degenerate hopf bifurcations in discontinuous planar systems abstract we study the stability of a singular point for planar discontinuous differential equations with a line of discontinuities this is done for the most generic cases by computing some kind of lyapunov constants our computations are based on the so called r p q generalized polar coordinates introduced by lyapunov and they are essentially different from the ones used in the smooth case these lyapunov constants are also used to generate limit cycles for some concrete examples

on hopf cyclicity of planar systems we investigate the maximal number of limit cycles which appear under perturbations in hopf bifurcations by using the first order melnikov function with multiple parameters

melnikov method for discontinuous planar systems abstract this paper treats the occurrence of homoclinic solutions in planar systems with discontinuous right hand side more precisely we deal with a t periodic perturbed system such that the unperturbed system is an autonomous possessing homoclinic orbit by means of the so called non smooth melnikov function there is shown the existence of a homoclinic solution for a perturbed system the non smooth melnikov function is derived and the method of how to find it in concrete problems is also introduced

degenerate hopf bifurcations in a family of ff type switching systems abstract the theory of hopf bifurcation for switching systems is different from that for smooth systems because the nonsmoothness may produce more limit cycles although some results were obtained and several versions of order were given for switching weak foci few works were found for switching centers or the general relationship between order and cyclicity in this paper we study degenerate hopf bifurcations in a family of ff type switching systems whose unperturbed upper system and lower one both have the same equilibrium of linear center type on the sole switching line we use fractional orders to uniform the concept of order of weak foci for both the smooth case and the switching one and generally discuss the relationship between the order of weak foci and the cyclicity and give the uniform upper bound for the cyclicity in terms of the bautin depth of the ideal generated by the switching lyapunov quantities moreover we investigate the independence of those switching lyapunov quantities we finally apply our results to a family of switching quadratic bautin systems and prove that the cyclicity is at least 5 and 8 in the cases of weak foci and centers respectively

vertical slot fishways mathematical modeling and optimal management fishways are the main type of hydraulic devices currently used to facilitate migration of fish past obstructions dams waterfalls rapids in rivers in this paper we present a mathematical formulation of an optimal control problem related to the optimal management of a vertical slot fishway where the state system is given by the shallow water equations the control is the flux of inflow water and the cost function reflects the need of rest areas for fish and of a water velocity suitable for fish leaping and swimming capabilities we give a first order optimality condition for characterizing the optimal solutions of this problem from a numerical point of view we use a characteristic galerkin method for solving the shallow water equations and we use an optimization algorithm for the computation of the optimal control finally we present numerical results obtained for the realistic case of a standard nine pools fishway

limit cycles bifurcate from centers of discontinuous quadratic systems like for smooth quadratic systems it is important to determine the maximum order of a fine focus and the cyclicity of discontinuous quadratic systems previously examples of discontinuous quadratic systems with five limit cycles bifurcated from a fine focus of order 5 have been constructed in this paper we construct a class of discontinuous quadratic systems with a fine focus of order 9 in addition by using a method similar to that developed by c christopher for smooth systems which allows one to estimate the cyclicity just from the lower order terms of lyapunov constants we show that the cyclicity of discontinuous quadratic systems is at least 9 thus improving on previous results

regularization methods for solving differential algebraic equations in m knorrenschild siam j numer anal 29 1992 1694 1715 a class of regularization methods for solving autonomous index 1 daes differential algebraic equations has been proposed in this paper we generalize it to non autonomous index 1 and higher index daes we will treat a daes as an infinitely stiff odes ordinary differential equations therefore the solutions of a daes can be considered as the limit of the solutions of the corresponding stiff odes in some sense some similar results with m knorrenschild 1992 loc cit are obtained

pseudospectral method for numerical solution of daes with an error estimation in e babolian m m hosseini reducing index and pseudospectral methods for differential algebraic equations appl math comput 140 2003 77 90 numerical solution of linear differential algebraic equations daes has been presented by pseudospectral method in this paper a new error estimation technique is proposed to pseudospectral method such that is well done for linear semi explicit daes when the daes has index 2 the obtained approximate solution can be improved by using the proposed error estimation technique furthermore with providing some examples the proposed error estimation method is dealt with numerically

a linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system abstract in this paper we study a class of planar piecewise smooth quadratic integrable non hamiltonian systems which have a center by using the averaging method we give an estimation of the number of limit cycles which bifurcate from the above periodic annulus under the polynomial perturbation of degree n our estimation is linear depending on n and it is at least twice the associated estimation of smooth systems

the focus center limit cycle bifurcation in symmetric 3d piecewise linear systems the birth of limit cycles in 3d three dimensional piecewise linear systems for the relevant case of symmetrical oscillators is considered a technique already used by the authors in planar systems is extended to cope with 3d systems where a greater complexity is involved under some given nondegeneracy conditions the corresponding theorem characterizing the bifurcation is stated in terms of the deviation from the critical value of the bifurcation parameter expressions in the form of power series for the period amplitude and the characteristic multipliers of the bifurcating limit cycle are also obtained the results are applied to accurately predict the birth of symmetrical periodic oscillations in a 3d electronic circuit genealogically related to the classical van der pol oscillator

bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems abstract the generalized homoclinic loop appears in the study of dynamics on piecewise smooth differential systems during the past two decades for planar piecewise smooth differential systems there are concrete examples showing that under suitable perturbations of a generalized homoclinic loop one or two limit cycles can appear but up to now there is no a general theory to study the cyclicity of a generalized homoclinic loop that is the maximal number of limit cycles which are bifurcated from it in this paper we provide some sufficient conditions on the cyclicity of some homoclinic loops especially we prove the existence of one or two limit cycles which are bifurcated from a generalized homoclinic loop of an unperturbed piecewise smooth differential systems also we obtain a class of piecewise differential system in which two limit cycles can be bifurcated from a generalized homoclinic loop with multiplicity one the phenomena cannot happen in the smooth differential systems finally we provide five concrete piecewise smooth differential systems showing the applications of our theories on this phenomenon

reducing index method for differential algebraic equations with constraint singularities this paper extends an earlier work appl math comput 140 2003 77 to differential algebraic equations with constraint singularities numerical solution of these problems is considered by pseudospectral method with domain decomposition and by giving a condition under which the general linear form problem can easily be transformed to the index reduced form by a simple formulation furthermore with providing some examples the aforementioned cases are dealt with numerically

low dimensional maps for piecewise smooth oscillators abstract dynamics of the piecewise smooth nonlinear oscillators is considered for which general methodology of reducing multidimensional flows to low dimensional maps is proposed this includes a definition of piecewise smooth oscillator and creation of a global iterative map providing an exact solution the global map is comprised of local maps which are constructed in the smooth sub regions of phase space to construct this low dimensional map it is proposed to monitor the points of intersections of a chosen boundary between smooth subspaces by a trajectory the dimension reduction is directly related to the dimension of the chosen boundary and the lower its dimension is the larger dimension reduction can be achieved full details are given for a drifting impact oscillator where the five dimensional flow is reduced to one dimensional 1d approximate analytical map first an exact two dimensional map has been formulated and analysed a further reduction to 1d approximate map is introduced and discussed standard nonlinear dynamic analysis reveals a complex behaviour ranging from periodic oscillations to chaos and co existence of multiple attractors accuracy of the constructed maps is examined by comparing with the exact solutions for a wide range of the system parameters

a modified spectral method for numerical solution of ordinary differential equations with non analytic solution in this article there is an attempt to introduce generally spectral methods for numerical solution of ordinary differential equations also to focus on those problems in which some coefficient functions or solution function is not analytic then by expressing weak and strong aspects of spectral methods to solve this kind of problems a modified spectral method which is more efficient than other spectral methods is suggested furthermore with providing some examples the aforementioned cases are dealt with numerically

adomian decomposition method for solution of differential algebraic equations solutions of differential algebraic equations is considered by adomian decomposition method in e babolian m m hosseini reducing index and spectral methods for differential algebraic equations j appl math comput 140 2003 77 and m m hosseini an index reduction method for linear hessenberg systems j appl math comput in press an efficient technique to reduce index of semi explicit differential algebraic equations has been presented in this paper adomian decomposition method is applied to reduced index problems the scheme is tested for some examples and the results demonstrate reliability and efficiency of the proposed methods

