normal forms for periodic systems the theory of normal forms for autonomous differential equations based on the lie bracket has proved to be an important technique in determining the behaviour of solutions of nonlinear differential equations near a stationary solution it has been widely used in the bifurcation theory for autonomous ordinary differential equations see e g 1 3 6 9 11 i in its simplest form the theory applies to an equation of the form

real versus complex robustness margin continuity as a smooth versus holomorphic singularity problem abstract it is well known that the real robustness margin can be discontinuous while the complex robustness margin is always continuous relative to problem data using some concepts from set valued analysis continuity of can be viewed as structural stability of the neutral stability region from this point of view the crucial issue is whether 0 j0 is a critical value of the return difference map this paper shows that the discrepancy between real and complex cases is due to the additional holomorphic property of the nyquist return difference mapping of the complex function the critical points of the nyquist map in the complex case are at most finite in number in contrast the critical points of the nyquist map of the real smooth case form generically a curve furthermore and more importantly in the complex case even when 0 j0 is critical the stability crossover is continuously deformed under the variation of certain parameters while in the real case the crossover could sustain a catastrophic change

number of limit cycles of the lienard equation in this paper we study a lienard system of the form x 5y2f x y52x where f x is an odd polyno mial we introduce a method that gives a sequence of algebraic approximations to the equation of each limit cycle of the system this sequence seems to converge to the exact equation of each limit cycle we obtain also a sequence of polynomials rn x whose roots of odd multiplicity are related to the number and location of the limit cycles of the system s1063 651x 97 02809 2

a recursive approach to compute normal forms normal forms are instrumental in the analysis of dynamical systems described by ordinary differential equations particularly when singularities close to a bifurcation are to be characterized however the computation of a normal form up to an arbitrary order is numerically hard this paper focuses on the computer programming of some recursive formulas developed earlier to compute higher order normal forms a computer program to reduce the system to its normal form on a center manifold is developed using the maple symbolic language however it should be stressed that the program relies essentially on recursive numerical computations while symbolic calculations are used only for minor tasks some strategies are proposed to save computation time examples are presented to illustrate the application of the program to obtain high order normalization or to handle systems with large dimension

on the failure of static stability analyses of nonconservative systems in regions of divergence instability abstract the stability of perfect bifurcational discrete dissipative systems under follower loads in regions of existence non existence of adjacent equilibria is thoroughly re examined in the light of recent progress in nonlinear dynamics a general theory for such nongradient systems described by autonomous ordinary differential equations is developed conditions for the existence of adjacent equilibria the stability of precritical critical and postcritical states as well as for different types of local bifurcations are established focusing attention on the interaction of geometric nonlinearities and vanishing damping new findings contradicting widely accepted results of the classical linear analysis are discovered in a small region of adjacent equilibria near a compound branching point which is explicitly determined an interaction of two consecutive postbuckling modes occurs related to the following phenomena in case of vanishing damping loss of stability may occur via a hopf dynamic bifurcation prior to static divergence buckling moreover the critical states of divergence instability may be associated with a double zero jacobian eigenvalue satisfying also the conditions of a hopf local bifurcation besides local dynamic bifurcations global bifurcations are also found an example is used to illustrate the qualitative findings

on codimension three bifurcations in the motion of articulated tubes conveying a fluid abstract we consider the planar motion of two articulated tubes conveying a fluid near a double degeneracy in the linearized dynamical equations in particular the interaction between hopf and pitchfork symmetric saddle node bifurcations and the effects of asymmetries on that interaction are studied in detail the fourth order system of nonlinear equations is reduced to a third order system using center manifold theory the method of averaging provides the normal form which can be considered as a planar vector field with three important parameters one each for the hopf bifurcation the pitchfork bifurcation and the asymmetry this second order system is shown to possess twenty three open sets in the parameter space each with qualitatively different phase portraits some of these portraits are presented along with bifurcation analysis and the physical implications of the results are discussed

