1,720,997 research outputs found
On a class of integrable time-dependent dynamical systems
No full textWe present some integrable time-dependent systems of classical dynamics, and we apply the results to the equation , with a positive nondecreasing differentiable function; some of the results are extended to the nonlinear
case. Moreover we investigate the conditions for the solutions to be bounded and we study their asymptotic behaviour
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 textPeriodic attractors for the varactor equation
No full textWe consider a class of differential equations, x + yx + x(2p) = f(wt), with p is an element of N and omega is an element of R-d, describing one-dimensional dissipative systems subject to a periodic forcing. For p = 1 the equation describes a resistor-inductor-varactor circuit, hence the name 'varactor equation'. We concentrate on the limit cycle described by the trajectory with the same period as the forcing; numerically, for g large enough, it appears to attract all trajectories which remain bounded in phase space. We find estimates for the basin of attraction of this limit cycle, which are good for large values of g. Also, we show that the results extend to the case of quasi-periodic forcing, provided the frequency vector omega satisfies a Diophantine condition - for instance, the Bryuno or the standard Diophantine condition
On 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 phenomena and attractive periodic solutions in the saturating inductor circuit
No full textIn this paper, we investigate bifurcation phenomena, such as those of the periodic solutions, for the 'unperturbed' nonlinear system G(x)x+beta x=0, with G(x) = (alpha +x(2))/ (1 + x(2)) and alpha > 1, beta > 0, when we add the two competing terms -f(t) + gamma x, with f(t) a time-periodic analytic 'forcing' function and gamma > 0 the dissipative parameter. The resulting differential equation G(x)x+beta x+yx-f(t) = 0 describes approximately an electronic system known as the saturating inductor circuit. For any periodic orbit of the unperturbed system, we provide conditions which give rise to the appearance of subharmonic solutions. Furthermore, we show that other bifurcation phenomena arise as there is a periodic solution with the same period as the forcing function f (t) which branches off the origin when the perturbation is switched on. We also show that such a solution, which encircles the origin, attracts the entire phase space when the dissipative parameter becomes large enough. We then compute numerically the basins of attraction of the attractive periodic solutions by choosing specific examples of the forcing function f (t), which are dictated by experience. We provide evidence showing that all the dynamics of the saturating inductor circuit is organized by the persistent subharmonic solutions and by the periodic solution around the origin
The effects of time-dependent dissipation on the basins of attraction for the pendulum with oscillating support
No full textWe consider a pendulum with vertically oscillating support and time-dependent damping coefficient which varies until reaching a finite final value. Although it is the final value which determines which attractors eventually exist, the sizes of the corresponding basins of attraction are found to depend strongly on the full evolution of the dissipation. In particular, we investigate numerically how dissipation monotonically varying in time changes the sizes of the basins of attraction. It turns out that, in order to predict the behaviour of the system, it is essential to understand how the sizes of the basins of attraction for constant dissipation depend on the damping coefficient. For values of the parameters where the systems can be considered as a perturbation of the simple pendulum, which is integrable, we characterise analytically the conditions under which the attractors exist and study numerically how the sizes of their basins of attraction depend on the damping coefficient. Away from the perturbation regime, a numerical study of the attractors and the corresponding basins of attraction for different constant values of the damping coefficient produces a much more involved scenario: changing the magnitude of the dissipation causes some attractors to disappear either leaving no trace or producing new attractors by bifurcation, such as period doubling and saddle-node bifurcation. Finally, we pass to the case of an initially non-constant damping coefficient, both increasing and decreasing to some finite final value, and we numerically observe the resulting effects on the sizes of the basins of attraction: when the damping coefficient varies slowly from a finite initial value to a different final value, without changing the set of attractors, the slower the variation the closer the sizes of the basins of attraction are to those they have for constant damping coefficient fixed at the initial value. Furthermore, if during the variation of the damping coefficient attractors appear or disappear, remarkable additional phenomena may occur. For instance, it can happen that, in the limit of very large variation time, a fixed point asymptotically attracts the entire phase space, up to a zero-measure set, even though no attractor with such a property exists for any value of the damping coefficient between the extreme values
The high-order Euler method and the spin-orbit model
No full textWe present an algorithm for the rapid numerical integration of smooth, time-periodic differential equations with small nonlinearity, particularly suited to problems with small dissipation. The emphasis is on speed without compromising accuracy and we envisage applications in problems where integration over long time scales is required; for instance, orbit probability estimation via Monte Carlo simulation. We demonstrate the effectiveness of our algorithm by applying it to the spin-orbit problem, for which we have derived analytical results for comparison with those that we obtain numerically. Among other tests, we carry out a careful comparison of our numerical results with the analytically predicted set of periodic orbits that exists for given parameters. Further tests concern the long-term behaviour of solutions moving towards the quasi-periodic attractor, and capture probabilities for the periodic attractors computed from the formula of Goldreich and Peale. We implement the algorithm in standard double precision arithmetic and show that this is adequate to obtain an
excellent measure of agreement between analytical predictions and the proposed fast algorithm
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
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