the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

almost sure convergence of a galerkin approximation for spdes of zakai type driven by square integrable martingales this work describes a galerkin type method for stochastic partial differential equations of zakai type driven by an infinite dimensional cadlag square integrable martingale error estimates in the semidiscrete case where discretization is only done in space are derived in l p and almost sure senses simulations confirm the theoretical results

pathwise approximation of stochastic differential equations on domains higher order convergence rates without global lipschitz coefficients we study the approximation of stochastic differential equations on domains for this we introduce modified ito taylor schemes which preserve approximately the boundary domain of the equation under consideration assuming the existence of a unique non exploding solution we show that the modified ito taylor scheme of order has pathwise convergence order e for arbitrary e 0 as long as the coefficients of the equation are sufficiently differentiable in particular no global lipschitz conditions for the coefficients and their derivatives are required this applies for example to the so called square root diffusions

universal scaling behaviour for iterated maps in the complex plane according to the theory of schroder and siegel certain complex analytic maps possess a family of closed invariant curves in the complex plane we have made a numerical study of these curves by iterating the map and have found that the largest curve is a fractal when the winding number of the map is the golden mean the fractal curve has universal scaling properties and the scaling parameter differs from those found for other types of maps also for this winding number there are universal scaling functions which describe the behaviour asn of theq n th iterates of the map whereq n is thenth fibonacci number

critical dynamics and trees abstract we show that the dependence of the critical perturbation of hamiltonian systems on the frequency v is related to the statistics of large planar trees using a complex map for illustration a critical function k v is given by the radii of convergence of power series whose coefficients are represented as sums over trees a preliminary approximation based on this relation leads to an expression for the critical function as the solution of a transcendental equation which appears to provide a reasonable approximation to the true function and to have the right structure near the dense set of singularities

restart could optimize the probability of success in a bernoulli trial the recently noticed ability of restart to reduce the expected completion time of first passage processes allows appealing opportunities for performance improvement in a variety of settings however complex stochastic processes often exhibit several possible scenarios of completion which are not equally desirable in terms of efficiency here we show that restart may have profound consequences on the splitting probabilities of a bernoulli like first passage process i e of a process which can end with one of two outcomes particularly intriguing in this respect is the class of problems where a carefully adjusted restart mechanism maximizes the probability that the process will complete in a desired way we reveal the universal aspects of this kind of optimal behavior by applying the general approach recently proposed for the problem of first passage under restart

monotonic continuous time random walks with drift and stochastic reset events in this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes we focus our attention on monotonic continuous time random walks with a constant drift the process increases between the reset events either by the effect of the random jumps or by the action of the deterministic drift as a result of all these combined factors interesting properties emerge like the existence for any drift strength of a stationary transition probability density function or the faculty of the model to reproduce power law like behavior general formulas for two extreme statistics the survival probability and the mean exit time are also derived to corroborate in an independent way the results of the paper monte carlo methods were used these numerical estimations are in full agreement with the analytical predictions

critical behavior of a kam surface i empirical results kolmogorov arnol d moser kam surfaces are studied in the context of a perturbed two dimensional twist map in particular we ask how a kam surface can disappear as the perturbation parameter is increased following greene we use cycles to numerically construct the kam curve and discover that at the critical coupling it shows structure at all length scales aspects of this structure are fitted by a scaling analysis critical indices and scaling functions are determined numerically some evidence is presented which suggests that the results are universal

taylor expansions of solutions of stochastic partial differential equations with additive noise the solution of a parabolic stochastic partial differential equation spde driven by an infinite dimensional brownian motion is in general not a semi martingale anymore and does in general not satisfy an ito formula like the solution of a finite dimensional stochastic ordinary differential equation sode in particular it is not possible to derive stochastic taylor expansions as for the solution of a sode using an iterated application of the ito formula consequently until recently only low order numerical approximation results for such a spde have been available here the fact that the solution of a spde driven by additive noise can be interpreted in the mild sense with integrals involving the exponential of the dominant linear operator in the spde provides an alternative approach for deriving stochastic taylor expansions for the solution of such a spde essentially the exponential factor has a mollifying effect and ensures that all integrals take values in the hilbert space under consideration the iteration of such integrals allows us to derive stochastic taylor expansions of arbitrarily high order which are robust in the sense that they also hold for other types of driving noise processes such as fractional brownian motion combinatorial concepts of trees and woods provide a compact formulation of the taylor expansions