application of optimal control theory to bioremediation abstract in situ bioremediation is a remediation technology in which the indigenous subsurface bacteria are stimulated by injecting compounds to provide food and energy the stimulated bacteria break down the target contaminants into less harmful substances the way the compounds are injected is a crucial component of the technology we use techniques from the theory of optimal control of distributed parameter systems to characterize an optimal injection function in a tube bioreactor the state system the set of equations that govern the evolution of bioremediation is a hybrid system consisting of both partial and ordinary differential equations

on hopf bifurcations of piecewise planar hamiltonian systems abstract in this paper we study the number of limit cycles appearing in hopf bifurcations of piecewise planar hamiltonian systems for the case that the hamiltonian function is a piecewise polynomials of a general form we obtain lower and upper bounds of the number of limit cycles near the origin respectively for some systems of special form we obtain the hopf cyclicity

melnikov method for homoclinic bifurcation in nonlinear impact oscillators based on an inverted pendulum impacting on rigid walls under external periodic excitation a class of nonlinear impact oscillators is discussed for its homoclinic bifurcation the melnikov method established for smooth dynamical systems is extended to be applicable to the nonsmooth one for nonlinear impact systems closed form solutions between impacts are generally unavailable the absence of closed form solutions makes difficulties in estimation of the gap between the stable manifold and unstable manifold in this paper we give a method to compute the melnikov functions up to the n t h order so as to obtain conditions of parameters for the persistence of homoclinic cycles which are formed via the identification given by the impact rule

algorithmic derivation of centre conditions the conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications we demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres we survey recent developments and illustrate our approach by means of examples

reducing index and pseudospectral methods for differential algebraic equations in this paper numerical solution of linear differential algebraic equations is considered by pseudospectral method also an index reduction technique for them is suggested for hessenberg index 2 system the index reduction technique is defined and its use is demonstrated we give a condition under which the general linear form problem can be easily transformed to the index reduced form by a simple formulation furthermore with providing some examples the aforementioned cases are dealt with numerically

isochronicity of centers in a switching bautin system abstract in this paper isochronicity of centers is discussed for a class of discontinuous differential system simply called switching system we give some sufficient conditions for the system to have a regular isochronous center at the origin and on the other hand construct a switching system with an irregular isochronous center at the origin we give a computation method for periods of periodic orbits near the center and use the method to discuss a switching bautin system for center conditions and isochronous center conditions we further find all of those systems which have an irregular isochronous center

limit cycles bifurcating from piecewise quadratic systems separated by a straight line abstract in this paper we are concerned with limit cycles bifurcating from piecewise quadratic systems separated by a straight line by means of the first order averaging theory of discontinuous differential systems and the argument principle we present an estimate of the maximum number of limit cycles bifurcating from the period annulus around the center of piecewise smooth systems

an algorithm for solving daes with mechanization in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of pade approximation method for solving differential algebraic equation systems was established and a new maple procedures mainproc was established too ten examples are presented to illustrate the implementation of the program

limit cycle bifurcations in perturbations of planar piecewise smooth systems with multiply lines of critical points abstract this paper investigates the limit cycle bifurcation problem of planar piecewise polynomial systems whose unperturbed system has a center at the origin and may have invariant multiply lines by melnikov function method this paper provides the upper and lower bounds for the poincare cyclicity in particular the exact cyclicities of two specific cases are given using analyzing method one of which has been discussed in the paper 7

theoretical and numerical analysis of an optimal control problem related to wastewater treatment in this work we deal with the design and management of wastewater treatment systems mainly the disposal of sea outfalls discharging polluting effluent from a sewerage system this problem is formulated as a pointwise optimal control problem with state and control constraints the main difficulties arise from the lack of regularity of the second member in the state system and from the pointwise constraints on the state variables we develop the theoretical analysis of the problem we propose an algorithm for its numerical resolution and finally we give results for a realistic problem posed in the ria of vigo spain