nonlinear dynamics of cantilevered extensible pipes conveying fluid abstract in this paper the nonlinear planar dynamics of a fluid conveying cantilevered pipe is investigated the centreline of the pipe is considered to be extensible i e coupled longitudinal and transverse displacements are considered the extended version of the lagrange equations for systems containing non material volumes is employed to derive the equations of motion resulting directly in a set of coupled nonlinear ordinary differential equations the pseudo arclength continuation technique along with direct time integration are used to solve these equations bifurcation diagrams of the system are constructed as the flow velocity is increased these diagrams are supplemented by time traces phase plane portraits and fast fourier transforms for some sets of system parameters as opposed to the case of an inextensible pipe an extensible pipe elongates in the axial direction as the flow velocity is increased from zero depending on the system parameters this static elongation can be considerable at the critical flow velocity the system loses stability via a supercritical hopf bifurcation emerging from the trivial solution for the transverse displacement and leading to a flutter

global and chaotic dynamics for a parametrically excited thin plate the global bifurcations and chaotic dynamics of a parametrically excited simply supported rectangular thin plate are analyzed the formulas of the thin plate are derived by von karman type equation and galerkin s approach the method of multiple scales is used to obtain the averaged equations based on the averaged equations theory of normal form is used to give the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues by maple program on the basis of the normal form global bifurcation analysis of the parametrically excited rectangular thin plate is given by a global perturbation method developed by kovacic and wiggins the chaotic motion of thin plate is found by numerical simulation

bifurcation phenomena near homoclinic systems a two parameter analysis the bifurcations of periodic orbits in a class of autonomous three variable nonlinear differential equation systems possessing a homoclinic orbit associated with a saddle focus with eigenvalues i where 1 sil nikov s condition are studied in a two parameter space the perturbed homoclinic systems undergo a countable set of tangent bifurcations followed by period doubling bifurcations leading to periodic orbits which may be attractors if 1 2 the accumulation rate of the critical parameter values at the homoclinic system is exp 2 a global mechanism for the onset of homoclinicity in strongly contractive flows is analyzed cusp bifurcations with bistability and hysteresis phenomena exist locally near the onset of homoclinicity a countable set of these cusp bifurcations with scaling properties related to the eigenvalues i of the stationary state are shown to occur in infinitely contractive flows in the two parameter space the periodic orbit attractor domain exhibits a spiral structure globally around the set of homoclinic systems in which all the different periodic orbits are continuously connected

the limited effectiveness of normal forms a critical review and extension of local bifurcation studies of the brusselator pde abstract the detection and unfolding of degenerate local bifuractions provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension using the brusselator partial differential equations pdes prigogine and lefever 1968 as motivation and main example we critically review this method we extend and correct previous calculations presenting explicit formulae from which normal forms accurate to third order may be computed and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold and with galerkin and finite difference approximations of the original pde while judicious use of symbolic manipulations makes feasible such high order center manifold and normal form calculations we show that the conclusions drawn from them are of limited use in understanding spatio temporal complexity and chaos as guckenheimer 1981 argued the method permits proof of existence of quasi periodic motions and under mild genericity assumptions sil nikov chaos sub shifts of finite type but the parameter and phase space ranges in which these results may be applied are extremely small

path integral approach to noncommutative space times we propose a path integral formulation of noncommutative generalizations of space time manifold in even dimensions characterized by a length scale p the commutative case is obtaind in the limit p 0

symbolic computation of normal forms for semi simple cases this paper presents a method and computer programs for computing the normal forms of ordinary differential equations whose jacobian matrix evaluated at an equilibrium involves semi simple eigenvalues the method can be used to deal with systems which are not necessarily described on a center manifold an iterative procedure is developed for finding the closed form expressions of the normal forms and associated nonlinear transformations computer programs using a symbolic computer language maple are developed to facilitate the application of the method the programs can be conveniently executed on a main frame a workstation or a pc machine without any interaction a number of examples are presented to demonstrate the applicability of the method and the computation efficiency of the maple programs