complexity of critical functions for hamiltonian systems transition to predominantly chaotic motion in hamiltonian systems with two degrees of freedom is described by a complicated function of frequency which is called the critical function a graph of this function is a fractal set with the local structure which is believed to depend only on the arithmetic nature of the frequency we calculated numerically fractal dimensions of this function for a few typical systems using the method of modular smoothing and an efficient algorithm for computation of the fractal dimensions the dimensions which measure the complexity of the fractal are indeed the same within the error bounds and are equal to the dimension of the exponent of the brjuno function which is a purely arithmetic function

critical functions for complex analytic maps critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area preserving maps the author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps by means of the majorant series method some rigorous estimates are given for complex area preserving maps like the semistandard map and the modulated singular map in particular the author makes use of the brjuno function to interpolate critical maps and proves that the convergence of the brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number the author also discusses the optimality of the rigorous bounds obtained

higher order pathwise numerical approximations of spdes with additive noise in this article the pathwise numerical approximation of semilinear parabolic stochastic partial differential equations spdes driven by additive noise is considered a new numerical scheme for the time and space discretization of such spdes is proposed and this scheme is shown to converge for such spdes faster than standard numerical schemes such as the linear implicit euler scheme or the linear implicit crank nicholson scheme the suggested scheme takes advantage of the smoothing effect of the dominant linear operator and of two linear functionals of the noise process of the spde the abstract setting under which the scheme is analyzed includes spdes driven by fractional brownian motions too

numerical computation of the asymptotic size of the rotation domain for the arnold family abstract we consider the arnold tongue of the arnold family of circle maps associated to a fixed diophantine rotation number the corresponding maps of the family are analytically conjugate to a rigid rotation this conjugation is defined on a maximal complex strip of the circle and after a suitable scaling the size of this strip is given by an analytic function of the perturbative parameter the main purpose of this paper is to perform a numerical accurate computation of this function and of its taylor expansion this allows us to verify previous theoretical results the rotation numbers we select are quadratic irrationals mainly the golden mean by introducing a nonstandard extrapolation process especially suited for the problem we compute all the quantities required rotation numbers arnold tongues fourier and taylor coefficients with high precision

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

the brjuno function continuously estimates the size of quadratic siegel disks if a is an irrational number yoccoz defined the brjuno function by a ni 0 a0 a1 i i i an 1 log 1 an where a0 is the fractional part of a and an 1 is the fractional part of 1 an the numbers a such that a i are called the brjuno numbers the quadratic polynomial p a z iu e2i az z2 has an indifferent fixed point at the origin if p a is linearizable we let r a be the conformal radius of the siegel disk and we set r a 0 otherwise yoccoz y proved that a i if and only if r a 0 and that the restriction of a iu a log r a to the set of brjuno numbers is bounded from below by a universal constant in bc2 we proved that it is also bounded from above by a universal constant in fact marmi moussa and yoccoz mmy conjecture that this function extends to r as a hi older function of exponent 1 2 in this article we prove that there is a continuous extension to r

phase transitions in optimal search times how random walkers should combine resetting and flight scales recent works have explored the properties of levy flights with resetting in one dimensional domains and have reported the existence of phase transitions in the phase space of parameters which minimizes the mean first passage time mfpt through the origin l kusmierz et al phys rev lett 113 220602 2014 here we show how actually an interesting dynamics including also phase transitions for the minimization of the mfpt can also be obtained without invoking the use of levy statistics but for the simpler case of random walks with exponentially distributed flights of constant speed we explore this dynamics both in the case of finite and infinite domains and for different implementations of the resetting mechanism to show that different ways to introduce resetting consistently lead to a quite similar dynamics the use of exponential flights has the strong advantage that exact solutions can be obtained easily for the mfpt through the origin so a complete analytical characterization of the system dynamics can be provided furthermore we discuss in detail how the phase transitions observed in random walks with resetting are closely related to several ideas recurrently used in the field of random search theory in particular to other mechanisms proposed to understand random search in space as mortal random walks or multiscale random walks as a whole we corroborate that one of the essential ingredients behind mfpt minimization lies in the combination of multiple movement scales regardless of their specific origin

