nontrivial periodic solutions for the asymptotically linear hamiltonian systems with resonance at infinity abstract in this paper we are concerned with the existence of nontrivial periodic solutions of an asymptotically linear hamiltonian systems when a kind of resonance condition is satisfied we prove several new existence results via the morse theory the garlerkin methods and the moslov index theory

morse theory and asymptotic linear hamiltonian system on considere l existence de solutions 2 periodiques des systemes hamiltoniens asymptotiquement quadratiques jz h z t ou j 0 i i 0 i in la matrice identite de n n

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

non stationary parametric oscillations the response of a single degree of freedom system with cubic non linearity to a non stationary principal parametric excitation is investigated the method of multiple scales is used to derive two first order ordinary differential equations for the evolution of the amplitude and phase of the response the evolution equations are numerically integrated for various sweeping rates of the amplitude and frequency of the excitation the results show that the non stationary response penetrates the instability regions and the higher the sweeping rate is the deeper the penetration is

a generalized local linearization principle for non linear dynamical systems tangent spaces in non linear dynamical systems are state dependent hence it is generally not possible to exactly represent a non linear dynamical system by a linear one over finite segments of the evolving trajectories in the phase space it is known from the well known theorem of hartman and grobman that a non linear oscillator admits a linearization within a neighbourhood of any hyperbolic fixed point in the sense that the linearized flow bears a topological conjugacy to the non linear flow within the same neighbourhood this linearization is based on the construction of the tangent space to the non linear vector field at the hyperbolic fixed point even though useful in bifurcation analysis such a linearization procedure cannot be successfully applied to identically simulate the non linear flow over the phase space because the functional form of the topological conjugacy is not known with this in mind an implicit approach to local linearization is evolved in this paper such that the tangent space of the linearized equation transversally intersects the tangent space to the non linear dynamical system at that point in the state space where the solution vector is desired several numerical schemes for the implementation of this implicit and local linearization procedure are explored and illustrated with several numerical results it is shown that the locally transversal linearization ltl procedure finally reduces the given set of non linear ordinary differential equations odes to a set of transcendental algebraic equations with the desired solution vector as the unknown the variables in the state space appear as unknown quantities in this approach finally one arrives at non linear poincare maps for the non linear oscillator in this scheme the characteristic time interval for the map is arbitrarily fixed the methodology is found to be quite versatile for handling non linear dynamical systems in particular it is verified that these schemes are capable of accurately predicting a wide spectrum of typically non linear response characteristics such as limit cycles multi periodicity almost periodicity and chaos for a limited class of response patterns the principle proposed has another advantage in that the time step for integrating the non linear odes need not be small an improved and a more general higher order version of the ltl method is also considered a distinct advantage of this higher version is in its improved ability for a closer simulation of phase dependent time histories where a difference in the initial conditions leads to a phase difference in the realized solution history

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

regularity of solutions to coercive and self controlling viscoplastic problems abstract we prove higher regularity of solutions to viscoplastic problems the main idea is to use the finite difference method which allows us to cancel the monotone nonlinearities our main assumption is that velocity is in l 2 h 1 this is true for all coercive and some special for example linear self controlling non coercive models

application of the incremental harmonic balance method to cubic non linearity systems abstract in this paper the formulation of the incremental harmonic balance ihb method is derived for a general system of differential equations with cubic non linearity which governs a wide range of engineering problems such as large amplitude vibration of beams or plates an incremental arc length method combined with a cubic extrapolation technique is adopted to trace the response curve automatically the stability of its periodic solutions can also be evaluated from the ihb formulation by multi variable floquet theory hsu s method is adopted for computing the transition matrix at the end of one period two numerical examples are presented which demonstrate the effectiveness of the ihb method and hsu s method in treating the non linear vibration of a strongly non linear multiple degree of freedom system

