fractal dimension of higher dimensional chaotic repellors abstract using examples we test formulae previously conjectured to give the fractal information dimension of chaotic repellors and their stable and unstable manifolds in typical dynamical systems in terms of the lyapunov exponents and the characteristic escape time from the repellor our main example a three dimensional chaotic scattering billiard yields a new structure for its invariant manifolds this system also provides an example of a system which is not typical and illustrates how perturbation to the system restores typicality and the applicability of the dimension formulae

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

variants of the deteriorated pss preconditioner for saddle point problems two new preconditioners which can be viewed as variants of the deteriorated positive definite and skew hermitian splitting preconditioner are proposed for solving saddle point problems the corresponding iteration methods are proved to be convergent unconditionally for cases with positive definite leading blocks the choice strategies of optimal parameters for the two iteration methods are discussed based on two recent optimization results for extrapolated cayley transform which result in faster convergence rate and more clustered spectrum compared with some preconditioners of similar structures the new preconditioners have better convergence properties and spectrum distributions in addition more practical preconditioning variants of the new preconditioners are considered numerical experiments are presented to illustrate the advantages of the new preconditioners over some similar preconditioners to accelerate gmres c 2018 elsevier ltd all rights reserved

intertwined basins of attraction of dynamical systems in this paper we investigate a kind of intertwining phenomenon and give a new definition of intertwined attractors of the dynamical systems and get some related results

strange nonchaotic attractors with wada basins a b s t r a c t we demonstrate strange nonchaotic attractors with wada basins snaswb and verify the abundance of snaswb in a quasiperiodically forced holmes map we identify the routes to the creation of the snaswb in a two parameter space the snaswb are characterized by the maximal lyapunov exponent by the estimation of the phase sensitivity exponent and by the singular continuous spectra we observe that the snas basins are totally wada basins in a large range of parameters the topological structures of the snas wada basins are distinguished by the basin cell method we investigate the underlying mechanism for the abundance of snaswb which is responsible for different types of basin cell in the absence of forcing this suggests that the snas cannot be predicted reliably for specific initial conditions these snaswb can thus be expected to occur more commonly in dynamical systems

a block product preconditioner for saddle point problems abstract in this paper a block product bp preconditioner is established for saddle point problems spectral properties of the bp preconditioned matrix are investigated a strategy for practical choice of quasi optimal parameter is given numerical results on saddle point linear systems arising from stokes problems and weighted least square problems show that the proposed bp preconditioner is more economic to implement within krylov subspace acceleration than some extensively studied preconditioners

wada basin boundaries and basin cells in dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is nonempty we consider a two dimensional diffeomorphism f that is f is an invertible map and both f and its inverse are differentiable with continuous derivatives which has at least three basins fractal basin boundaries contain infinitely many periodic points generally only finitely many of these periodic points are outermost on the basin boundary that is accessible from a basin for many systems all accessible points lie on stable manifolds of periodic points a point x on the basic boundary is a wada point if every open neighborhood of x has a nonempty intersection with at least three different basins we call the boundary of a basin a wada basin boundary if all its points are wada points our main goal is to have definitions and hypotheses for wada basin boundaries that can be verified by computer the basic notion basin cell will play a fundamental role in our results for numerical verifications assuming each accessible point on the boundary of a basin b is on the stable manifold of some periodic orbit we show that partial derivative b over bar is a wada basin boundary if the unstable manifold of each of its accessible periodic orbits intersects at least three basins in addition we find conditions for basins b 1 b 2 b n n greater than or equal to 3 to satisfy partial derivative b over bar 1 partial derivative b over bar 2 partial derivative b over bar n our results provide numerically verifiable conditions guaranteeing that the boundary of a basin is a wada basin boundary our examples make use of an existing numerical procedure for finding the accessible periodic points on the basin boundary and another procedure for plotting stable and unstable manifolds to verify the existence of wada basin boundaries