modular smoothing and kam tori abstract transformation properties of kam tori under the action of a modular group on frequencies are analysed and used for calculating the kam tori of a hamiltonian system with one and a half degrees of freedom first we derive a relation between the kam torus with a frequency v and the kam torus with the frequency v 1 v 1 v where 1 v is the integer part of 1 v given by a continuous function l1 v in the double limit v m n and k 0 where m n is a rational frequency and k is a perturbation parameter the function l1 v was introduced by buric percival and vivaldi and also gives the transformation properties of the critical parameter as a function of the frequency this function provides a zeroth order modular smoothing approximation to the invariant tori which is better than a perturbation expansion the first order corrections to this approximation can be calculated from a function r1 v that appears to be smooth

optimal first arrival times in levy flights with resetting we consider the diffusive motion of a particle performing a random walk with levy distributed jump lengths and subject to a resetting mechanism bringing the walker to an initial position at uniformly distributed times in the limit of an infinite number of steps and for long times the process converges to superdiffusive motion with replenishment we derive a formula for the mean first arrival time mfat to a predefined target position reached by a meandering particle and we analyze the efficiency of the proposed searching strategy by investigating criteria for an optimal a shortest possible mfat

on the relationship between the bruno function and the breakdown invariant tori we study the ratio epsilon c omega exp eta b omega where epsilon c omega is the breakdown threshold function for invariant tori eta is a parameter and b omega is the bruno function which is purely arithmetic i e it only depends by the continued fractions expansion of number omega we consider the standard map as a model and we focus our analysis on the exponential decay of the chaotic regions close to an invariant torus with diophantine rotation frequency our numerical experiments together with some heuristic consideration show that epsilon c omega exp eta b omega is not a continuous function on diophantine numbers omega for all values of eta

ito s and tanaka s type formulae for the stochastic heat equation the linear case in this paper we consider the linear stochastic heat equation with additive noise in dimension one then using the representation of its solution x as a stochastic convolution of the cylindrical brownian motion with respect to an operator valued kernel we derive ito s and tanaka s type formulae associated to x

the transition to chaos for a special solution of the area preserving quadratic map abstract following previous work on chaotic boundaries of half plane hamiltonian maps a special solution of the area preserving quadratic map is introduced and investigated the breakdown of regular bounded motion on invariant curves is found from the radius of convergence of a power series whose successive terms oscillate wildly due to the presence of small divisors previous techniques for taming such series are found to be insufficient and new ones are introduced it is found that half plane hamiltonian maps appear to have certain universal features and that the chaotic boundary has similarities to the boundaries of siegel domains of complex conformal maps the chaotic boundary function c has some interesting new features which are not fully understood

generalized integrating factor methods for stiff pdes the integrating factor if method for numerical integration of stiff nonlinear pdes has the disadvantage of producing large error coefficients when the linear term has large norm we propose a generalization of the if method and in particular construct multistep type methods with several orders of magnitude improved accuracy we also consider exponential time differencing etd methods and point out connections with a particular application of the commutator free lie group methods we present a new fourth order etdrk method with improved accuracy the methods considered are compared in several numerical examples

critical kam circles and the brjuno function the authors consider a complex hamiltonian map and show how to approximate the critical value of the perturbation parameter at which a given kam circle disappears by means of a function the brjuno function which only depends on the continued fraction expansion of the rotation number

robust random search with scale free stochastic resetting a new model of search based on stochastic resetting is introduced wherein rate of resets depends explicitly on time elapsed since the beginning of the process it is shown that rate inversely proportional to time leads to paradoxical diffusion which mixes self similarity and linear growth of the mean square displacement with nonlocality and non gaussian propagator it is argued that such resetting protocol offers a general and efficient search boosting method that does not need to be optimized with respect to the scale of the underlying search problem e g distance to the goal and is not very sensitive to other search parameters both subdiffusive and superdiffusive regimes of the mean squared displacement scaling are demonstrated with more general rate functions

b series and order conditions for exponential integrators we introduce a general format of numerical ode solvers which include many of the recently proposed exponential integrators we derive a general order theory for these schemes in terms of b series and bicolored rooted trees to ease the construction of specific schemes we generalize an idea of zennaro em math comp 46 1986 pp 119 133 and define natural continuous extensions in the context of exponential integrators this leads to a relatively easy derivation of some of the most popular recently proposed schemes the general format of schemes considered here makes use of coefficient functions which will usually be selected from some finite dimensional function spaces we will derive lower bounds for the dimension of these spaces in terms of the order of the resulting schemes finally we illustrate the presented ideas by giving examples of new exponential integrators of orders 4 and 5

renormalization group analysis of quasi periodicity in analytic maps analytic maps of the form f z e 2 pi i omega z mathcal o z 2 display quasiperiodicity when satisfies a diophantine condition quasiperiodic motion is confined to a neighborhood of the origin known as a siegel domain the boundary of this domain obeys universal scaling relations in this paper we investigate these scaling relations through a renormalization group analysis and we discuss singularities and asymptotic form of the scaling function