detection and remediation of stagnation in the nelder mead algorithm using a sufficient decrease condition the nelder mead algorithm can stagnate and converge to a nonoptimal point even for very simple problems in this note we propose a test for sufficient decrease which if passed for all iterations will guarantee convergence of the nelder mead iteration to a stationary point if the objective function is smooth and the diameters of the nelder mead simplices converge to zero failure of this condition is an indicator of potential stagnation as a remedy we propose a new step which we call an oriented restart that reinitializes the simplex to a smaller one with orthogonal edges whose orientation is determined by an approximate descent direction from the current best point we also give results that apply when the objective function is a low amplitude perturbation of a smooth function we illustrate our results with some numerical examples

convergence properties of the nelder mead simplex method in low dimensions the nelder mead simplex algorithm first published in 1965 is an enormously popular direct search method for multidimensional unconstrained minimization despite its widespread use essentially no theoretical results have been proved explicitly for the nelder mead algorithm this paper presents convergence properties of the nelder mead algorithm applied to strictly convex functions in dimensions 1 and 2 we prove convergence to a minimizer for dimension 1 and various limited convergence results for dimension 2 a counterexample of mckinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the nelder mead algorithm converges to a nonminimizer it is not yet known whether the nelder mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions

limit cycles for discontinuous quadratic differential systems with two zones abstract in this paper we study the maximum number of limit cycles given by the averaging theory of first order for discontinuous differential systems which can bifurcate from the periodic orbits of the quadratic isochronous centers x y x 2 y x x y and x y x 2 y 2 y x 2 x y when they are perturbed inside the class of all discontinuous quadratic polynomial differential systems with the straight line of discontinuity y 0 comparing the obtained results for the discontinuous with the results for the continuous quadratic polynomial differential systems this work shows that the discontinuous systems have at least 3 more limit cycles surrounding the origin than the continuous ones

on the origin and bifurcations of stick slip oscillations a recently proposed model of macroscopic friction is investigated using methods of dynamical systems analysis particular emphasis is put on the bifurcations associated with the appearance of stick

sequential regularization methods for higher index daes with constraint singularities the linear index 2 case standard stabilization techniques for higher index differential algebraic equations daes often involve elimination of the algebraic solution components this may not work well if there are singularity points where the constraint jacobian matrix becomes rank deficient this paper proposes instead a sequential regularization method srm a functional iteration procedure for solving problems with isolated singularities which have smooth differential solution components for linear index 2 daes we consider both initial value problems ivps and boundary value problems bvps the convergence of the srm is described and proved in detail we believe that this is the first convergence proof for any method for daes with this type of constraint singularities moreover the regularization parameter in our method is not necessarily very small so the srm is an important improvement over usual regularization methods various aspects of the subsequent numerical discretization of the regularized problems are discussed as well and some numerical verifications are carried out

continuation method for computing certain singular points for index 1 differential algebraic equations we will use the arclength continuation method to compute certain singular points for index 1 daes backward difference method will be employed to descretize the problem we will show that the resulting numerical iterative matrix is nonsingular in the presence of the singularity some numerical examples will be given to show the efficiency of the resulting augmented system

bifurcation of limit cycles by perturbing a piecewise linear hamiltonian system with a homoclinic loop abstract in this paper we study limit cycle bifurcations for a kind of non smooth polynomial differential systems by perturbing a piecewise linear hamiltonian system with the center at the origin and a homoclinic loop around the origin by using the first melnikov function of piecewise near hamiltonian systems we give lower bounds of the maximal number of limit cycles in hopf and homoclinic bifurcations and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in e in the case when the degree of perturbing terms is low we obtain a precise result on the number of zeros of the first melnikov function