a new numeric analytical principle for nonlinear deterministic and stochastic dynamical systems a new implicit local linearization method termed as the locally transversal linearization ltl is developed in the present paper for a numeric analytical integration of nonlinear dynamical systems under deterministic and or stochastic excitations the ltl procedure is developed in such a way that at a chosen time instant the ltl solution manifold transversally intersects the nonlinear solution manifold at that particular point or cross section in the state space where the solution vector needs to be obtained in general construction of the ltl equation is non unique and thus it is generally needed to substitute the conditionally linear ltl based solution into the given nonlinear ordinary differential equation ode at a specified time instant to ensure a transversal intersection in the present study however it is demonstrated that a transversal intersection at a given time instant may as well be ensured by just choosing a few specific forms of the ltl equation thereby eliminating the need for substitution this new variant of ltl is particularly suitable for discretization of stochastic differential equations sdes in that one simultaneously avoids dealing with formal derivatives of wiener processes while satisfying the nonlinear sde at discrete time points the stochastic ltl procedure is adaptable for obtaining strong pathwise solutions of a general nonlinear dynamical system with continuous but not necessarily differentiable vector fields under additive and or multiplicative excitations a specific advantage of the ltl method is that for a class of response regime herein referred to as the phase independent regime it can obtain an accurate description of response irrespective of the choice of the time step and without a propagation of local errors presently the method has been numerically illustrated for strong solutions of the hardening duffing oscillator under deterministic sinusoidal or additive stochastic white noise excitations or their combination limited numerical results on periodic solutions chaos with or without noise and stochastic resonance are included in this study the results presented are indicative of the sample path accuracy consistency stochastic numerical stability and a partial insensitivity to time step sizes of the proposed method

global stability analysis of fuzzy controllers using cell mapping methods linear and nonlinear stability analysis of a nonlinear fuzzy logic controller is considered the linear analysis is carried out by lyapunov s first method i e by determination of the eigenvalues of the linearized system the nonlinear global stability analysis is executed by a combination of two methods simple and interpolated cell mapping examples of controllers with a complex dynamic behavior are presented these include controllers with multiple solutions such as i stable equilibrium point and stable limit circle ii five unstable equilibrium points and three stable limit circles and iii examples where the linear analysis is completely insufficient and only nonlinear investigation provides desirable results the structure of domains of attraction of stable solutions is determined comparison of the results obtained by the cell mapping method with those provided by direct numerical integration are presented

the response of two degree of freedom systems with quadratic non linearities to a combination parametric resonance abstract the response of two degree of freedom systems with quadratic non linearities to a combination parametric resonance in the presence of two to one internal resonances is investigated the method of multiple scales is used to construct a first order uniform expansion yielding four first order non linear ordinary differential equations governing the modulation of the amplitudes and the phases of the two modes steady state responses and their stability are computed for selected values of the system parameters the effects of detuning the internal resonance detuning the parametric resonance the phase and magnitude of the second mode parametric excitation and the initial conditions are investigated the first order perturbation solution predicts qualitatively the trivial and non trivial stable steady state solutions and illustrates both the quenching and saturation phenomena in addition to the steady state solutions other periodic solutions are predicted by the perturbation amplitude and phase modulation equations these equations predict a transition from constant steady state non trivial responses to limit cycle responses hopf bifurcation some limit cycles are also shown to experience period doubling bifurcations the perturbation solutions are verified by numerically integrating the governing differential equations

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

the response of two degree of freedom systems with quadratic non linearities to a parametric excitation the method of multiple scales is used to analyze the response of two degree of freedom systems with quadratic non linearities to a parametric harmonic excitation having the frequency four ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when 2 2 1 and either 2 1 or 2 2 where 1 and 2 are the linear undamped natural frequencies of the system two critical values 1 and 2 of the amplitude f of the excitation are identified in the analysis when f 2 the amplitude of the directly excited mode grows exponentially with time according to the linear analysis whereas the amplitudes of both modes achieve steady state constant values irrespective of the initial amplitudes according to the non linear analysis when f 1 the motion decays to zero according to both the linear and non linear analyses when 1 f 2 the motion decays to zero according to the linear analysis whereas it achieves a periodic steady state or decays to zero depending on the initial amplitudes according to the non linear analysis this is an example of subcritical instability when 2 2 the steady state value of the higher mode which is directly excited is a constant that is independent of the excitation of amplitude f whereas the amplitude of the lower mode which is indirectly excited through internal resonance grows with the excitation amplitude f this is another example of saturation