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

a mild ito formula for spdes this article introduces a new somehow mild ito type formula for the solution process of a stochastic partial differential equation of evolutionary type

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

diffusion with stochastic resetting at power law times what happens when a continuously evolving stochastic process is interrupted with large changes at random intervals ensuremath tau distributed as a power law ensuremath sim ensuremath tau ensuremath 1 ensuremath alpha phantom rule 0 28em 0ex ensuremath alpha g0 modeling the stochastic process by diffusion and the large changes as abrupt resets to the initial condition we obtain exact closed form expressions for both static and dynamic quantities while accounting for strong correlations implied by a power law our results show that the resulting dynamics exhibits a spectrum of rich long time behavior from an ever spreading spatial distribution for ensuremath alpha l1 to one that is time independent for ensuremath alpha g1 the dynamics has strong consequences on the time to reach a distant target for the first time we specifically show that there exists an optimal ensuremath alpha that minimizes the mean time to reach the target thereby offering a step towards a viable strategy to locate targets in a crowded environment

explicit exponential runge kutta methods for semilinear parabolic problems the aim of this paper is to analyze explicit exponential runge kutta methods for the time integration of semilinear parabolic problems the analysis is performed in an abstract banach space framework of sectorial operators and locally lipschitz continuous nonlinearities we commence by giving a new and short derivation of the classical nonstiff order conditions for exponential runge kutta methods but the main interest of our paper lies in the stiff case by expanding the errors of the numerical method in terms of the solution we derive new order conditions that form the basis of our error bounds for parabolic problems we show convergence for methods up to order four and we analyze methods that were recently presented in the literature these methods have classical order four but they do not satisfy some of the new conditions therefore an order reduction is expected we present numerical experiments which show that this order reduction in fact arises in practical examples based on our new conditions we finally construct methods that do not suffer from order reduction

integral fluctuation theorems for stochastic resetting systems we study the stochastic thermodynamics of resetting systems violation of microreversibility means that the well known derivations of fluctuations theorems break down for dynamics with resetting despite that we show that stochastic resetting systems satisfy two integral fluctuation theorems the first is the hatano sasa relation describing the transition between two steady states the second integral fluctuation theorem involves a functional that includes both dynamical and thermodynamic contributions we find that the second law like inequality found by fuchs et al for resetting systems europhys lett 113 60009 2016 euleej0295 507510 1209 0295 5075 113 60009 can be recovered from this integral fluctuation theorem with the help of jensen s inequality

strong convergence rate of splitting schemes for stochastic nonlinear schrodinger equations abstract in this paper we show that solutions of stochastic nonlinear schrodinger nls equations can be approximated by solutions of coupled splitting systems based on these systems we propose a new kind of fully discrete splitting schemes which possess algebraic strong convergence rates for stochastic nls equations key ingredients of our approach are using the exponential integrability and stability of the corresponding splitting systems and numerical approximations in particular under very mild conditions we derive the optimal strong convergence rate o n 2 1 2 of the spectral splitting crank nicolson scheme where n and denote the dimension of the approximate space and the time step size respectively

hamiltonian maps in the complex plane abstract the standard map is a nonintegrable discrete time analog of the vertical pendulum detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency the functional dependence of the coordinate q on the canonical angle variable is analtyically continued into the complex plane where natural boundaries are found at constant absolute values of im the boundaries represent the appearance of chaotic motion in the complex plane when the domain of analyticity shrinks to zero the kam invariant curve is destroyed two independent numerical methods with fourier analysis in the angle variable were used one based on a variation annihilation method and the other on a double expansion the results were further checked by direct solution of the complex equations of motion the numerically simpler but intrinsically complex semipendulum and semistandard map are also studied we conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or kam surfaces of general nonintegrable systems with analytic hamiltonians