bifurcations in nonsmooth dynamical systems a review is presented of the one parameter nonsmooth bifurcations that occur in a variety of continuous time piecewise smooth dynamical systems motivated by applications a pragmatic approach is taken to defining a discontinuity induced bifurcation dib as a nontrivial interaction of a limit set with respect to a codimension one discontinuity boundary in phase space only dibs that are local are considered that is bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows three classes of systems are considered involving either state jumps jumps in the vector field or jumps in some derivative of the vector field a rich array of dynamics are revealed involving the sudden creation or disappearance of attractors jumps to chaos bifurcation diagrams with sharp corners and cascades of period adding for each kind of bifurcation identified where possible a kind of normal form or discontinuity mapping dm is given together with a canonical example and an application the goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators impact oscillators dc dc converters and problems in control theory

adomian decomposition method for solution of nonlinear differential algebraic equations in m m hosseini adomian decomposition method with chebyshev polynomials appl math comput in press an efficient modification of the adomian decomposition method was presented by using chebyshev polynomials also in m m hosseini adomian decomposition method for solution of differential algebraic equations j comput appl math in press solution of linear differential algebraic equations was considered by adomian decomposition method in this paper standard and modified adomian decomposition methods are applied to non linear differential algebraic equations the schemes are tested for some examples and the results demonstrate reliability and efficiency of the proposed methods

on normal form calculations in impact oscillators normal form calculations are useful for analysing the dynamics close to bifurcations however the application to non smooth systems is a topic for current research here we consider a class of imp

on hopf bifurcation in non smooth planar systems abstract as we know for non smooth planar systems there are foci of three different types called focus focus ff focus parabolic fp and parabolic parabolic pp type respectively the poincare map with its analytical property and the problem of hopf bifurcation have been studied in coll et al 2001 3 and filippov 1988 6 for general systems and in zou et al 2006 13 for piecewise linear systems in this paper we also study the problem of hopf bifurcation for non smooth planar systems obtaining new results more precisely we prove that one or two limit cycles can be produced from an elementary focus of the least order order 1 for foci of ff or fp type and order 2 for foci of pp type theorem 2 3 different from the case of smooth systems for piecewise linear systems we prove that 2 limit cycles can appear near a focus of either ff fp or pp type theorem 3 3

limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential systems abstract in this paper we consider the quadratic isochronous centers perturbed inside piecewise polynomial differential systems of arbitrary degree n with the straight line of discontinuity x 0 the main concerns are the number of zeros of the first order melnikov functions and the estimate of the number of limit cycles bifurcating from the period annuli for quadratic isochronous centers s 1 s 2 and s 3 we will provide a sharp upper bound for the number of zeros of the first order melnikov functions for quadratic isochronous center s 4 we give a rough estimate however when the problem is reduced to perturbations inside polynomial differential systems our result for s 4 will improve that in li et al 2000 12 significantly moreover we will reveal out some equivalence between the first order melnikov function method and the first order averaging method for investigating the number of limit cycles of piecewise polynomial differential systems

bifurcation of periodic orbits in a class of planar filippov systems abstract in this paper we discuss the perturbations of a general planar filippov system with exactly one switching line when the system has a limit cycle we give a condition for its persistence when the system has an annulus of periodic orbits we give a condition under which limit cycles are bifurcated from the annulus we also further discuss the stability and bifurcations of a nonhyperbolic limit cycle when the system has an annulus of periodic orbits we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching

numerical solution of linear differential algebraic equations in appl math comput 140 2003 77 and appl math comput in press a reducing and modified reducing index methods have been proposed for some cases of semi explicit daes differential algebraic equations in this paper these methods are generalized to more cases also numerical implementation of generalized methods will be presented through pseudospectral method with domain decomposition in addition aforementioned methods will be considered by some examples

control of near grazing dynamics in impact oscillators a method is presented for controlling the persistence of a local attractor near a grazing periodic trajectory in a piecewise smooth dynamical system in the presence of discontinuous jumps in the state associated with intersections with system discontinuities in particular it is shown that a discrete linear feedback strategy may be employed to retain the existence of an attractor near the grazing trajectory such that the deviation of the attractor from the grazing trajectory goes to zero as the system parameters approach those corresponding to grazing contact the implementation relies on a local analysis of the near grazing dynamics using the concept of discontinuity mappings numerical results are presented for a linear and a nonlinear oscillator