1,720,982 research outputs found
Nonlinear perturbations of Hill's systems and the pendulum with variable length
No full textThe stability of the trivial solution to the nonlinear Hill equation has been extensively studied in the literature. In this paper, we provide an approach mainly based on the application of KAM theory to relate the stability of the nonlinear equation to the stability of the linearised equation. In addition, we extend the stability result to the case where a quasi-periodic perturbation is added to the periodic forcing of Hill's equation. We focus on the pendulum with variable length, both because of its physical interest and for the sake of concreteness, but the analysis may be extended to any nonlinear Hill system. Therefore, rather than looking for optimal estimates, which strongly depend on the considered system, we emphasise the general strategy for studying the stability of both the linearised equations and the full nonlinear equations
Lindstedt series for perturbations of isochronous systems: a review of the general theory
No full textOn the dynamics of a vertically driven damped planar pendulum
No full textResults on the dynamics of the planar pendulum with parametric vertical time-periodic forcing are reviewed and extended. Numerical methods are employed to study the various dynamical features of the system about its equilibrium positions.
Furthermore, the dynamics of the system far from its equilibrium points is systematically investigated by using phase portraits and Poincar\ ́e sections. The attractors and the associated basins of attraction are computed. We also calculate the Lyapunov exponents to show that for some parameter values the dynamics of the pendulum shows sensitivity to initial conditions
Bifurcation curves of subharmonic solutions and Melnikov theory under degeneracies
No full textWe study perturbations of a class of analytic two-dimensional autonomous systems with perturbations depending periodically on time; for instance one can imagine a periodically driven or forced system with one degree of freedom. In the first part of the paper, we revisit a problem considered by Chow and Hale on the existence of subharmonic solutions. In the analytic setting, under more general ( weaker) conditions on the perturbation, we prove their results on the bifurcation curves dividing the region of non-existence from the region of existence of subharmonic solutions. In particular our results apply also when one has degeneracy to the first order - i.e. when the subharmonic Melnikov function is identically constant. Moreover we can deal as well with the case in which degeneracy persists to arbitrarily high orders, in the sense that suitable generalizations to higher orders of the subharmonic Melnikov function are also identically constant. The bifurcation curves consist of four branches joining continuously at the origin, where each of them can have a singularity ( although generically they do not). The branches can form a cusp at the origin: we say in this case that the curves are degenerate as the corresponding tangent lines coincide. The method we use is completely different from that of Chow and Hale, and it is essentially based on the proof of convergence of the perturbation theory. It also allows us to treat the Melnikov theory in degenerate cases in which the subharmonic Melnikov function is either identically vanishing or has a zero which is not simple. This is investigated at length in the second part of the paper. When the subharmonic Melnikov function has a non-simple zero, we consider explicitly the case where there exist subharmonic solutions, which, although not analytic, still admit a convergent fractional series in the perturbation parameter
Explicit Estimates on the Torus for the Sup-norm and the Crest Factor of Solutions of the Modified Kuramoto–Sivashinky Equation in One and Two Space Dimensions
No full textWe consider the Modified Kuramoto–Sivashinky Equation (MKSE) in one and two space dimensions and we obtain explicit and accurate estimates of various Sobolev norms of the solutions. In particular, by using the sharp constants which appear in the functional interpolation inequalities used in the analysis of partial differential equations, we evaluate explicitly the sup-norm of the solutions of the MKSE. Furthermore we introduce and then compute the so-called crest factor associated with the above solutions. The crest factor provides information on the distortion of the solution away from its space average and therefore, if it is large, gives evidence of strong turbulence. Here we find that the time average of the crest factor scales like λ(2d-1)/8 for λ large, where λ is the bifurcation parameter of the source term and d= 1 , 2 is the space dimension. This shows that strong turbulence cannot be attained unless the bifurcation parameter is large enough
Globally and locally attractive solutions for quasi-periodically forced systems
No full textWe consider a class of differential equations, x + yx + g(x) = f(omega t), with omega is an element of R-d, describing one-dimensional dissipative systems subject to a periodic or quasi-periodic (Diophantine) forcing. We study existence and properties of trajectories with the same quasi-periodicity as the forcing. For g(x) = x(2p+1), p is an element of N, we show that, when the dissipation coefficient is large enough, there is only one such trajectory and that it describes a global attractor. In the case of more general nonlinearities, including g(x) = x(2) (describing the varactor equation), we find that there is at least one trajectory which describes a local attractor
Frequency locking in an injection-locked frequency divider equation
No full textWe consider a model for a resonant injection-locked frequency divider, and study analytically the locking onto rational multiples of the driving frequency. We provide explicit formulae for the width of the plateaux appearing in the devil’s staircase structure of the lockings, and in particular show that the largest plateaux correspond to even integer values for the ratio of the frequency of the driving signal to the frequency of the output signal. Our results prove the experimental and numerical results available in the literature
On the stability of the upside-down pendulum with damping
No full textA rigorous analysis is presented in order to show that, in the presence of friction, the upward equilibrium position of the vertically driven pendulum, with a small nonvanishing
damping term, becomes asymptotically stable when the period of the forcing is below an appropriate threshold value. As a by-product we obtain an analytic expression of the solution for initial data close enough to the equilibrium position
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