complexity in hamiltonian driven dissipative chaotic dynamical systems the existence of symmetry in chaotic dynamical systems often leads to one or several low dimensional invariant subspaces in the phase space we demonstrate that complex behaviors can arise when the dynamics in the invariant subspace is hamiltonian but the full system is dissipative in particular an infinite number of distinct attractors can coexist these attractors can be quasiperiodic strange nonchaotic and chaotic with different positive lyapunov exponents finite perturbations in initial conditions or parameters can lead to a change from nonchaotic attractors to chaotic attractors however arbitrarily small perturbations can lead to dynamically distinct chaotic attractors this work demonstrates that the interplay between conservative and dissipative dynamics can give rise to rich complexity even in physical systems with a few degrees of freedom

pressure and fractal indices for the gibbs measures of hyperbolic julia sets the authors give a simple algorithm to compute the topological pressure for the julia sets j of hyperbolic rational endomorphism this allows one to compute the spectra of the generalised dimensions and entropies with respect to the whole class of gibbs measures on j

solution of the boundary value problem for optimal escape in continuous stochastic systems and maps topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps a topological method is introduced that simplifies the solution of boundary value problems the activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path the method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps

integrity measures quantifying the erosion of smooth and fractal basins of attraction abstract the phase portraits of a dissipative dynamical system are characterized by the existence of one or more stable attractors which typically include point equilibria periodic oscillations harmonic and subharmonic quasi periodic solutions and chaotic attractors each attractor is embedded in its own domain or basin of attraction bounded by a separatrix associated with an unstable saddle solution under the variation of a control parameter as the attractors move and bifurcate the basins also undergo corresponding changes and metamorphoses these changes in size and shape are usually continuous but can be discontinuous as when an attractor vanishes along with its basin at a saddle node bifurcation associated with the homoclinic tangling of the invariant manifolds of the saddle solution basin boundaries can also change in nature from smooth to fractal in this paper the escape of a driven oscillator from a cubic potential well as an archetypal example is used to explore the engineering significance of the basin erosions that occur under increased forcing various measures of engineering integrity of the constrained attractor are introduced a global measure assesses the overall basin area a local measure assesses the distance from the attractor to the basin boundary and a velocity measure is related to the size of impulse that could be sustained without failure since engineering systems may be subjected to pulse loads of finite duration attention is given to both the absolute and transient basin boudaries the significant erosion of these at homoclinic tangencies is particularly highlighted in the present study the fractal basins having a severely reduced integrity under all three criteria

a simple feedback control for a chaotic oscillator with limited power supply we proposed a simple feedback control method to suppress chaotic behavior in oscillators with limited power supply the small amplitude controlling signal is applied directly to the power supply system so as to alter the characteristic curve of the driving motor numerical results are presented showing the method efficiency for a wide range of control parameters moreover we have found that for some parameters this kind of control may introduce coexisting periodic attractors with complex basins of attraction and therefore serious problems with predictability of the final state the system will asymptote to

indeterminate sub critical bifurcations in parametric resonance in this paper we show that an indeterminate sub critical bifurcation with an unpredictable outcome arises in typical parametrically excited systems this occurs when the trivial solution is located on a fractal boundary as it loses its stability at a sub critical bifurcation here in the presence of even infinitesimal noise the outcome is indeterminate in the sense that we cannot predict on to which co existing attractor the system will settle

on the organisation of transient chaos application to irregular scattering it is shown how the local structure of chaotic repellers being responsible for transient chaotic behaviour is deduced from the properties of hyperbolic periodic orbits relations between static and dynamical multifractal spectra with respect to the natural invariant measure on the repeller are derived for invertible maps of the plane the results obtained for maps with unit jacobian apply to hamiltonian systems with two degrees of freedom which exhibit the phenomenon of irregular scattering and are characterised by an exponential decay of trapping probability

some properties of mixing repellers applying the thermodynamic formalism to mixing repellers for regular maps the author obtains some rigorous relations between the dimensions of the repellers and the dynamic properties of the transformation on them for a class of one dimensional expanding maps these quantities can be approximated to any desired accuracy by means of the corresponding variables for linear expanding maps which can be computed exactly finally a mathematical foundation and some properties of the generalised dimensions are given