a first regularity result for the armstrong frederick cyclic hardening plasticity model with cosserat effects abstract the purpose of this article is to prove the holder continuity up to the boundary of the displacement vector and the microrotation matrix for the quasistatic rate independent armstrong frederick cyclic hardening plasticity model with cosserat effects this model is of non monotone and non associated type in the case of two space dimensions we use the hole filling technique of widman and morrey s dirichlet growth theorem

response of self excited two degree of freedom systems to multifrequency excitations the response to multifrequency excitation of a two degree of freedom self excited system is analyzed by using the method of multiple scales three cases of resonance are considered superharmonic primary and subharmonic also considered is the case of simultaneous superharmonic and subharmonic resonance steady state and stability analyses are carried out for each case and numerical results are presented showing the influences of the several parameters

nontrivial periodic solutions for strong resonance hamiltonian systems via a local linking theorem in this paper we first establish an existence result of critical points for a class of functionals defined on hilbert spaces by using a local linking idea then as an application of the existence result we obtain the existence of periodic solutions of strong resonance hamiltonian systems which are asymptotically linear both at infinity and at origin

an investigation of stability of a control surface with structural nonlinearities in supersonic flow using zubov s method it is well known that the presence of nonlinearities may significantly affect the aeroelastic response of an aerospace vehicle structure in this paper the aeroelastic behaviour at high mach numbers of an all moving control surface with a nonlinearity in the root support is investigated very often under certain flight conditions a stable equilibrium point corresponding to zero displacement of the structure together with an unstable limit cycle arising from a sub critical hopf bifurcation results from the presence of the nonlinearity the dynamic aeroelastic response to external excitation is also of interest and when sinusoidal forcing is applied the stable equilibrium point may then be replaced by a periodic attractor and the limit cycle by an unstable multi periodic solution with or without this forcing there is an attractor which will possess a domain of attraction in this paper the problem of estimating these domains of attraction is tackled using zubov s method in the absence of forcing the method is applied directly to the aeroelastic equations while for the forced system the method of averaging is applied to approximate the aeroelastic equations by an autonomous system the behaviour of the system with forcing is also investigated for flight speeds below a threshold which may occur where the unstable limit cycle of the unforced system disappears in this regime the nonlinear system may nevertheless still possess multiple attractors and their domains of attraction are investigated again using an averaged form of the aeroelastic equations in this study the nonlinearity in the root support was assumed to be due to a cubic hardening restoring moment the zubov approach which always yields conservative estimates was shown to be capable of rapidly giving a good indication of stability domain boundaries under many conditions although this investigation focuses on an aeroelastic system the general form of equations considered arises in many other settings so that the approach would be relevant to a whole range of engineering applications

the response of a single degree of freedom system with quadratic and cubic non linearities to a fundamental parametric resonance abstract the response of a one degree of freedom system with quadratic and cubic non linearities to a principal parametric resonance is investigated the method of multiple scales is used to determine the equations that describe to second order the modulation of the amplitude and phase with time about one of the foci these equations are used to determine the fixed points and their stability the perturbation results are verified by integrating the governing equation with use of a digital computer and an analogue computer for small excitation amplitudes the analytical results are in excellent agreement with the numerical solutions the large amplitude responses are investigated by using both a digital and an analogue computer the cases of single and double well potentials are investigated systems with double well potentials exhibit complicated dynamic behaviors including period multiplying and demultiplying bifurcations and chaos in some cases a bifurcated response coexists with another periodic attractor and a chaotic attractor coexists with a periodic attractor long time histories phase planes poincare maps fractal basin maps and spectra of the response are presented a bifurcation diagram of many solutions in the excitation amplitude excitation frequency plane is also presented

