provide a complete answer concerning the existence and uniqueness of the bvp the analysis relies on a number of ingredients a geometric singular perturbation framework for a reduction to a singular bvp a reduction of the singular bvp to a matrix eigenvalue problem a relation between the matrix eigenvalues and zeros of a meromorphic function and an application of the cauchy argument principle for identifying zeros of the meromorphic function once the zeros of the meromorphic function in a stripe are determined an explicit solution of the singular bvp is available it is expected that this work would be useful for studies of other pnp systems

turning points and traveling waves in fitzhugh nagumo type equations abstract consider the following fitzhugh nagumo type equation u t u x x f u w w t g u w where f u w u u a w 1 u for some smooth function a w and g u w u w by allowing a w to cross zero and one the corresponding traveling wave equation possesses special turning points which result in very rich dynamics in this work we examine the existence of fronts backs and pulses solutions in particular the co existence of different fronts will be discussed

geometric singular perturbation theory with real noise abstract we prove the existence of families of random invariant manifolds for singularly perturbed systems of ordinary differential equations with sufficiently small real noise we use these invariant manifolds to prove a random version of the inclination theorem or exchange lemma

some results of the degenerate fractional differential system with delay an analytic study on linear systems of degenerate fractional differential equations with constant coefficients is presented we discuss the existence and uniqueness of solutions for the initial value problem of linear degenerate fractional differential systems the exponential estimation of the degenerate fractional differential system with delay and sufficient conditions for the finite time stability for the system are obtained finally an example is provided to illustrate the effectiveness of the presented analytical approaches

one dimensional steady state poisson nernst planck systems for ion channels with multiple ion species abstract the one dimensional poisson nernst planck pnp system is a basic model for ion flow through membrane channels if the debye length is much smaller than the characteristic radius of the channel the pnp system can be treated as a singularly perturbed system we provide a geometric framework for the study of the steady state pnp system involving multiple types of ion species with multiple regions of piecewise constant permanent charge special structures of this particular problem are revealed which together with the general framework allows one to reduce the existence and multiplicity of singular orbits to a system of nonlinear algebraic equations near each singular orbit an application of the exchange lemma from the geometric singular perturbation theory gives rise to the existence and local uniqueness of a solution of the singular boundary value problem a new phenomenon on multiplicity and spatial behavior of steady states involving three or more types of ion species is discovered in an example the phenomenon cannot occur when only two types of ion species are involved

caputo and related fractional derivatives in singular systems abstract by using the caputo c fractional derivative and two recently defined alternative versions of this derivative the caputo fabrizio cf and the atangana baleanu ab fractional derivative firstly we focus on singular linear systems of fractional differential equations with constant coefficients that can be non square matrices or square singular we study existence of solutions and provide formulas for the case that there do exist solutions then we study the existence of unique solution for given initial conditions several numerical examples are given to justify our theory

geometric singular perturbation approach to steady state poisson nernst planck systems boundary value problems of a one dimensional steady state poisson nernst planck pnp system for ion flow through a narrow membrane channel are studied by assuming the ratio of the debye length to a characteristic length to be small the pnp system can be viewed as a singularly perturbed problem with multiple time scales and is analyzed using the newly developed geometric singular perturbation theory within the framework of dynamical systems the global behavior is first studied in terms of limiting fast and slow systems it is rather surprising that a complete set of integrals is discovered for the nonlinear limiting fast system this allows a detailed description of the boundary layers for the problem the slow system itself turns out to be a singularly perturbed one too which indicates that the singularly perturbed pnp system has three different time scales a singular orbit zeroth order approximation of the boundary value problem is identified based on the dynamics of limiting fast and slow

a note on property of the mittag leffler function abstract recently the authors have found in some publications that the following property 0 1 of mittag leffler function is taken for granted and used to derive other properties 0 1 e a t s e a t e a s t s 0 where a is a real constant and 0 in this note it is proved that the above property is unavailable unless 1 or a 0 moreover a new equality on e a t is developed whose limit state as 1 is just the property 0 1

exact and discretized stability of the bagley torvik equation abstract the paper discusses stability and asymptotic properties of a two term linear fractional differential equation involving the bagley torvik equation as the particular case these properties are analysed for the exact as well as numerical solutions obtained from the grunwald letnikov discretization of the studied differential equation as the main results precise descriptions of the exact and discretized stability regions are presented including the decay rate of the solutions these results enable us among others to observe similarities and distinctions between the asymptotic behaviour of the classical and fractional damping models