fractal boundaries in open hydrodynamical flows signatures of chaotic saddles we introduce the concept of fractal boundaries in open hydrodynamical flows based on two gedanken experiments carried out with passive tracer particles colored differently it is shown that the signature for the presence of a chaotic saddle in the advection dynamics is a fractal boundary between regions of different colors the fractal parts of the boundaries found in the two experiments contain either the stable or the unstable manifold of this chaotic set we point out that these boundaries coincide with streak lines passing through appropriately chosen points as an illustrative numerical experiment we consider a model of the von k arm an vortex street a time periodic two dimensional flow of a viscous fluid around a cylinder

asymptotic behavior of matrix coefficients of admissible representations introduction 869 1 generalities on reductive groups 873 2 the infinitesimal cartan decomposition 876 3 the t radial components 880 4 differential equations satisfied by sphereical functions 885 5 asymptotic behavior of spherical functions on a 888 6 asymptotic behavior of spherical functions along the walls of a 892 7 leading characters and growth estimates on the group 898 8 admissible representations and their matrix coefficients 906 appendix 917

computing the measure of nonattracting chaotic sets abstract recently a numerical method called the pim triple algorithm has been formulated for generating long orbits on nonattracting chaotic sets we investigate under which conditions the measure for a nonattracting invariant set generated by the pim triple algorithm agrees with the natural measure

skew cyclic displacements and inversions of two innovative patterned matrices in this paper we deal mainly with a class of column upper plus lower cupl toeplitz matrices without toeplitz structure which are close to the toeplitz matrices in the sense that their 1 1 cyclic displacements coincide with cyclic displacement of some toeplitz matrices by constructing the corresponding displacement of the matrices we derive the formulas on representation of the inverses of the cupl toeplitz matrices in the form of sums of products of factor 1 1 circulants and 1 1 circulants furthermore through the relation between the cupl toeplitz matrices and the cupl hankel matrices the inverses of the cupl hankel matrices can be obtained as well

indeterminate bifurcational phenomena in hardening systems indeterminate bifurcations are emerging as an important ingredient of nonlinear dissipating dynamical systems in this paper we show that indeterminate bifurcations with an unpredictable outcome are a typical ingredient of nonlinear hardening systems such phenomena are clearly important concepts in the theory of nonlinear resonance and their discovery complements the earlier work carried out on softening systems as an illustrative example we examine the dynamics of a parametrically excited hardening system and show that when the trivial solution is located on a highly intertwined basin boundary its loss of stability gives a dynamic jump whose outcome is indeterminate in the sense that we cannot predict on to which coexisting attractor the system will settle

structure in the parameter dependence of order and chaos for the quadratic map many dynamical systems are thought to exhibit windows of attracting periodic behaviour for arbitrarily small perturbations from parameter values yielding chaotic attractors this structural instability of chaos is particularly well documented and understood for the case of the one dimensional quadratic map in this paper we attempt to numerically characterize the global parameter space structure of the dense set of periodic windows occurring in the chaotic regime of the quadratic map in particular we use scaling techniques to extract information on the probability distribution of window parameter widths as a function of period and location of the window in parameter space we also use this information to obtain the uncertainty exponent which is a quantity that globally characterizes the ability to identify chaos in the presence of small parameter uncertainties

a hybridized iterative algorithm of the bicorstab and gpbicor methods for solving non hermitian linear systems in this study we derive a new iterative algorithm including its preconditioned version which is a hybridized variant of the biconjugate a orthogonal residual stabilized bicorstab method and the generalized product type solvers based on bicor gpbicor method the proposed method which is named gpbicor m similarly to the gpbicg m method proposed by fujino 2002 can be regarded as an extension of the bicorstab2 method introduced by zhao and huang 2013 inspired by fujino s idea for improving the bicgstab2 method in the established gpbicor m method the parameters computed by the bicorstab method are chosen at successive m iteration steps and afterwards the parameters of the gpbicor method are utilized in the subsequent iteration steps therefore the proposed method can inherit the low computational cost of bicorstab and the attractive convergence of gpbicor extensive numerical convergence results on selected real and complex matrices are shown to assess the performances of the proposed gpbicor m method also against other popular non hermitian krylov subspace methods