nonlinear dynamics of a pipe conveying pulsating fluid with combination principal parametric and internal resonances abstract nonlinear dynamics of a hinged hinged pipe conveying pulsatile fluid subjected to combination and principal parametric resonance in the presence of internal resonance is investigated the system has geometric cubic nonlinearity due to stretching effect out of immovable support conditions at both ends the pipe conveys fluid at a velocity with a harmonically varying component over a constant mean velocity for appropriate choice of system parameters the natural frequency of the second mode is approximately three times that of the first mode for a range of mean flow velocity activating a three to one internal resonance the analysis is carried out using the method of multiple scales by directly attacking the governing nonlinear integro partial differential equations and the associated boundary conditions the set of first order ordinary differential equations governing the modulation of amplitude and phase is analyzed numerically for combination parametric resonance and principal parametric resonance stability bifurcation and response behavior of the pipe are investigated the amplitude and frequency detuning of the harmonic velocity perturbation are taken as the control parameters the system exhibits response in the directly excited and indirectly excited modes due to modal interaction dynamic response of the system is presented in the form of phase plane trajectories poincare maps and time histories a wide array of dynamical behavior is observed illustrating the influence of internal resonance

homoclinic orbits and mixed mode oscillations in far from equilibrium systems nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated we study the bifurcation sequences of the mixed mode oscillations generated by the homoclinicity which are shown to belong to two different types depending on the nature of the liapunov numbers of the basic periodic orbit a detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation chaotic cool flame oscillations are also observed in the same model when the control parameter crosses a critical value this chaotic set of trajectories becomes globally unstable and forms a cantor like hyperbolic repellor and the ignition mechanism generates ahomoclinic tangency to the cantor set of trajectories the complex bifurcation diagram may be globally reconstructed from a one dimensional dynamical system thanks to the strong contractivity of thermokinetics it is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system

a perturbation analysis of interactive static and dynamic bifurcations the instability behavior of a nonlinear autonomous system in the vicinity of a coincident critical point which leads to interactions between static and dynamic bifurcations is studied the critical point considered is characterized by a simple zero and a pair of pure imaginary eigenvalues of the jacobian and the system contains two independent parameters the static and dynamic bifurcations and quasiperiodic motions resulting from the interaction of the bifurcation modes and the associated invariant tori are analyzed by a novel unification technique that is based on an intrinsic perturbation procedure divergence boundary dynamic bifurcation boundary secondary bifurcations and invariant tori are determined explicitly two illustrative examples concerning control systems are presented

on asymptotic expansions of twisted products the series development of the quantum mechanical twisted product is studied the series is shown to make sense as a moment asymptotic expansion of the integral formula for the twisted product either pointwise or in the distributional sense depending on the nature of the factors a condition is given that ensures convergence and is stronger than previously known results possible applications are examined

center manifolds normal forms and bifurcations of vector fields with application to coupling between periodic and steady motions abstract we study the instabilities known to aeronautical engineers as flutter and divergence mathematically these states correspond to bifurcations to limit cycles and multiple equilibrium points in a differential equation making use of the center manifold and normal form theorems we concentrate on the situation in which flutter and divergence become coupled and show that there are essentially two ways in which this is likely to occur in the first case the system can be reduced to an essential model which takes the form of a single degree of freedom nonlinear oscillator this system which may be analyzed by conventional phase plane techniques captures all the qualitative features of the full system we discuss the reduction and show how the nonlinear terms may be simplified and put into normal form invariant manifold theory and the normal form theorem play a major role in this work and this paper serves as an introduction to their application in mechanics repeating the approach in the second case we show that the essential model is now three dimensional and that far more complex behavior is possible including nonperiodic and chaotic motions throughout we take a two degree of freedom system as an example but the general methods are applicable to multi and even infinite degree of freedom problems