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

combination parametric resonance leading to periodic and chaotic response in two degree of freedom systems with quadratic non linearities the dynamic response of a parametrically excited non linear system with two degrees of freedom is studied the system of equations which describes the compliant motion of a robotic manipulator exhibits internal resonance and is excited at a frequency near a primary resonance the effect of coupling is studied in detail non zero periodic motions are observed to co exist with a stable equilibrium state over a range of detuning near the resonant frequency loss of stability of the periodic response is observed at points of vertical tangency corresponding to a jump in the response while for some of the cases hopf bifurcation in the response corresponding to amplitude modulated motions is observed in the latter case period doubling of the amplitude of response is observed with a monotonic change in detuning eventually leading to chaotic motion in the system near exact tuning to the parametric resonance a quenching of the chaotic motion is found leaving only the stable zero equilibrium state for the system over some detuning interval

homoclinic solutions for a class of second order hamiltonian systems abstract some new results for the existence of homoclinic orbits are obtained for the second order hamiltonian systems u t v t u t f t where t r u r n and v c 1 r r n r v t x k t x w t x is t periodic with respect to t t 0 and f r r n is a continuous and bounded function our assumptions on k and f are rather loose

combined analytical and numerical approaches in dynamic stability analyses of engineering systems abstract dynamic stability is a widely studied area that has attracted many researchers from various disciplines although dynamic stability is usually associated with mechanics theoretical physics or other natural and technical disciplines it is also relevant to social economic and philosophical areas of our lives therefore it is useful to occasionally highlight the general aspects of this amazing area to present some relevant examples and to evaluate its position among the various branches of rational mechanics from this perspective the aim of this study is to present a brief review concerning the dynamic stability problem its basic definitions and principles important phenomena research motivations and applications in engineering the relationships with relevant systems that are prone to stability loss encountered in other areas such as physics other natural sciences and engineering are also noted the theoretical background which is applicable to many disciplines is presented in this paper the most frequently used dynamic stability analysis methods are presented in relation to individual dynamic systems that are widely discussed in various engineering branches in particular the lyapunov function and exponent procedures routh hurwitz lienard and other theorems are outlined together with demonstrations the possibilities for analytical and numerical procedures are mentioned together with possible feedback from experimental research and testing the strengths and shortcomings of these approaches are evaluated together with examples of their effective complementing of each other the systems that are widely encountered in engineering are presented in the form of mathematical models the analyses of their dynamic stability and post critical behaviour are also presented the stability limits bifurcation points quasi periodic response processes and chaotic regimes are discussed the limit cycle existence and stability are examined together with their separating roles as attractors and repulsers two levels of stability loss recovery of the system is possible or final collapse is inevitable as can be observed in softening systems are noted time limited excitation and relevant transition effects e g seismic excitation are also discussed together with the evaluation of possible system reliability improvement the dynamic stability investigation of two degrees of freedom aero elastic systems in a linear formulation using several approaches is briefly highlighted further systems modelling problems that arise in transport engineering are also outlined a few hints for applications are given some open problems and possible future research strategies are outlined

the response of two degree of freedom systems with quadratic and cubic non linearities to multifrequency parametric excitations abstract the response of two d o f systems with quadratic and cubic non linearities to multi frequency parametric excitations is determined by using the method of multiple scales four first order ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when principal parametric resonances of both modes and combination resonances of the additive and difference type occur simultaneously in all cases the steady state solutions and their stability are determined numerical results depicting the various resonances are presented

periodic and subharmonic solutions of second order hamiltonian systems abstract some solvability conditions of periodic solutions and subharmonic solutions are obtained for a class of new superquadratic non autonomous second order hamiltonian systems by the minimax methods in critical point theory