origin of chaotic transients in excitatory pulse coupled networks we develop an approach to understanding long chaotic transients in networks of excitatory pulse coupled oscillators our idea is to identify a class of attractors sequentially active firing saf attractors in terms of the temporal event structure of firing and receipt of pulses then all attractors can be classified into two groups saf attractors and non saf attractors we establish that long transients typically arise in the transitional region of the parameter space where the saf attractors are collectively destabilized bifurcation behavior of the saf attractors is analyzed to provide a detailed understanding of the long irregular transients although demonstrated using pulse coupled oscillator networks our general methodology may be useful in understanding the origin of transient chaos in other types of networked systems an extremely challenging problem in nonlinear dynamics and complex systems

intermingled basins in coupled lorenz systems we consider a system of two identical linearly coupled lorenz oscillators presenting synchronization of chaotic motion for a specified range of the coupling strength we verify the existence of global synchronization and antisynchronization attractors with intermingled basins of attraction such that the basin of one attractor is riddled with holes belonging to the basin of the other attractor and vice versa we investigated this phenomenon by verifying the fulfillment of the mathematical requirements for intermingled basins and obtained scaling laws that characterize quantitatively the riddling of both basins in this system

on the application of normal forms near attracting fixed points of dynamical systems abstract it is shown on an integrable example in the plane that normal form solutions need not converge over the full basin of attraction of fixed points of dissipative dynamical systems their convergence breaks down at a singularity in the complex time plane of the exact solutions of the problem however as is demonstrated on a nonintegrable example with 3 dimensional phase space the region of convergence of normal forms can be large enough to extend almost to a nearby hyperbolic fixed point whose invariant manifolds embrace the attracting fixed point forming a complicated basin boundary thus in such problems normal forms are shown to be useful in practice as a tool for finding large regions of initial conditions for which the solutions are necessarily attracted to the fixed point at t

intertwined basins of attraction generated by the stable manifold of a saddle point this article deals with the intertwined basins of attraction we give a necessary and sufficient condition for the existence of intertwined basins of attraction

noise induced escape through a fractal basin boundary we study noise induced escape within a discrete dynamical system that has two co existing chaotic attractors in phase space separated by a locally disconnected fractal basin boundary it is shown that escape occurs via a unique accessible point on the fractal boundary the structure of escape paths is determined by the original saddles forming the homoclinic structure of the system and by their hierarchical interrelations

a dynamical meaning of fractal dimension when two atlractors of a dynamical system have a common basin boundary b small changes in initial conditions which lie near b can result in radically different long term behavior of the trajectory a quantitative description of this phenomenon is obtained in terms of the fractal dimension of the basin boundary b

on the intertwined basins of attraction for planar flows in this paper we discuss the intertwined basins of attraction one of the simplest cases of obstruction to predictability some sufficient conditions to guarantee the existence of intertwined basins of attraction are given

on hss based sequential two stage method for non hermitian saddle point problems abstract for large sparse saddle point problems with symmetric positive definite 1 1 block li et al studied an efficient iterative method see li et al 2011 25 by making use of the same preconditioning technique and a new matrix splitting based on the hermitian and skew hermitian splitting hss of the 1 1 block of the preconditioned non hermitian saddle point systems an efficient sequential two stage method is proposed for solving the non hermitian saddle point problems theoretical analysis shows the proposed iterative method is convergent and that the spectral radius of iterative matrix monotonically decreases and tends to 0 as the iterative parameter approaches infinity numerical experiments arising from naiver stokes problem are provided to show that the new iterative method is feasible effective and robust

indeterminate trans critical bifurcations in parametrically excited systems this paper may be regarded as a companion paper to a recent article m s soliman j m t thompson proc r soc lond a 438 511 518 1992 that addressed the phenomena of indeterminate sub critical bifurcations in typical parametrically excited systems this earlier work focused on the principal subharmonic resonance which occurs when the excitation frequency is almost twice the linear natural frequency in contrast we here focus on the fundamental harmonic resonance that arises when the two frequencies are almost the same we show that under this condition indeterminate trans critical bifurcations with unpredictable outcome can arise when the trivial solution is located on a fractal boundary as it loses its stability at the bifurcation the outcome is then unpredictable even under infinitesimal perturbations from the trivial solution

