p matrices are just sufficient abstract the classes of sufficient matrices and of p matrices have recently arisen in connection with the linear complementarity problem it is known that p matrices are sufficient we show that conversely every sufficient matrix is a p matrix

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

sufficient matrices and the linear complementarity problem abstract we pose and answer two questions about solutions of the linear complementarity problem lcp the first question is concerned with the conditions on a matrix m which guarantee that for every vector q the solutions of the lcp q m are identical to the karush kuhn tucker points of the natural quadratic program associated with q m in answering this question we introduce the class of row sufficient matrices the transpose of such a matrix is what we call column sufficient the latter matrices turn out to furnish the answer to our second question which asks for the conditions on m under which the solution set of q m is convex for every q in addition to these two main results we discuss the connections of these twonew matrix classes with other well known matrix classes in linear complementarity theory

on the uniqueness of solutions to linear complementarity problems this paper characterizes the classu of all realn n matricesm for which the linear complementarity problem q m has a unique solution for all realn vectorsq interior to the conek m of vectors for which q m has any solution at all it is shown that restricting the uniqueness property to the interior ofk m is necessary because whenu the problem q m has infinitely many solutions ifq belongs to the boundary of intk m it is shown thatm must have nonnegative principal minors whenu andk m is convex finally it is shown that whenm has nonnegative principal minors only one of which is 0 andk m r n thenu andk m is a closed half space

mechanisms for the hard bubbling transition in symmetrically coupled chaotic systems we investigate mechanisms for the hard bubbling transition in symmetrically coupled one dimensional maps a transition from strong to weak synchronization occurs via a first period doubling or pitchfork transverse bifurcation of a periodic saddle embedded in the synchronous chaotic attractor the consequence of such transverse bifurcations depends on the existence of an absorbing area controlling the global dynamics in the presence of an absorbing area a subcritical transverse bifurcation is found to give rise to abrupt appearance of transient intermittent bursts with large amplitude for this case any small parameter mismatch that is unavoidable in real systems induces a hard bubbling transition via an interior crisis through a detailed numerical analysis we present explicit examples for the subcritical period doubling and pitchfork transverse bifurcations leading to hard bubbling

jordan type inequalities involving the bessel and modified bessel functions in this work two general forms of jordan s inequalities for the bessel and modified bessel functions are established and proved by using l hospital s rule for monotonicity and some properties of the spherical bessel and the modified spherical bessel functions of the first kind the applications of the results above give two new infinite series for the bessel and modified bessel functions

new characterizations of row sufficient matrices in dealing with a linear complementarity problem much depends on knowing that the matrix through which the particular lcp is defined belongs to a suitable matrix class two such classes are su the so called sufficient matrices and l which were introduced in r w cottle j s pang v venkateswaran sufficient matrices and the linear complementarity problem linear algebra appl 114 115 1989 231 249 b c eaves the linear complementarity problem manage sci 17 1971 612 634 respectively in an earlier article i adler r w cottle s verma sufficient matrices belong to l math prog 106 2006 391 401 the authors proved that su is a subclass of l by definition the class su is the intersection of two distinct classes rsu the row sufficient matrices and csu the column sufficient matrices in the present work we strengthen the aforementioned inclusion by showing that all row sufficient matrices belong to l using what we call structural properties of certain matrix classes we add to the existing characterizations of rsu in r w cottle s m guu two characterizations of sufficient matrices linear algebra appl 170 1992 65 74 s m guu r w cottle on a subclass of p0 linear algebra appl 223 224 1995 325 335 h valiaho criteria for sufficient matrices linear algebra appl 233 1996 109 129 this line of inquiry was inspired by asking what must be true of a row sufficient l matrix we establish three new characterizations of rsu in terms of the matrix classes l e0 and q0 and the structural properties of sign change invariance completeness and fullness the new characterizations of rsu provide new characterizations of su by adjoining a fourth structural property we call reflectiveness