are there critical points on the boundaries of singular domains generalizing a result of e ghys we prove a general theorem that implies that if a rational functionf of the riemann sphere of degree 2 leaves invariant a singular domainc a disk or a ring on which the rotation number off satisfies a diophantine condition provided that on bar c f is injective then each boundary component ofc contains critical point off the injectivity condition is always satisfied for singular disks associated to linearizable periodic elliptic points off z z n a withne n 2 andae we also show that the singular disks associated to periodic elliptic points off z e az that satisfy a diophantine condition are unbounded in in the end of the paper we give a survey of the theory of iteration of entire functions of

effects of refractory period on stochastic resetting we consider a stochastic process undergoing resetting after which a random refractory period is imposed in this period the process is quiescent and remains at the resetting position using a first renewal approach we compute exactly the stationary position distribution and analyse the emergence of a delta peak at the resetting position in the case of a power law distribution for the refractory period we find slow relaxation we generalise our results to the case when the resetting period and the refractory period are correlated by computing the laplace transform of the survival probability of the process and the mean first passage time i e the mean time to completion of a task we also compute exactly the joint distribution of the active and absorption time to a fixed target

pathwise convergence of an efficient scheme for spdes with non globally lipschitz nonlinearity abstract this paper aims to extend the scheme proposed in jentzen et al 2011 for stochastic partial differential equations spdes with global lipschitz coefficients to non global lipschitz coefficients we investigate the pathwise convergence of the scheme for a class of semilinear parabolic spdes with non globally lipschitz nonlinearity we show first that the scheme is convergent uniformly in time and space under lipschitz assumption on the nonlinearity of the spde then we obtain the convergence in the case of non globally lipschitz nonlinearity via a localization technique compared to the scheme introduced in jentzen 2009 for spdes under non global lipschitz coefficients the scheme considered in this paper is simpler in the sense that the former uses two linear functionals of the noise while the latter uses one

order conditions of stochastic runge kutta methods by b series in this paper general order conditions and a global convergence proof are given for stochastic runge kutta methods applied to stochastic ordinary differential equations sodes of stratonovich type this work generalizes the ideas of b series as applied to deterministic ordinary differential equations odes to the stochastic case and allows a completely general formalism for constructing high order stochastic methods either explicit or implicit some numerical results will be given to illustrate this theory

optimal stochastic restart renders fluctuations in first passage times universal stochastic restart may drastically reduce the expected run time of a computer algorithm expedite the completion of a complex search process or increase the turnover rate of an enzymatic reaction these diverse first passage time fpt processes seem to have very little in common but it is actually quite the other way around here we show that the relative standard deviation associated with the fpt of an optimally restarted process i e one that is restarted at a constant nonzero rate which brings the mean fpt to a minimum is always unity we interpret further generalize and discuss this finding and the implications arising from it

diffusion with resetting in bounded domains we consider the one dimensional diffusion in a bounded domain with stochastic resetting we start our analysis by presenting a method to derive the master equation for different resetting mechanisms in the next step we compute the non equilibrium steady state for a special case of this differential equation then we consider the existence of an absorbing point in the system and calculate the mean time to absorption of the diffusive particle by the target numerical and analytical calculations of the optimal resetting rate reveal a second order phase transition finally we discuss different special cases of the presented problem

a method for accurate stability bounds in a small denominator problem the author considers the problem of obtaining realistic lower bounds for the siegel radius recent advances of the analysis of siegel disks allows one to give a very accurate numerical algorithm based on rigorous results he finds that for non quadratic polynomial maps the maximal siegel radius might correspond to rotation numbers different from the golden mean

the brjuno functions and their regularity properties we show that various possible versions of the brjuno function based on different kinds of continued fraction developments are all equivalent and we study their regularity l p bmo and holder properties through a systematic analysis of the functional equation which they fulfill

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

first passage under stochastic resetting in an interval we consider a brownian particle diffusing in a one dimensional interval with absorbing end points we study the ramifications when such motion is interrupted and restarted from the same initial configuration we provide a comprehensive study of the first passage properties of this trapping phenomena we compute the mean first passage time and derive the criterion on which restart always expedites the underlying completion we show how this set up is a manifestation of a success failure problem we obtain the success and failure rates and relate them with the splitting probabilities namely the probability that the particle will eventually be trapped on either of the boundaries without hitting the other one numerical studies are presented to support our analytic results