classification and unfolding of degenerate hopf bifurcations with o 2 symmetry no distinguished parameter abstract the hopf bifurcation in the presence of o 2 symmetry is considered when the bifurcation breaks the symmetry the critical imaginary eigenvalues have multiplicity two and generically there are two primary branches of periodic orbits which bifurcate simultaneously in applications these correspond to rotating traveling waves and standing waves using equivariant singularity theory a classification of all such bifurcations up to and including codimension three is presented no distinguished parameter is assumed the universal unfoldings reveal the existence of both 2 tori and 3 tori corresponding to quasiperiodic waves with two and three independent frequencies respectively

sensitivities and linear stability analysis around a double zero eigenvalue a general multiparameter system admitting a double zero eigenvalue at a critical equilibrium point is considered a sensitivity analysis of the critical eigenvalues is performed to explore the neighborhood of the critical point in the parameter space because the coalescence of the eigenvalues implies that the jacobian matrix is defective or nilpotent well suited techniques of perturbation analysis must be employed to evaluate the eigenvalues and the eigenvector sensitivities different asymptotic methods are used based on perturbations both of the eigenvalue problem and the characteristic equation the analysis reveals the existence of a generic nonsingular case and of a nongeneric singular case however even in the generic case a codimension 1 subspace exists in the parameter space on which a singularity occurs by the use of the relevant asymptotic expansions linear stability diagrams arebuiltup and different bifurcation mechanisms divergence hopf doubledivergence doubledivergence hopf degeneratehopf arehighlighted the problem of e nding a uniqueexpression uniformly valid in the wholespaceis then addressed it isfound that a second degreealgebraic equation governsthebehaviorofthecriticaleigenvalues it also permits clarie cation of the geometrical meaning of the unfolding parameters which has been discussed in literature for the takens bogdanova bifurcation finally a mechanical system loaded by nonconservative forces and exhibiting a double zero bifurcation is studied as an example

global study of rayleigh duffing oscillators in this paper we investigate the global dynamics of rayleigh duffing oscillators with global parameters including equilibria at both finity and infinity existences and coexistence of limit cycles and homoclinic loops in fact this oscillator will occur hopf bifurcations homoclinic bifurcations and double limit cycle bifurcations moreover we find that the homoclinic bifurcation of this oscillator is special which is a gluing bifurcation the global bifurcation diagram and all phase portrait are given and numerical simulations are shown to verify our analysis finally

the spectral action for moyal planes extending a result of vassilevich we obtain the asymptotic expansion for the trace of a spatially regularized heat operator l f e t where is a generalized laplacian defined with moyal products and l f is moyal left multiplication the moyal planes corresponding to any skewsymmetric matrix being spectral triples the spectral action introduced in noncommutative geometry by chamseddine and connes is computed this result generalizes the connes lott action previously computed by gayral for symplectic

continuity properties of the real complex structured singular value the structured singular value function mu is defined with respect to a given uncertainty set this function is continuous if the uncertainties are allowed to be complex however if some uncertainties are required to be real then it can be discontinuous it is shown that mu is always upper semi continuous and conditions are derived under which it is also lower semi continuous with these results the real parameter robustness problem is reexamined a related although not equivalent problem is formulated which is always continuous and the relationship between the new problem and the original real mu m problem is made explicit a numerical example and results obtained via this related problem are presented

applications of statistical physics to the oil industry predicting oil recovery using percolation theory in this paper we apply scaling laws from percolation theory to the problem of estimating the time for a uid injected into an oil eld for the purposes of recovering the oil to breakthrough into a production well the main contribution is to show that percolation theory when applied to a realistic model can be used to obtain the same results as calculated in a more conventional way but signicantly more quickly specically we found that a previously proposed scaling form for the breakthrough time distribution when applied to a real oil eld is in good agreement with more time consuming simulation results consequently these methods can be used in practical engineering circumstances to aid decision making for real eld problems c 1999 elsevier science b v all rights reserved

a study of the limit cycles associated with a generalized codimension 3 lienard oscillator in this paper melnikov and petrov methods are used to study the limit cycles associated with the degenerate bifurcations of a representative generalized codimension 3 lienard oscillator it is shown that the coexistence of multiple limit cycles is possible and that jumping phenomena exist between limit cycles moreover it is found that the stable limit cycles oscillations and the unstable limit cycles oscillations can occur alternatively