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

the response of multidegree of freedom systems with quadratic non linearities to a harmonic parametric resonance abstract an analysis is presented of the response of multidegree of freedom systems with quadratic non linearities to a harmonic parametric excitation in the presence of an internal resonance of the combination type 3 2 1 where the n are the linear natural frequencies of the systems in the case of a fundamental resonance of the third mode i e 3 where is the frequency of the excitation one can identify two critical values 1 and 2 where 2 1 of the amplitude f of the excitation the value f 2 corresponds to the transition from stable to unstable solutions when f 2 the motion grows exponentially with time according to the linear theory but the non linearity limits the motion to a finite amplitude steady state the amplitude of the third mode which is directly excited is independent of f whereas the amplitudes of the first and second modes which are indirectly excited through the internal resonance are functions of f when 1 f 2 the motion decays or achieves a finite amplitude steady state depending on the initial conditions according to the non linear theory whereas it decays to zero according to the linear theory this is an example of subcritical instability in the case of a fundamental resonance of either the first or second mode the trivial response is the only possible steady state when f 2 the motion decays to zero according to both linear and non linear theories when f 2 the motion grows exponentially with time according to the linear theory but it is aperiodic according to the non linear theory experiments are being planned to check these theoretical results

parametrically excited non linear systems a comparison of two methods subharmonic resonance of two degree of freedom systems with cubic non linearities to multifrequency parametric excitations in the presence of three to one internal resonance is investigated two approximate methods the multiple scales and the generalized synchronization are used to construct first order non linear ordinary differential equations governing the modulation of the amplitudes and phases steady state solutions and their stability are computed for selected values of the system parameters the results obtained by the two methods are in excellent agreement numerical solutions are carried out and graphical representations of the results are presented and discussed

notes on periodic solutions of subquadratic second order systems some existence theorems are obtained for periodic solutions of the subquadratic second order systems by the minimax methods in critical point theory

parametric excitation of two internally resonant oscillators abstract the response of two degree of freedom systems with quadratic non linearities to a principal parametric resonance in the presence of two to one internal resonances is investigated the method of multiple scales is used to construct a first order uniform expansion yielding four first order non linear ordinary differential averaged equations governing the modulation of the amplitudes and the phases of the two modes these equations are used to determine steady state responses and their stability when the higher mode is excited by a principal parametric resonance the non trivial steady state value of its amplitude is a constant that is independent of the excitation amplitude whereas the amplitude of the lower mode which is indirectly excited through the internal resonance increases with the amplitude of the excitation however in addition to poincare type bifurcations this response exhibits a hopf bifurcation leading to amplitude and phase modulated motions when the lower mode is excited by a principal parametric resonance the averaged equations exhibit both poincare and hopf bifurcations in some intervals of the parameters the periodic solutions of the averaged equations in the latter case experience period doubling bifurcations leading to chaos

the response of non linear single degree of freedom systems to multifrequency excitations the method of multiple scales is used to analyze the response of single degree of freedom systems with cubic non linearities to excitations that involve multiple frequencies two first order ordinary differential equations are derived for the evolution of the amplitude and phase with damping non linearity and all possible resonances conditions for the existence and stability of steady state solutions are determined these results are used to suggest simple means of controlling or minimizing the large oscillations these means may take the form of adding non resonant loads that shift the natural frequency of the system or adding another resonant load having the proper frequencies amplitudes and phases

basin boundaries in asymmetric vibrations of a circular plate abstract in order to investigate further nonlinear asymmetric vibrations of a clamped circular plate under a harmonic excitation we reexamine a primary resonance studied by yeo and lee corrected solvability conditions for non linear asymmetric vibrations of a circular plate journal of sound and vibration 257 2002 653 665 in which at most three stable steady state responses one standing wave and two traveling waves are observed to exist further examination however tells that there exist at most five stable steady state responses one standing wave and four traveling waves two of the traveling waves lose their stability by hopf bifurcation and have a sequence of period doubling bifurcations leading to chaos when the system has five attractors three equilibrium solutions one standing wave and two traveling waves and two chaotic attractors two modulated traveling waves the basin boundaries of the attractors on the principal plane are obtained also examined is how basin boundaries of the modulated motions quasi periodic and chaotic motions evolve as a system parameter varies the basin boundaries of the modulated motions turn out to have the fractal nature

parametrically excited non linear multidegree of freedom systems with repeated natural frequencies abstract a method for analyzing multidegree of freedom systems having a repeated natural frequency subjected to a parametric excitation is presented attention is given to the ordering of the various terms linear and non linear in the governing equations the analysis is based on the method of multiple scales as a numerical example involving a parametric resonance panel flutter is discussed in detail in order to illustrate the type of results one can expect to obtain with this analysis some of the analytical results are verified by a numerical integration of the governing equations