q deformed statistical mechanical property in the dynamics of trajectories en route to the feigenbaum attractor we demonstrate that the dynamics toward and within the feigenbaum attractor combine to form a q deformed statistical mechanical construction the rate at which ensemble trajectories converge to the attractor and to the repellor is described by a q entropy obtained from a partition function generated by summing distances between neighboring positions of the attractor the values of the q indices involved are given by the unimodal map universal constants while the thermodynamic structure is closely related to that formerly developed for multifractals as an essential component in our demonstration we expose in great detail the features of the dynamics of trajectories that either evolve toward the feigenbaum attractor or are captured by its matching repellor the dynamical properties of the family of periodic superstable cycles in unimodal maps are seen to be key ingredients for the comprehension of the discrete scale invariance features present at the period doubling transition to chaos elements in our analysis are the following i the preimages of the attractor and repellor of each of the supercycles appear entrenched into a fractal hierarchical structure of increasing complexity as period doubling develops ii the limiting form of this rank structure results in an infinite number of families of well defined phase space gaps in the positions of the feigenbaum attractor or of its repellor iii the gaps in each of these families can be ordered with decreasing width in accordance with power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions iv the power law with log periodic modulation associated with the rate of approach of trajectories toward the attractor and to the repellor is explained in terms of the progression of gap formation v the relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated vi a mean field evaluation of the atypical partition function a thermodynamic interpretation of the time evolution process and a crossover to ordinary exponential statistics are given we make clear the dynamical origin of the anomalous thermodynamic framework existing at the feigenbaum attractor

a nonsmooth coulomb friction oscillator abstract a forced coulomb friction oscillator whose frictional force is allowed to vary with displacement is analyzed geometrically the equation of motion for the oscillator is piecewise linear we geometrically observe the nature of the flow in each region of solvability and then see how these solutions interact at the boundary of the regions the dynamics of the flow is viewed in terms of a map on the boundary between the regions for chaotic motion we geometrically construct the strange attractor and show that its exact behavior is that of a one dimensional map the following dynamical properties arise from the nonsmooth nature of the coulomb friction law the flow may not be invertible the flow may reach its attractor in finite time the dimension of the attractor may be less than or equal to two embeddings of an observable may not be diffeomorphic to the full phase flow

prey predator systems with intertwined basins of attraction in this article we show the existence of intertwined basins of attraction for a class of prey predator systems which improves the extant results by deleting some crucial and hard testing conditions

on basin boundaries in this paper we make a study of a class of basin boundaries we show that an efficient characterization of a given element of this class can be constructed with segments of differentiable curves

extreme phase sensitivity in systems with fractal isochrons abstract sensitivity to initial conditions is usually associated with chaotic dynamics and strange attractors however even systems with quasi periodic dynamics can exhibit it in this context we report on the fractal properties of the isochrons of some continuous time asymptotically periodic systems we define a global measure of phase sensitivity that we call the phase sensitivity coefficient and show that it is an invariant of the system related to the capacity dimension of the isochrons similar results are also obtained with discrete time systems as an illustration of the framework we compute the phase sensitivity coefficient for popular models of bursting neurons suggesting that some elliptic bursting neurons are characterized by isochrons of high fractal dimensions and exhibit a very sensitive unreliable phase response

quasipotential approach to critical scaling in noise induced chaos when a dynamical system exhibits transient chaos and a nonchaotic attractor as in a periodic window noise can induce a chaotic attractor in particular when the noise amplitude exceeds a critical value the largest lyapunov exponent of the attractor of the system starts to increase from zero while a scaling law for the variation of the lyapunov exponent with noise was uncovered previously it is mostly based on numerical evidence and a heuristic analysis this paper presents a more general approach to the scaling law one based on the concept of quasipotentials besides providing deeper insights into the problem of noise induced chaos the quasipotential approach enables previously unresolved issues to be addressed the fractal properties of noise induced chaotic attractors and applications to biological systems are also discussed

basins of wada abstract we describe situations in which there are several regions more than two with the wada property namely that each point that is on the boundary of one region is on the boundary of all we argue that such situations arise even in studies of the forced damped pendulum where it is possible to have three attractor regions coexisting and the three basins of attraction have the wada property