the newhouse set has a positive hausdorff dimension the newhouse phenomenon of infinitely many coexisting periodic attractors is studied in its simplest form one shows that the corresponding parameter set the newhouse set jn has a strictly positive hausdorff dimension this result is stronger than that of tedeschini lalli and yorke commun math phys 106 635 1986 concerning the lebesgue measure of the newhouse set and is complementary to our knowledge on the topological properties ofjn namely it is a residual set hence uncountable and everywhere dense in a parameter interval

uncertainty relation in quantum mechanics with quantum group symmetry the commutation relations uncertainty relations and spectra of position and momentum operators were studied within the framework of quantum group symmetric heisenberg algebras and their bargmann fock representations as an effect of the underlying noncommutative geometry a length and a momentum scale appear leading to the existence of nonzero minimal uncertainties in the positions and momenta the usual quantum mechanical behavior is recovered as a limiting case for not too small and not too large distances and momenta

characterization and efficient computation of the structured singular value the concept of structured singular value was recently introduced by doyle 1 as a tool for the analysis and synthesis of feedback systems with structured uncertainties it is a key to the design of control systems under joint robustness and performance specifications and it very nicely complements the h infty approach to control system design in this paper it is shown that the structured singular value can be obtained as the solution of several smooth optimization problems properties of these optimization problems are exhibited leading to a fast algorithm that always yields the structured singular value for block structures of size no larger than 3 and often does for block structures of larger size

a system with three limit cycles appearing in a hopf bifurcation and dying in a homoclinic bifurcation the cusp of order 4 abstract in this paper we give an example of a family of polynomial vector fields with three limit cycles appearing simultaneously on a hopf bifurcation h of order 3 and vanishing simultaneously in a homoclinic loop bifurcation hl of order 3 the region with three limit cycles is a topological 3 simplex the system is a generalization of bogdanov s system at the same time we give the bifurcation diagram for the universal unfolding of the cusp of order 4 this bifurcation diagram is a cone it contains a cone on the bifurcation diagram with the three limit cycles inside a 3 simplex region plus saddle node and cusp bifurcations of lower order

worldline approach to noncommutative field theory the study of the heat trace expansion in non commutative field theory has shown the existence of moyal non local seeley dewitt coefficients which are related to the uv ir mixing and manifest in some cases the non renormalizability of the theory we show that these models can be studied in a worldline approach implemented in phase space and arrive at a master formula for the n point contribution to the heat trace expansion this formulation could be useful in understanding some open problems in this area as the heat trace expansion for the non commutative torus or the introduction of renormalizing terms in the action as well as for generalizations to other non local operators

imperfection sensitivity of interacting hopf and steady state bifurcations and their classification abstract the bifurcation problem of interacting time periodic and stationary solutions of nonlinear evolution equations with double degeneracy is discussed in terms of singularity and imperfect bifurcation theory a complete classification up to symmetry covariant contact equivalence and codimension three of generic perturbations of interacting hopf and steady state bifurcations is presented the sensitivity of the bifurcation diagrams to imperfections is analyzed normal forms describing sequences of secondary and tertiary bifurcations leading to motions on tori are determined a variety of phenomena such as gaps in hopf branches periodic motions not stably connected to steady states and the formation of islands is discovered which one can expect to find in perturbed evolution equations on pure geometric grounds implications for physical systems are discussed

on finite 4d quantum field theory in non commutative geometry the truncated 4 dimensional spheres4 and the action of the self interacting scalar field on it are constructed the path integral quantization is performed while simultaneously keeping theso 5 symmetry and the finite number of degrees of freedom the usual field theory uv divergences are manifestly absent