catastrophic bifurcation from riddled to fractal basins most existing works on riddling assume that the underlying dynamical system possesses an invariant subspace that usually results from a symmetry in realistic applications of chaotic systems however there exists no perfect symmetry the aim of this paper is to examine the consequences of symmetry breaking on riddling in particular we consider smooth deterministic perturbations that destroy the existence of invariant subspace and identify as a symmetry breaking parameter is increased from zero two distinct bifurcations in the first case the chaotic attractor in the invariant subspace is transversely stable so that the basin is riddled we find that a bifurcation from riddled to fractal basins can occur in the sense that an arbitrarily small amount of symmetry breaking can replace the riddled basin by fractal basins we call this a catastrophe of riddling in the second case where the chaotic attractor in the invariant subspace is transversely unstable so that there is no riddling in the unperturbed system the presence of a symmetry breaking no matter how small can immediately create fractal basins in the vicinity of the original invariant subspace this is a smooth fractal basin boundary metamorphosis we analyze the dynamical mechanisms for both catastrophes of riddling and basin boundary metamorphoses derive scaling laws to characterize the fractal basins induced by symmetry breaking and provide numerical confirmations the main implication of our results is that while riddling is robust against perturbations that preserve the system symmetry riddled basins of chaotic attractors in the invariant subspace on which most existing works are focused are structurally unstable against symmetry breaking perturbations

a constructive characterization of q 0 matrices with nonnegative principal minors in 2 we characterized the class of matrices with nonnegative principla minors for which the linear complementarity problem always has a solution that class is contained in the one we study here our main result gives a finitely testable set of necessary and sufficient conditions under which a matrix with nonnegative principal minors has the property that if a corresponding linear complementarity problem is feasible then it is solvable in short we constructively characterize the matrix class known asqo po

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

sufficient matrices belong to l in this paper we establish a significant matrix class inclusion that seems to have been overlooked in the literature of the linear complementarity problem we show that p the class of sufficient matrices is a subclass of l in the course of demonstrating this inclusion we introduce other new matrix classes that forge interesting new connections between known matrix classes

riddling and chaotic synchronization of coupled piecewise linear lorenz maps abstract we investigate the parametric evolution of riddled basins related to synchronization of chaos in two coupled piecewise linear lorenz maps riddling means that the basin of the synchronized attractor is shown to be riddled with holes belonging to another basin in an arbitrarily fine scale which has serious consequences on the predictability of the final state for such a coupled system we found that there are wide parameter intervals for which two piecewise linear lorenz maps exhibit riddled basins globally or locally which indicates that there are riddled basins in coupled lorenz equations as previously suggested by numerical experiments the use of piecewise linear maps makes it possible to prove rigorously the mathematical requirements for the existence of riddled basins

fractal distribution of floaters on a fluid surface and the transition to chaos for random maps abstract the long time spatial distribution of particles floating on the surface of a confined fluid whose flow velocity has complicated time dependence is considered it is shown that this distribution can be either a fractal or else can clump at several or one discrete points the transition from the latter type of distribution to the former occurs when the lyapunov exponent characterizing the particle motion passes through zero from negative values to positive values the characteristic features of this type of transition are investigated using random maps it is shown that near the transition there are extremely intermittent temporal fluctuations in the particle cloud and their scaling with a parameter is elucidated

amplitude death in the absence of time delays in identical coupled oscillators we study the dynamics of oscillators that are mutually coupled via dissimilar or conjugate variables and find that this form of coupling leads to a regime of amplitude death analytic estimates are obtained for coupled landau stuart oscillators and this is supplemented by numerics for this system as well as for coupled lorenz oscillators time delay does not appear to be necessary to cause amplitude death when conjugate variables are employed in coupling identical systems coupled chaotic oscillators also show multistability prior to amplitude death and the basins of the coexisting attractors appear to be riddled this behavior is quantified an appropriately defined uncertainty exponent in the coupled lorenz system is shown to be zero

different types of chaos synchronization in two coupled piecewise linear maps dynamics of a four parameter family of two dimensional piecewise linear endomorphisms which consist of two linearly coupled one dimensional maps is considered we show that under analytically given conditions chaotic behavior in both maps can be synchronized depending on the coupling the parameters chaotic attractor s synchronized state is characterized by different types of stability s1063 651x 96 11409 4

a general form of jordan type double inequality for the generalized and normalized bessel functions in this note a general form of jordan type double inequality involving the generalized and normalized bessel functions is presented and then some recent results concerning generalized and sharp work of jordan s inequality are extended at the same time the applications of the results above give two new infinite series for sinx x and sinhx x

optimal periodic orbits of chaotic systems invariant sets embedded in a chaotic attractor can generate time averages that differ from the average generated by typical orbits on the attractor motivated by two different topics namely controlling chaos and riddled basins of attraction we consider the question of which invariant set yields the largest optimal value of an average of a given smooth function of the system state we present numerical evidence and analysis which indicate that the optimal average is typically achieved by a low period unstable periodic orbit embedded in the chaotic attractor

naimark sacker bifurcations in linearly coupled quadratic maps abstract we report exact analytical expressions locating the 0 1 1 2 and 2 4 bifurcation curves for a prototypical system of two linearly coupled quadratic maps of interest is the precise location of the parameter sets where naimark sacker bifurcations occur starting from a non diagonal period 2 orbit this result is the key to understand the onset of synchronization in networks of quadratic maps

six new redheffer type inequalities for circular and hyperbolic functions in this paper six new redheffer type inequalities involving circular functions and hyperbolic functions are established