fractals and dynamical chaos in a random 2d lorentz gas with sinks abstract two dimensional random lorentz gases with absorbing traps are considered in which a moving point particle undergoes elastic collisions on hard disks and annihilates when reaching a trap in systems of finite spatial extension the asymptotic decay of the survival probability is exponential and characterized by an escape rate which can be related to the average positive lyapunov exponent and to the dimension of the fractal repeller of the system for infinite systems the survival probability obeys a stretched exponential law of the form p c t exp ct1 2 the transition between the two regimes is studied and we show that for a given trap density the non integer dimension of the fractal repeller increases with the system size to finally reach the integer dimension of the phase space nevertheless the repeller remains fractal we determine the special scaling properties of this fractal

noise reduction finding the simplest dynamical system consistent with the data abstract a novel method is described for noise reduction in chaotic experimental data whose dynamics are low dimensional in addition we show how the approach allows experimentalists to use many of the same techniques that have been essential for the analysis of nonlinear systems of ordinary differential equations and difference equations

analysis of chaotic saddles in low dimensional dynamical systems the derivative nonlinear schrodinger equation abstract in this paper we present a computational study of nonattracting chaotic sets known as chaotic saddles in a low dimensional dynamical system describing stationary solutions of the derivative nonlinear schrodinger equation a driven dissipative model for alfven waves these chaotic saddles have gaps which are filled at chaotic transitions such as a saddle node bifurcation and an interior crisis we give a detailed explanation of how to numerically determine the chaotic saddles and describe how a chaotic attractor after an interior crisis point can be decomposed into two chaotic saddles dynamically connected by a set of coupling unstable periodic orbits created by a gap filling explosion after the crisis this coupling between two chaotic saddles is responsible for the intermittent dynamics displayed by the chaotic system after the interior crisis

the dimension of the maximal measure for a polynomial map let f c c be a polynomial of degree d 2 the most interesting dynamics of f takes place on its julia set j f an invariant compact perfect set which can be defined for example as the closure of the repelling periodic points brolin constructed the equilibrium distribution m on j f an ergodic measure which is also the weak limit as n x of a probability measure distributed equally over the dn nth inverse images of almost any point we shall prove the following

the transition to chaotic attractors with riddled basins abstract recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another attractor basin arbitrarily nearby the basin is riddled with holes here we consider the dynamics near the transition to this situation as a parameter is varied using a simple analyzable model we obtain the characteristic behaviors near this transition numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered

periodic orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system chaotic dynamical systems with two or more attractors lying on invariant subspaces may provided certain mathematical conditions are fulfilled exhibit intermingled basins of attraction each basin is riddled with holes belonging to basins of the other attractors in order to investigate the occurrence of such phenomenon in dynamical systems of ecological interest two species competition with extinction we have characterized quantitatively the intermingled basins using periodic orbit theory and scaling laws the latter results agree with a theoretical prediction from a stochastic model and also with an exact result for the scaling exponent we derived for the specific class of models investigated we discuss the consequences of the scaling laws in terms of the predictability of a final state extinction of either species in an ecological experiment

weighted toeplitz regularized least squares computation for image restoration the main aim of this paper is to develop a fast algorithm for solving weighted toeplitz regularized least squares problems arising from image restoration based on augmented system formulation we develop new hermitian and skew hermitian splitting hss preconditioners for solving such linear systems the advantage of the proposed preconditioner is that the blurring matrix weighting matrix and regularization matrix can be decoupled such that the resulting preconditioner is not expensive to use we show that for a preconditioned system that is derived from a saddle point structure of size m n times m n the preconditioned matrix has an eigenvalue at 1 with multiplicity n and the other m eigenvalues of the form 1 lambda with lambda 1 we also study how to choose the hss parameter to minimize the magnitude of lambda and therefore the krylov subspace method applied to solving the preconditioned system converges very quickly experimental results for image restoration problems are re