three dimensional dynamics of a fluid conveying cantilevered pipe fitted with an additional spring support and an end mass abstract the three dimensional 3 d dynamics of a fluid conveying cantilevered pipe fitted with an end mass and additional intra span spring support is investigated in this paper both theoretically and experimentally the main objective is to examine how the dynamics of a cantilevered pipe with additional spring support is modified by the presence of a small mass attached at the free end in the theoretical study the nonlinear three dimensional equations of motion are discretized via galerkin s method and the resulting equations are solved by a finite difference method for the cases studied the system was found to lose stability by planar flutter as the flow velocity is increased beyond that point a sequence of higher order bifurcations ensue involving 2 d and 3 d periodic and quasiperiodic motions as well as chaotic ones in the experiments performed with elastomer pipes and water flow similarly 2 d and 3 d periodic quasiperiodic and chaotic oscillations were observed theory and experiments have been shown to be in good qualitative and quantitative agreement the experimental behaviour is illustrated by video clips electronic annexes moreover the effects of i small stiffness imperfections and ii excitation by a point force are explored theoretically in a preliminary way

a method of constructing cycles without contact around a weak focus has a weak focus then limit cycles can be made to appear in its neighborhood by varying slightly the right sides 11 frommer 2 bautin 3 chin et al 4 and tung 5 are examples of excellent expositions of poincare s idea it is necessary to go from numerical examples to general theory in the first part of this paper we present a method of constructing cycles without contact around a weak focus which makes clear poincart s idea used in the above work 2 51 in the second part another method to analyze global phase portraits is used quadratic systems with at least four limit cycles are produced by means of both the above methods this contradicts the results of petrovskii and landis 6

chaotic oscillations of the autonomous system of a constrained pipe conveying fluid abstract experiments have shown that a cantilevered pipe conveying fluid the limit cycle motions of which beyond the hopf bifurcation interact with motion limiting non linear constraints exhibits region of chaotic motions in this paper the planar dynamics of the flexible pipe system are examined theoretically by means of a two degree of freedom four dimensional analytical model exploring the existence of chaotic oscillations in the parameter space of this autonomous system calculations of the lyapunov exponents phase portraits of the oscillation bifurcation diagrams and poincare maps establish definitively the existence of chaotic motions the route to chaos is shown to be via period doubling bifurcations the effect of some key parameters on the chaotic regions is investigated

heat kernel and number theory on nc torus the heat trace asymptotics on the noncommutative torus where generalized laplacians are made out of left and right regular representations is fully determined it turns out that this question is very sensitive to the number theoretical aspect of the deformation parameters the central condition we use is of a diophantine type more generally the importance of number theory is made explicit in a few examples we apply the results to the spectral action computation and revisit the uv ir mixing phenomenon for a scalar theory although we find non local counterterms in the nc phi 4 theory on mathbb t 4 we show that this theory can be made renormalizable at least at one loop and maybe even beyond

path integral and noncommutative poisson brackets we find that in presence of noncommutative poisson brackets the relation between lagrangian and hamiltonian is modified we discuss this property by using the path integral formalism for non relativistic systems we apply this procedure to the harmonic oscillator with a minimal length

complex analytic theory of the μ function in this paper we consider the determinant of the multivariable return difference nyquist map crucial in defining the complex m function as a holomorphic function defined on a polydisk of uncertainty the key property of holomorphic functions of several complex variables that is crucial in our argument is that it is an open mapping from this single result only we show that in the diagonal perturbation case all preimage points of the boundary of the horowitz template are included in the distinguished boundary of the polydisk in the block diagonal perturbation case where each block is norm bounded by one a preimage of the boundary is shown to be a unitary matrix in each block finally some algebraic geometry together with the weierstrass preparation theorem allows us to show that the deformation of the crossover under holomorphic variations of certain parameters is continuous q 1999 academic press

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

the number of limit cycle bifurcation diagrams for the generalized mixed rayleigh lienard oscillator abstract this paper investigates the generalized mixed rayleigh lienard oscillator with highly nonlinear terms not restrict to the number of limit cycles this analysis considers mainly the number of limit cycle bifurcation diagrams of the system first the singularity theory approach is applied to the first order averaged approximation of the system with lower order nonlinear terms to reveal all possible bifurcation diagrams by summarizing the generating rule and structural distinction of different bifurcation diagrams a numerical procedure is then developed calculation suggests that the number of bifurcation diagrams increase very fast as the order of nonlinear terms lastly numerical simulations are adopted to approve the analytical results