periodic solutions to second order hamiltonian systems in an unbounded potential well in this paper we give some results on the existence of periodic solutions to the second order hamiltonian system where and is an open set of n with non empty bounded complement n we suppose v t x is periodic in t v t x as x and v is super or sub quadratic as x

infinitesimal elastic plastic cosserat micropolar theory modelling and global existence in the rate independent case in this article we investigate the regularizing properties of generalized continua of cosserat micropolar type in the elasto plastic case we propose an extension of classical infinitesimal elasto plasticity to include consistently non dissipative micropolar effects it is shown that the new model is thermodynamically admissible and allows for a unique global in time solution of the corresponding rate independent initial boundary value problem the method of choice is the yosida approximation and a passage to the limit

bifurcation and stability of a two tower system under wind induced parametric external and self excitation a two degree of freedom nonlinear system describing the dynamics of two towers exposed to turbulent wind flow and linked by a nonlinear viscous device is considered the steady component of the wind is responsible for self excitation while the turbulent part causes both parametric and external excitations considered in a specific resonance condition the possible occurrence of multiple hopf bifurcations is taken into account the periodic and quasi periodic solutions are studied after applying a perturbation scheme the effect of the viscous device on the dynamics of the structure is analyzed in order to investigate its effectiveness in mitigating the oscillations of the two independent towers

a twist condition and periodic solutions of hamiltonian systems abstract in this paper we investigate existence of nontrivial periodic solutions to the hamiltonian system hs j z h t z z r 2 n under a general twist condition for the hamiltonian function in terms of the difference of the conley zehnder index at the origin and at infinity we establish existence of nontrivial periodic solutions compared with the existing work in the literature our results do not require the hamiltonian function to have linearization at infinity our results allow interactions at infinity between the hamiltonian and the linear spectra the general twist condition raised here seems to resemble more the spirit of poincare s last geometric theorem

dynamical poroplasticity model existence theory for gradient type nonlinearities with lipschitz perturbations abstract in this article we study existence theory to a non coercive fully dynamic model of poroplasticity with the non homogeneous boundary conditions where the constitutive function is a continuous element of class lm it is a sum of a maximal monotone map g and a globally lipschitz map l without any additional growth conditions we are able to prove the existence of a solution such that the inelastic constitutive equation is satisfied in the measure valued sense moreover if g is a gradient of a differentiable convex function then there exists a solution such that the constitutive equation is satisfied almost everywhere

response of two degree of freedom systems to multifrequency parametric excitations abstract the method of multiple scales is used to analyze the response of two degree of freedom systems to multifrequency parametric excitations the equations describing the modulation of the amplitudes with time are derived for the case of four simultaneous resonances main resonances of the two modes combination resonance of the difference type and combination resonance of the summed type when only two simultaneous resonances exist the modulation equations can be transformed into a system of four real first order differential equations with constant coefficients whose solution depends on the eigenvalues of a four by four constant coefficient matrix when three or more simultaneous resonances exist the modulation equations cannot be made autononous in general and hence one may not be able to solve them except numerically

a semi analytical locally transversal linearization method for non linear dynamical systems an numeric analytical implicit and local linearization methodology called the locally transversal linearization ltl is developed in the present paper for analyses and simulations of non linear oscillators the ltl principle is based on deriving the locally linearized equations in such a way that the tangent space of the linearized equations transversally intersects that of the given non linear dynamical system at that particular point in the state space where the solution vector is sought for purposes of numerical implementation two different numerical schemes namely ltl 1 and ltl 2 schemes based on the ltl methodology are presented both ltl 1 and ltl 2 procedures finally reduce the given set of non linear ordinary differential equations odes to a set of transcendental algebraic equations valid over a short interval of time or over a short segment of the evolving trajectories as projected on the phase space while in the ltl 1 scheme the desired solution vector at a forward time point enters the linearized differential equations as an unknown parameter in the ltl 2 scheme a set of unknown residues enters the linearized system as parameters a limited set of examples involving a few well known single degree of freedom sdof non linear oscillators indicate that the ltl methodology is capable of accurately predicting many complicated non linear response patterns including limit cycles quasi periodic orbits and even strange attractors copyright 2001 john wiley sons ltd