on the differential equations satisfied by weighted orbital integrals weighted orbital integrals are distributions on reductive groups over local fields appearing both in the local and global trace formulas there are associated invariant distributions which play the same role in the invariant trace formulas in the case of real groups the fourier transforms of these distributions satisfy a system of differential equations as a step towards determining those fourier transforms we show that this system is holonomic and has a simple singularity at infinity we deduce that any solution has a series expansion and is a linear combination of certain canonical solutions for some groups of small rank we solve the recursion formula for the coefficients explicitly

two new variants of the hss preconditioner for regularized saddle point problems two new preconditioners which can be viewed as improved variants of the hermitian and skew hermitian splitting hss preconditioner are presented for regularized saddle point problems the unconditionally convergent properties of the corresponding iteration methods are deduced and the selection of optimal iteration parameters resulting in fast convergence for the two iteration methods are discussed moreover eigenvalue bounds of the preconditioned matrices and upper bounds of the degree of their minimal polynomials are obtained compared with the hss preconditioner they are much better approximations to the coefficient matrix of the saddle point problem and they have better convergence properties and spectrum distributions numerical experiments from the stokes problem are presented which illustrates the advantages of the two preconditioners over the hss preconditioner and some other preconditioners to accelerate the convergence rate of gmres

basins of attraction in piecewise smooth hamiltonian systems piecewise smooth hamiltonian systems arise in physical and engineering applications for such a system typically an infinite number of quasiperiodic attractors coexist here we use the term attractors to indicate invariant sets to which typically initial conditions approach as a result of the piecewise smoothness of the underlying system these attractors are therefore characteristically different from the attractors in dissipative dynamical systems we find that the basins of attraction of different attractors exhibit a riddledlike feature in that they mix with each other on arbitrarily fine scales this practically prevents prediction of attractors from specific initial conditions and parameters the mechanism leading to the complicated basin structure is found to be characteristically different from those reported previously for similar basin structure in smooth dynamical systems we demonstrate the phenomenon using a class of electronic relaxation oscillators with voltage protection and provide a theoretical explanation

on the eigenvalues of a class of saddle point matrices we study spectral properties of a class of block 2 2 matrices that arise in the solution of saddle point problems these matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system we give conditions for having a positive real spectrum and for ensuring diagonalizability of the matrix in particular we show that these properties hold for the discrete stokes operator and we discuss the implications of our characterization for augmented lagrangian formulations for krylov subspace solvers and for certain types of preconditioners

cyclic displacements and decompositions of inverse matrices for cupl toeplitz matrices abstract in this paper we deal mainly with a class of cupl toeplitz matrices 32 without toeplitz structure which are close to the toeplitz matrices in the sense that their 1 1 cyclic displacements coincide with cyclic displacement of some toeplitz matrices by constructing the corresponding displacement of the matrices we derive the formulas on representation of the inverses of the cupl toeplitz matrices in the form of sums of products of factor row first plus last right circulants furthermore the inverses of the cupl hankel matrices can be also obtained which benefits from the relation between the cupl toeplitz matrices and the cupl hankel matrices

unpredictability of the asymptotic attractors in phase coupled oscillators an array of phase coupled oscillators may exhibit multiple coexisting chaotic and nonchaotic attractors the system of coupled circle maps is such an example we demonstrate that it is common for this type of system to exhibit an extreme type of final state sensitivity in both parameter and phase space numerical computations reveal that there exist substantial regions of the parameter space where arbitrarily small perturbations in parameters or initial conditions can alter the asymptotic attractor of the system completely consequently asymptotic attractors of the system cannot be predicted reliably for specific parameter values and initial conditions

analog computation with dynamical systems a b s t r a c t physical systems exhibit various levels of complexity their long term dynamics may converge to fixed points or exhibit complex chaotic behavior this paper presents a theory that enables to interpret natural processes as special purpose analog computers since physical systems are naturally described in continuous time a definition of computational complexity for continuous time systems is required in analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems discrete or continuous in time on the basis of an intrinsic time scale of the system dissipative dynamical systems are classified into the computational complexity classes pd co rpd npd and exp t corresponding to their standard counterparts according to the complexity of their long term behavior the complexity of chaotic attractors relative to regular ones leads to the conjecture pa fi npj continuous time flows have been proven useful in solving various practical problems our theory provides the tools for an algorithmic analysis of such flows as an example we analyze the continuous hopfield network 1998 elsevier science b v all rights reserved