complex brjuno functions 1 1 the real brjuno function let r q and let pn qn n 0 be the sequence of the convergents of its continued fraction expansion a brjuno number is an irrational number such that n 0 log qn 1 qn the importance of brjuno numbers comes from the study of one dimensional analytic small divisors problems in the case of germs of holomorphic diffeomorphisms of one complex variable with an indifferent fixed point extending a previous result of siegel s brjuno proved br1 br2 that all germs with linear part e are linearizable if is a brjuno number conversely the third author proved that this condition is also necessary yo1 similar results hold for the local conjugacy of analytic diffeomorphisms of the circle kh yo2 yo3 and for some area preserving maps ma da1 including the standard family da2 bg1 bg2 the set of brjuno numbers is invariant under the action of the modular group pgl 2 z and it can be characterized as the set where the brjuno function b r q r is finite this arithmetical function is z periodic and satisfies the remarkable functional equation

continuous time random walk under time dependent resetting continuous time random walks of a particle that is randomly reset to an initial position are considered the distribution of the waiting time between the reset events is represented as a sum of an arbitrary number of exponentials the governing equation of this stochastic process is established the mean first passage time to a particular position is calculated it is shown that anomalous subdiffusion has a significant impact on the shape of the stationary state

on the splitting up method and stochastic partial differential equations we consider two stochastic partial differential equations du varepsilon t l ru varepsilon t f r t dv varepsilon t r m k u varepsilon t g k t circ dy t k qquad hspace 5pt varepsilon 0 1 driven by the same multidimensional martingale y yk and by different increasing processes vr0 vr1 r 1 2 d1 where lr and mk are second and first order partial differential operators and stands for the stratonovich differential we estimate the moments of the supremum in t of the sobolev norms of u1 t u0 t in terms of the supremum of the differences break vr0t vr1t hence we obtain moment estimates for the error of a multistage splitting up method for stochastic pdes in particular for the equation of the unnormalized conditional density in nonlinear filtering

improved kam estimates for the siegel radius for the siegel center problem we explore the possibility of improving the kam estimates with a view to possible extensions to hamiltonian systems the use of a suitable norm and explicit perturbative computations allow estimates to within a factor 2 of the siegel radius for the quadratic map

diffusion with optimal resetting we consider the mean time to absorption by an absorbing target of a diffusive particle with the addition of a process whereby the particle is reset to its initial position with rate r we consider several generalisations of the model of m r evans and s n majumdar 2011 diffusion with stochastic resetting phys rev lett 106 160601 i a space dependent resetting rate r x ii resetting to a random position z drawn from a resetting distribution p z iii a spatial distribution for the absorbing target pt x as an example of i we show that the introduction of a non resetting window around the initial position can reduce the mean time to absorption provided that the intial position is sufficiently far from the target we address the problem of optimal resetting that is minimising the mean time to absorption for a given target distribution for an exponentially decaying target distribution centred at the origin we show that a transition in the optimal resetting distribution occurs as the target distribution narrows

exponential rosenbrock type methods we introduce a new class of exponential integrators for the numerical integration of large scale systems of stiff differential equations these so called rosenbrock type methods linearize the flow in each time step and make use of the matrix exponential and related functions of the jacobian in contrast to standard integrators the methods are fully explicit and do not require the numerical solution of linear systems we analyze the convergence properties of these integrators in a semigroup framework of semilinear evolution equations in banach spaces in particular we derive an abstract stability and convergence result for variable step sizes this analysis further provides the required order conditions and thus allows us to construct pairs of embedded methods we present a third order method with two stages and a fourth order method with three stages respectively the application of the required matrix functions to vectors are computed by krylov subspace approximations we briefly discuss these implementation issues and we give numerical examples that demonstrate the efficiency of the new integrators

random walk with random resetting to the maximum position we study analytically a simple random walk model on a one dimensional lattice where at each time step the walker resets to the maximum of the already visited positions to the rightmost visited site with a probability r and with probability 1 r it undergoes symmetric random walk i e it hops to one of its neighboring sites with equal probability 1 r 2 for r 0 it reduces to a standard random walk whose typical distance grows as n for large n in the presence of a nonzero resetting rate 0 r 1 we find that both the average maximum and the average position grow ballistically for large n with a common speed v r moreover the fluctuations around their respective averages grow diffusively again with the same diffusion coefficient d r we compute v r and d r explicitly we also show that the probability distribution of the difference between the maximum and the location of the walker becomes stationary as n however the approach to this stationary distribution is accompanied by a dynamical phase transition characterized by a weakly singular large deviation function we also show that r 0 is a special critical point for which the growth laws are different from the r 0 case and we calculate the exact crossover functions that interpolate between the critical r 0 and the off critical r 0 behavior for finite but large n