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

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

on the complexity of computing the handicap of a sufficient matrix the class of sufficient matrices is important in the study of the linear complementarity problem lcp some interior point methods ipm s for lcp s with sufficient data matrices have complexity polynomial in the bit size of the matrix and its handicap in this paper we show that the handicap of a sufficient matrix may be exponential in its bit size implying that the known complexity bounds of interior point methods are not polynomial in the input size of the lcp problem we also introduce a semidefinite programming based heuristic that provides a finite upper bond on the handicap for the sub class of mathcal p matrices where all principal minors are positive

symmetry breaking bifurcation for coupled chaotic attractors the authors consider transitions from synchronous to asynchronous chaotic motion in two identical dissipatively coupled one dimensional mappings they show that the probability density of the asymmetric component satisfies a scaling law the exponent in this scaling law varies continuously with the distance from the bifurcation point and is determined by the spectrum of local lyapunov exponents of the uncoupled map finally they show that the topology of the invariant set is rather unusual though the attractor for supercritical coupling is a line it is surrounded by a strange invariant set which is dense in a two dimensional neighbourhood of the attractor

determining the handicap of a sufficient matrix abstract any linear complementarity problem with a sufficient matrix can be solved by means of the unified interior point method the complexity bound of the method is the better the smaller the so called handicap of the matrix is we propose a method for determining the handicap of a sufficient matrix and show that a sufficient matrix and its transpose have the same handicap

wilker and huygens type inequalities for the generalized trigonometric and for the generalized hyperbolic functions abstract wilker and huygens type inequalities for the generalized trigonometric and hyperbolic functions are established main results of this paper are proven with the aid of two inequalities which involve a few parameters for special values of the parameters obtained results simplify to the known ones which are already published in mathematical literature

role of invariant and minimal absorbing areas in chaos synchronization in this paper the method of critical curves a tool for the study of the global dynamical properties of two dimensional noninvertible maps is applied to the study of chaos synchronization and related phenomena of riddling blowout and on off intermittency a general procedure is suggested in order to obtain the boundary of a particular two dimensional compact trapping region called absorbing area containing the onedimensional chaotic set on which synchronized dynamics occur the main purpose of the paper is to show that only invariant and minimal absorbing areas are useful to characterize the global dynamical behavior of the dynamical system when a milnor attractor with locally riddled basin or a chaotic saddle exists and may strongly influence the effects of riddling and blowout bifurcations some examples are given for a system of two coupled logistic maps and some practical methods and numerical tricks are suggested in order to ascertain the properties of invariance and minimality of an absorbing area some general considerations are given concerning the transition from locally riddled to globally riddled basins and the role of the absorbing area in the occurrence of such transition is discussed s1063 651x 98 11611 2

two characterizations of sufficient matrices abstract two characterizations are given for the class of sufficient matrices defined by cottle pang and venkateswaran the first is a direct translation of the definition into linear programming terms the second can be thought of as a generalization of a theorem of t d parsons on p matrices

loss of synchronization in coupled rossler systems this paper considers the loss of synchronization for a system of two coupled rossler oscillators bifurcation curves for the transverse destabilization of low periodic orbits embedded in the synchronized chaotic state are obtained and we show that desynchronization for a pair of symmetrically coupled identical rossler systems is associated with different orbits undergoing transverse pitchfork or period doubling bifurcations the transverse destabilization of the period 1 orbit is examined in detail and we follow the sequence of bifurcations that the asynchronous periodic cycles undergo in the presence of an asymmetry in the coupling the transverse period doubling bifurcation remains essentially the same the transverse pitchfork bifurcation on the other hand is transformed into a transcritical riddling bifurcation if the interacting rossler oscillators have different parameter values the non generic character of the pitchfork bifurcation leads it to be replaced by a saddle node bifurcation occurring off the symmetric sub space finally we show how the transverse stability properties of the equilibrium point can be used to obtain approximative analytical results for the transverse stability of the coupled chaotic oscillators 2001 elsevier science b v all rights reserved