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

zeroes of complete elliptic integrals for 1 2 resonance abstract we give an estimate for the number of zeroes of the elliptic integral i h a x y dx b x y dy along the ovals h x y h h x y y 2 2 x 2 x 4 in terms of the degrees of the symmetric polynomials a and b

bifurcation analysis of double pendulum with a follower force abstract the stability and bifurcations of a double pendulum with a follower force are considered the focus is placed on a doubly degenerate system which possesses a zero eigenvalue and a pair of pure imaginary eigenvalues that is coupled flutter and divergence bifurcation the local qualitative behavior of the system is examined in the neighborhood of the degenerate system the four dimensional equations of motion are reduced to the two dimensional ones by using some qualitative reduction theory for dynamical systems and it is shown that two distinct bifurcations can occur in the system according to the ratio of two damping coefficients of the double pendulum numerical simulations have been performed using the original four dimensional equations to confirm the analytical results some global forms of behavior of the solutions have also been investigated numerically

homoclinic tangencies in the dynamics of articulated pipes conveying fluid abstract numerical evidence is presented showing homoclinic tangencies to periodic orbits occuring in a four dimensional system of ordinary differential equations that describes the dynamics of a pair of articulated pipes conveying fluid bifurcation sequences of mixed mode oscillations are discovered in two separate regions of parameter space in one region the equations have z 2 symmetry and a symmetric pair of heteroclinic tangencies is implied in both cases satisfactory agreement is found between the observed bifurcation sequences and the local theory of homoclinic tangencies global branches of mixed mode orbits are computed it is discovered that some orbits are generated by either sil nikov or figure of eight homoclinic orbits to stationary points whereas others form isolas that feature characteristic towers of saddle node and period doubling bifurcations

global stabilty analysis of a classical non conservative system under a follower load abstract the bifurcations critical conditions and global stability of the classical two degree of freedom ziegler s dissipative model under a partial follower load are thoroughly discussed with the aid of a complete non linear dynamic analysis attention is mainly focused on the range of values of the non conservativeness loading parameter for which adjacent equilibria do not exist various characteristic findings are established by studying the nature of the jacobian matrix eigenvalues associated with the above non linear autonomous dissipative system it is found that the system exhibits stable hopf bifurcations global stability for the entire region of non adjacent equilibria while the classical linear stability analysis leads to flutter instability local instability moreover in an explicit general form the significant effect of damping on the critical dynamic bifurcation load is established on the basis of the latter the critical loads for the case of vanishing damping are found to be completely different from those of the classical local stability analysis all findings of this analysis are checked against numerical solution of the equations of motion

analysis of non linear dynamics and bifurcations of a double pendulum abstract in this paper the dynamic behaviour of a double pendulum system in the vicinity of several compound critical points is explored through both analytical and numerical approaches four types of critical points are considered which are characterized by a double zero eigenvalue a simple zero and a pair of pure imaginary eigenvalues and two pairs of pure imaginary eigenvalues including resonant and non resonant cases with the aid of normal form theory the explicit expressions for the critical bifurcation lines leading to incipient and secondary bifurcations are obtained possible bifurcations leading to 2 d and 3 d tori are also investigated closed form stability conditions of the bifurcation solutions are presented a time integration scheme is used to find the numerical solutions for these bifurcation cases which agree with the analytical results finally numerical simulation is also applied to obtain double period cascading bifurcations leading to chaos