pathwise convergent higher order numerical schemes for random ordinary differential equations random ordinary differential equations rodes are ordinary differential equations odes with a stochastic process in their vector field they can be analysed pathwise using deterministic calculus but since the driving stochastic process is usually only holder continuous in time the vector field is not differentiable in the time variable so traditional numerical schemes for odes do not achieve their usual order of convergence when applied to rodes nevertheless deterministic calculus can still be used to derive higher order numerical schemes for rodes via integral versions of implicit taylor like expansions the theory is developed systematically here and applied to illustrative examples involving brownian motion and fractional brownian motion as the driving processes

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the frequency

transformation properties of perturbation expansions around elliptic fixed points transformation properties of perturbation expansions of vibrational quasi periodic orbits upon the modular transformations of frequencies are studied using simple but nontrivial examples the siegel complex quadratic map and special solutions of the real area preserving quadratic map it is shown that the transformation properties are similar to those in the previously studied case of the rotational invariant tori except for some special features of the real map related to the atypical nature of 1 3 resonance in this case

dynamical transition in the temporal relaxation of stochastic processes under resetting a stochastic process when subject to resetting to its initial condition at a constant rate generically reaches a nonequilibrium steady state we study analytically how the steady state is approached in time and find an unusual relaxation mechanism in these systems we show that as time progresses an inner core region around the resetting point reaches the steady state while the region outside the core is still transient the boundaries of the core region grow with time as power laws at late times with new exponents alternatively at a fixed spatial point the system undergoes a dynamical transition from the transient to the steady state at a characteristic space dependent timescale t x we calculate analytically in several examples the large deviation function associated with this spatiotemporal fluctuation and show that generically it has a second order discontinuity at a pair of critical points characterizing the edges of the inner core these singularities act as separatrices between typical and atypical trajectories our results are verified in the numerical simulations of several models such as simple diffusion and fluctuating one dimensional interfaces

michaelis menten reaction scheme as a unified approach towards the optimal restart problem we study the effect of restart and retry on the mean completion time of a generic process the need to do so arises in various branches of the sciences and we show that it can naturally be addressed by taking advantage of the classical reaction scheme of michaelis and menten stopping a process in its midst only to start it all over again may prolong leave unchanged or even shorten the time taken for its completion here we are interested in the optimal restart problem i e in finding a restart rate which brings the mean completion time of a process to a minimum we derive the governing equation for this problem and show that it is exactly solvable in cases of particular interest we then continue to discover regimes at which solutions to the problem take on universal details independent forms which further give rise to optimal scaling laws the formalism we develop and the results obtained can be utilized when optimizing stochastic search processes and randomized computer algorithms an immediate connection with kinetic proofreading is also noted and discussed

overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space time noise we consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space time white noise we introduce a new numerical scheme for the time discretization of the finite dimensional galerkin stochastic differential equations which we call the exponential euler scheme and show that it converges in the strong sense faster than the classical numerical schemes such as the linear implicit euler scheme or the crank nicholson scheme for this equation with the general noise in particular we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1 3 which exceeds the barrier order 1 6 for numerical schemes using only basic increments of the noise process reported previously by contrast our scheme takes advantage of the smoothening effect of the laplace operator and of a linear functional of the noise and therefore overcomes this order barrier

large deviations for markov processes with resetting markov processes restarted or reset at random times to a fixed state or region in space have been actively studied recently in connection with random searches foraging and population dynamics here we study the large deviations of time additive functions or observables of markov processes with resetting by deriving a renewal formula linking generating functions with and without resetting we are able to obtain the rate function of such observables characterizing the likelihood of their fluctuations in the long time limit we consider as an illustration the large deviations of the area of the ornstein uhlenbeck process with resetting other applications involving diffusions random walks and jump processes with resetting or catastrophes are discussed

universal small scale structure near the boundary of siegel disks of arbitrary rotation number abstract we present numerical evidence that for analytic maps of the complex plane with a siegel disk of any irrational rotation number and a quadratic critical point on its boundary the small scale structure near the critical point can be described asymptotically by a universal two parameter family we propose an explanation in terms of an ergodic attractor for a renormalisation operator