three dimensional optical billiard chaotic scattering we report on the first experimental measurements taken from an easily studied visually remarkable chaotic scattering system that we call an optical billiard in particular we measure the fractal dimension of the basin boundaries generated by the reflection of light from various configurations of polished reflecting spheres we find that our experimentally determined fractal dimension values agree well with the values we find from simulations we also find that depending on the configuration of the spheres the boundary can be either a nowhere differentiable surface or else can possess the topological wada property whereby any basin boundary point is simultaneously on the boundary of four distinct basins other configurations yielding other topologies of the chaotic scattering set are suggested for future study

on chaos and fractal behavior in a generalized duffing s system abstract in this paper three duffing s equations their attractors and their initial condition maps are examined for elements of universality in the literature all three equations have been examined briefly before but their behavior has only been incompletely described this paper more completely examines these system and answers important questions concerning the reason for chaotic system behavior

new variant of the hss iteration method for weighted toeplitz regularized least squares problems from image restoration abstract for fast solving weighted toeplitz least squares problems from image restoration ng and pan 2014 studied a new hermitian and skew hermitian splitting nhss preconditioner in this paper a generalization of the nhss preconditioner and the corresponding iterative method are presented convergence for the new iteration method is studied and optimal choice of the parameters is discussed bounds on the eigenvalues and the corresponding eigenvector distributions are proposed the degree of the minimal polynomial of the preconditioned matrix is obtained numerical experiments arising from image restoration are provided which show the proposed iteration method is effective and confirm our theoretical results are correct

basin boundary metamorphoses changes in accessible boundary orbits abstract basin boundaries sometimes undergo sudden metamophoses these metamorphoses can lead to the conversion of a smooth basin boundary to one which is fractal or else can cause a fractal or else can basin boundary to suddenly jump in size and change its character although remaining fractal for an invertible map in the plane there may be an infinite number of saddle periodic orbits in a basin boundary that is fractal nonetheless we have found that typically only one of them can be reached or accessed directly from a given basin the other periodic orbits are buried beneath infinitely many layers of the fractal structure of the boundary the boundary metamorphoses which we investigate are characterized by a sudden replacement of the basin boundary s accessible orbit

fractal control boundaries of driven oscillators and their relevance to safe engineering design when a metastable damped oscillator is driven by strong periodic forcing the catchment basin of constrained motions in the space of the starting conditions x 0 0 develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds the four dimensional basin in the phase control space spanned by x 0 0 f where f is the magnitude and the frequency of the excitation will likewise acquire a fractal boundary and we here explore the engineering significance of the control cross section corresponding for example to x 0 0 0 the fractal boundary in this section is a failure locus for a mechanical or electrical system subjected while resting in its ambient equilibrium state to a sudden pulse of excitation we assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well absolute and transient basins are examined giving control space maps analogous to familiar pictures of the mandelbrot set generalizing from this prototype study it is argued that in engineering design against boat capsize or earthquake damage for example a study of safe basins should augment and perhaps entirely replace conventional analysis of the steady state attracting solutions

dynamic bifurcations in non stationary systems transitions with an unpredictable outcome in nonlinear non stationary systems dynamic bifurcations result in a transition to a qualitatively new state in this paper we examine how the dynamics of transition of such systems may be assessed using the concept of transient basins of attraction we delineate the phenomenon of indeterminate dynamic bifurcations where it is shown that the response after the system passes through critical parameter values may be extremely sensitive to the choice of initial conditions or parameter states this new form of unpredictability in systems whose parameters vary with time is clearly an important concept to be assimilated in the theory of non stationary dynamics

a modified product preconditioner for indefinite and asymmetric generalized saddle point matrices in this paper we present a new modified product preconditioner mp for a class of large sparse linear systems with indefinite and asymmetric matrices the eigenvalue distribution and form of the eigenvectors of the presented new preconditioned matrix and its minimal polynomial are investigated some numerical experiments illustrate that the proposed new preconditioner performs better than block diagonal preconditioner block triangular preconditioner constraint preconditioner and product preconditioner