bifurcations and chaotic motions in the autonomous system of a restrained pipe conveying fluid the stability and dynamics of a cantilevered pipe conveying fluid with motion limiting constraints and a linear spring support have been investigated emphasis is placed on analyzing local qualitative behavior of the system in the neighborhood of a doubly degenerate point using some qualitative reduction methods of dynamical system theory the four dimensional differential equation of motion is reduced to a two dimensional one and then the possible motions of the pipe are predicted through analyzing bifurcations of the solution to the reduced equation of motion the unfolding result is found to be in good agreement with the result obtained using the numerical method it is also found that there exist the quasi periodic motions and route to chaos through breakup of the quasi periodic torus surface in some parameter region of the system which differs from that of periodic doubling bifurcation route found earlier in this system numerical simulations have been performed using the four dimensional equation of motion to confirm the analytical results

flow induced bifurcations to three dimensional oscillatory motions in continuous tubes three dimensional motions of a cantilever tube carrying an incompressible fluid and having rotational symmetry about the vertical axis are examined for bifurcating oscillatory solutions the system behavior depends on two parameters rho the flow velocity and beta the mass ratio of the fluid and the tube as the flow rate is increased past a critical value the zero solution becomes unstable by a double pair of complex conjugate eigenvalues of the linearized operator crossing the imaginary axis using ideas from the center manifold theory and the method of integral averaging the governing partial differential equations are reduced to a set of four nonlinear ordinary differential equations it is then shown that the system bifurcates into two qualitatively distinct oscillatory motions these oscillatory solutions themselves for some values of the mass ratio beta bifurcate into more complex amplitude modulated oscillations

feynman path integral on the non commutative plane we formulate the feynman path integral on a non commutative plane using coherent states the propagator for a free particle exhibits uv cutoff induced by the parameter of non commutativity

a note on some differences between real and complex stability radii abstract real and complex stability radii measure the robustness of stable linear systems under real and complex parameter perturbations respectively in this note we point out some basic differences between them in particular we investigate their different behaviour with respect to system interconnections and changes in system data

local birth of homoclinic chaos abstract local generation of chaotic behaviours associated with a sil nikov homoclinic orbit is studied near the codimension two bifurcation at the double stationary oscillatory instability of eigenvalues 0 we develop a perturbative calculation of the sil nikov homoclinic orbit and of its bifurcation locus starting from a conservative vector field flat nonanalytic behaviours due to non axisymmetric perturbations of the normal form are discussed and we show that the codimension one locus of sil nikov homoclinicity is unavoidable near bicriticality under certain conditions the results are applied to rossler s model quasiperiodic regimes born at the double instability are also studied and illustrated in the rossler model

bifurcations in a two degree of freedom elastic system with follower forces abstract the stability and bifurcations of a two degree of freedom elastic system subjected to follower forces are studied attention is mainly concentrated on local behavior small amplitude motions about a centre equilibrium for the parameter values in the neighborhood of some point on the stability boundaries by using centre manifold theory and some codimension two bifurcations it is shown that a number of distinct bifurcations may occur in such a system according to the ratio of two damping coefficients numerical integrations of the full equations of motion have been performed to confirm the analytical aspects of the local solutions as well as to understand some aspects of the global behavior of the solutions

the cusp of order n we study the bifurcation diagram of the family x y y x2 0 1 y 2xy 3x3y xly where l 3 n 1 2 n 2 and i is small when l 1 it corresponds to the bogdanov takens bifurcation

lienard systems limit cycles and melnikov theory lienard systems constitute a general class of two dimensional autonomous systems among which the van der pol equation is found recently giacomini and neukirch phys rev e 57 3809 1997 introduced a sequence of polynomials whose roots are related to the number and location of limit cycles of lienard systems we show that in the limit of these sequences the same information is given by a polynomial which melnikov theory associates with a given lienard system and discuss the relationship existing among them s1063 651x 98 00201 3

modal analysis of mechanical systems with impact non linearities limitations to a modal superposition this paper presents an attempt to generalize the modal superposition formula to mechanical systems with impact type non linearities following the procedure introduced for smooth non linearities the study is restricted to simple one and two degree of freedom systems with a unilateral constraint on one of the degrees of freedom for which the response can be analytically determined generalized frequencies modes and masses are built in the procedure the results obtained for various sets of parameters point out some limitations to the validity of a general modal superposition formula

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3