existence of nontrivial periodic solutions for a class of resonance hamiltonian systems abstract in this paper we study the asymptotically linear hamiltonian systems jz h t z with resonance at infinity via the morse theory and the galerkin methods and obtain some new existence results about the nontrivial periodic solutions of this system

stability and primary simultaneous resonance of harmonically excited non linear spring pendulum system the considered system is a harmonically excited non linear spring pendulum which simulates the ship roll motion the method of multiple time scale perturbation is applied to solve the non linear differential equations describing the system up to and including the fourth order approximation all possible resonance cases were extracted at this approximation order this work is limited only to the primary resonance cases the stability of the system is investigated using both frequency response equations and phase plane method the effects of the different parameters on system behavior are studied numerically variation of some equations parameters leads to the bending of the frequency response curves and hence to the jump phenomenon it is quite clear that some of the simultaneous primary resonance cases are undesirable in the design of such system as they represent some of the worst behavior of the system such cases should be avoided as working conditions for the system some recommendations regarding the different parameters of the system are reported

an asymptotically linear non cooperative elliptic system with lack of compactness in this paper we consider an asymptotically linear non cooperative elliptic system at resonance the functional associated to the system is strongly indefinite and so the morse index of each critical point is infinite to overcome this difficulty we will use morse theory for strongly indefinite functionals developed by abbondandolo we shall prove that if the system has a non resonant solution the morse index of which is different from the morse index at infinity then there exists another solution

on resonance hamiltonian systems without the palais smale condition abstract in this paper we study periodic solutions of resonance hamiltonian systems by combining conley index theory and galerkin approximation method we prove the existence of periodic solutions of hamiltonain systems where both the palais smale condition and the strong resonance method fail

a global analysis of an harmonically excited spring pendulum system with internal resonance abstract a non linear system can often have multiple attractors in this case it is important to study the global behavior of the system by determining the global domain of attraction of each attractor this paper is concerned with the global behavior of an harmonically excited spring pendulum system with internal resonance before studying the global behavior the local behaviors of the system around its steady state periodic solutions are examined two non linearly coupled second order non autonomous ordinary differential equations governing the spring pendulum system are transformed into a four dimensional autonomous system in amplitude and phase variables next the stability of equilibrium states of the autonomous system corresponding to steady state periodic solutions of the non autonomous system is examined when certain resonance conditions are satisfied the spring pendulum system has a very complex behavior including jump phenomena and hopf bifurcation global domains of attraction for the spring pendulum system are investigated when two stable steady state periodic solutions exist the interpolated mapping method a numerical method for determining global domains of attraction is presented so as to be applied to systems with dimension higher than two then the domains of attraction for a specific set of parameter values are obtained by the method it is shown that the interpolated mapping method can be an efficient tool for analyzing the global behaviors of higher dimensional systems also examined is how the global domains of attraction evolve as the forcing frequency is varied across critical values at which jump phenomena occur it is discovered that there exists a special plane in the four dimensional state space which contains all the attractors by using this plane called the principal plane the global domains of attraction can be discussed more effectively results show that knowledge of this evolution is essential to better understand the mechanism of the jump phenomena

the armstrong frederick cyclic hardening plasticity model with cosserat effects abstract we propose an extension of the cyclic hardening plasticity model formulated by armstrong and frederick which includes micropolar effects our micropolar extension establishes coercivity of the model which is otherwise not present we study then existence of solutions to the quasistatic rate independent armstrong frederick model with cosserat effects which is however still of non monotone non associated type in order to do this we need to relax the pointwise definition of the flow rule into a suitable weak energy type inequality it is shown that the limit in the yosida approximation process satisfies this new solution concept the limit functions have a better regularity than previously known in the literature where the original armstrong frederick model has been studied