invariant manifolds and singularly perturbed boundary value problems boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers this paper addresses the question of when singular solutions to such prob

on systems of linear fractional differential equations with constant coefficients abstract this paper deals with the study of linear systems of fractional differential equations such as the following system 1 y a x y b x where y is the riemann liouville or the caputo fractional derivative of order 0 2 a x a 11 x a 1 n x a n 1 x a nn x b x b 1 x b n x are matrices of known real functions in a way analogous to the usual case we show how a generalized matrix exponential function and certain fractional green function in connection with the mittag leffler type functions would allow us to obtain an explicit representation of the general solution to the system 1 when a is a constant matrix

exchange lemmas 1 deng s lemma abstract deng s lemma gives estimates on the behavior of solutions of ordinary differential equations in the neighborhood of a partially hyperbolic equilibrium we prove a generalization in which partially hyperbolic equilibrium is replaced by normally hyperbolic invariant manifold

identifying the successive blumenthal getoor indices of a discretely observed process this paper studies the identification of the levy jump measure of a discretely sampled semimartingale we define successive blumenthal getoor indices of jump activity and show that the leading index can always be identified but that higher order indices are only identifiable if they are sufficiently close to the previous one even if the path is fully observed this result establishes a clear boundary on which aspects of the jump measure can be identified on the basis of discrete observations and which cannot we then propose an estimation procedure for the identifiable indices and compare the rates of convergence of these estimators with the optimal rates in a special parametric case which we can compute explicitly 1 introduction let x be a one dimensional semimartingale defined on a finite time interval 0 t our objective is to make some progress toward the identification of the jump measure of x at high frequency the motivation for what follows has its roots in a family of econometric problems which can be stated as follows we observe a single path of x but not fully although other observation

diversity of traveling wave solutions in fitzhugh nagumo type equations abstract in this work we consider the diversity of traveling wave solutions of the fitzhugh nagumo type equations u t u x x f u w w t e g u w where f u w u u a w 1 u for some smooth function a w and g u w u w when a w crosses zero and one the corresponding profile equation possesses special turning points which result in very rich dynamics in w liu e van vleck turning points and traveling waves in fitzhugh nagumo type equations j differential equations 225 2006 381 410 liu and van vleck examined traveling waves whose slow orbits lie only on two portions of the slow manifold and obtained the existence results by using the geometric singular perturbation theory based on the ideas of their work we study the co existence of different traveling waves whose slow orbits could involve all portions of the slow manifold there are more complicated and richer dynamics of traveling waves than those of w liu e van vleck turning points and traveling waves in fitzhugh nagumo type equations j differential equations 225 2006 381 410 we give a complete classification of all different fronts of traveling waves and provide an example to support our theoretical analysis

testing whether jumps have finite or infinite activity we propose statistical tests to discriminate between the finite and infinite activity of jumps in a semimartingale discretely observed at high frequency the two statistics allow for a symmetric treatment of the problem we can either take the null hypothesis to be finite activity or infinite activity when implemented on high frequency stock returns both tests point toward the presence of infinite activity jumps in the data

exchange lemmas 2 general exchange lemma abstract exchange lemmas are used in geometric singular perturbation theory to track flows near normally hyperbolic invariant manifolds we prove a general exchange lemma and show that it implies versions of existing exchange lemmas for rectifiable slow flows and loss of stability turning points

analytic study on linear neutral fractional differential equations this paper is devoted to investigating linear neutral fractional differential equations with constant coefficients the existence uniqueness and iterative formula of the solution are obtained meanwhile dependence of the solution on initial value and the general solution represented by a fundamental solution matrix are discussed several examples are given to illustrate the applications of our results some conclusions in the literature are extended greatly

on stable manifolds for planar fractional differential equations in this paper we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations the construction of this stable manifold is based on the associated lyapunov perron operator an example is provided to illustrate the result

numerical solution of fractional integro differential equations by a hybrid collocation method fractional integro differential equations have been recently solved by many methods such as adomian decomposition method differential transform method collocation method and taylor expansion approach in this paper a hybrid collocation method is used which combines a non polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval a theoretical analysis for the convergence order of the method is presented some numerical examples are given which confirm the theoretical results

the sil nikov problem exponential expansion strong λ lemma c1 linearization and homoclinic bifurcation on introduit une nouvelle theorie appelee le developpement exponentiel et on explore ses interactions avec la theorie de la variete invariante locale les lemmes et la theorie de linerarisation reguliere