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

on a subclass of p0 abstract in 1966 fiedler and ptak wrote the first systematic investigation of the matrix class p 0 consisting of all real square matrices with nonnegative principal minors in this note we focus on a particular subclass of p 0 that has arisen within the field of linear complementarity theory namely the class su of sufficient matrices our principal result is that su contains another class p we conjecture that the latter two classes are in fact the same

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

transverse instability and riddled basins in a system of two coupled logistic maps riddled basins denote a characteristic type of fractal domain of attraction that can arise when a chaotic motion is restricted to an invariant subspace of total phase space an example is the synchronized motion of two identical chaotic oscillators the paper examines the conditions for the appearance of such basins for a system of two symmetrically coupled logistic maps we determine the regions in parameter plane where the transverse lyapunov exponent is negative the bifurcation curves for the transverse destabilization of low periodic orbits embedded in the chaotic attractor are obtained and we follow the changes in the attractor and its basin of attraction when scanning across the riddling and blowout bifurcations it is shown that the appearance of transversely unstable orbits does not necessarily lead to an observable basin riddling and that the loss of weak stability when the transverse lyapunov exponent becomes positive does not necessarily destroy the basin of attraction instead the symmetry of the synchronized state may break and the attractor may spread into two dimensional phase space

template analysis of a nonlinear dynamo in this paper we extend our previous template analysis of a self exciting faraday disc dynamo with a linear series motor to the case of a nonlinear series motor this introduces two additional nonlinear symmetry breaking terms into the governing dynamo equations we investigate the consequences for the identification of a possible template on which the unstable periodic orbits upos lie by computing gauss linking numbers between pairs of upos we show that their values are not incompatible with those for a template for the lorenz attractor for its classic parameter values

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

universality for the parameter mismatching effect on weak synchronization in coupled chaotic systems to examine the universality for the parameter mismatching effect on weak chaotic synchronization we study coupled multidimensional invertible systems such as the coupled henon maps and coupled pendula by generalizing the method proposed in coupled one dimensional 1d noninvertible maps we introduce the parameter sensitivity exponent to measure the degree of the parameter sensitivity of a weakly stable synchronous chaotic attractor in terms of the parameter sensitivity exponents we characterize the effect of the parameter mismatch on the intermittent bursting and the basin riddling occurring in the regime of weak synchronization it is thus found that the scaling exponent for the average characteristic time i e the average interburst time and the average chaotic transient lifetime for both the bubbling and riddling cases is given by the reciprocal of the parameter sensitivity exponent as in the simple system of coupled 1d maps hence the reciprocal relation i e 1 seems to be universal in the sense that it holds in typical coupled chaotic systems of different nature

stability and transverse manifolds of periodic points for coupled maps abstract we consider the domains of stability of periodic points for linear internal and external coupled maps we obtain the approximation expressions of the transverse manifolds at periodic points which show that the transverse manifolds of periodic points are asymptotically elliptic paraboloids we point out that the action of maps on the transverse manifold has the symmetry only in two coupled maps we study in detail the hereditary properties of domains of stability of period doubling points for the internal coupling of an arbitrary one dimensional map with the help of its quadratic approximation and show that these domains follow the universal rules similar to the feigenbaum universality rules for one dimensional maps

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

riddling bifurcations in coupled piecewise linear maps abstract a mechanism for riddling bifurcations in the system of coupled piecewise linear maps is described we give sufficient conditions for the occurrence of locally and globally riddled basins based on the properties of absorbing areas of the chaotic attractors on the invariant manifold it is also shown that riddled basins are preserved upon bifurcation of the chaotic attractors as long as the attractor after bifurcation is located in the absorbing area of the attractor before bifurcation

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

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

a further refinement of a jordan type inequality and its application abstract by introducing taylor polynomials a new sharpened and generalized version of jordan s inequality is established the result is then used to obtain a substantially more refined inequality of jordan type moreover an application of the results presented here toward the improvement of the yang le inequality is also considered in this paper

some improvements and generalizations of steffensen s integral inequality abstract in this paper the authors present several interesting results on improvements of steffensen s integral inequality moreover an error in a generalization of steffensen s inequality due to mercer p r mercer extensions of steffensen s inequality j math anal appl 246 2000 325 329 is corrected