The Melnikov function for prediction of Smale horseshoe chaos is applied to the rf-driven Josephson junction. Linear and quadratic damping resistors are considered. In latter case analytic solution including dc bias used obtain an improved threshold curve onset chaos. compared new computational solutions. technique provides a good, but slightly low, estimate threshold.

Results on the dynamics of planar pendulum with parametric vertical timeperiodic forcing are reviewed and extended. Numerical methods employed to study various dynamical features system about its equilibrium positions. Furthermore, far from points is systematically investigated by using phase portraits Poincaréesections. The attractors associated basins attraction computed. We also calculate Lyapunov exponents show that for some parameter values shows sensitivity initial conditions.

In this paper, we make a detailed study of the spin–orbit dynamics Mercury, as predicted by realistic model that has been recently introduced in series papers mainly Efroimsky and Makarov. We present numerical analytical results concerning nature librations Mercury's spin 3:2 resonance. The provide evidence are quasi-periodic time, consisting slow oscillation, with an amplitude order arcminutes, superimposed on 88-d libration. This contrasts recent astronomical observations hence suggests...

We consider a class of ordinary differential equations describing one-dimensional analytic systems with quasi-periodic forcing term and in the presence damping. In limit large damping, under some generic non-degeneracy condition on force, there are solutions which have same frequency vector as term. prove that such Borel summable at origin when is either any number or two-dimensional ratio its components an irrational constant type. first case proof given simplifies provided previous work...

We consider a class of second order ordinary differential equations describing one-dimensional systems with quasi-periodic analytic forcing term and in the presence damping. As physical application one can think resistor-inductor-varactor circuit periodic (or quasi-periodic) function, even if range applicability theory is much wider. In limit large damping we look for solutions which have same frequency vector term, study their analyticity properties inverse coefficient. find that already...

We present a comprehensive study of interpolation inequalities for periodic functions with zero mean, including the existence and asymptotic expansions extremals, best constants, various remainder terms, etc. Most attention is paid to critical (logarithmic) Sobolev inequality in two-dimensional case, although number results concerning constants algebraic case different space dimensions are also obtained.

A set of ladder inequalities for the 2d and 3d forced Navier-Stokes equations on a periodic domain (0, L)d is developed, leading to natural definition length scales. The authors discuss what happens these scales if intermittent fluctuations in vorticity field occur, they consider how compare those derived from attractor dimension number determining modes. Their methods are based estimates ratios norms which appear play role make many calculations comparatively easy. In cannot preclude...

We consider parametrically forced Hamiltonian systems with one-and-a-half degrees of freedom and study the stability dynamics when frequency forcing is relatively high or low. show that, provided sufficiently high, Kolmogorov-Arnold-Moser (KAM) theorem may be applied even amplitude far away from perturbation regime. A similar result obtained for low frequency, but in that case we need to not too large; however, are still able amplitudes which outside In addition, find numerically stable very...

We consider a model for resonant injection-locked frequency divider, and study analytically the locking onto rational multiples of driving frequency. provide explicit formulae width plateaux appearing in devil's staircase structure lockings, particular show that largest correspond to even integer values ratio signal output signal. Our results prove experimental numerical available literature.

We consider dissipative one-dimensional systems subject to a periodic force. As model system, particularly suited for numerical analysis, we investigate the driven cubic oscillator in presence of friction, and study numerically how time-varying friction affects dynamics. find that, if damping coefficient increases time up final constant value, then basins attraction leading resonances are larger than they would have been had fixed at that value since beginning. From quantitative point view,...

The differential equation with f ( t ) positive, periodic and continuous is studied. After describing some physical applications of this equation, we construct a variety invariant sets for it, thereby partitioning the phase plane into regions in which solutions grow without bound also those bounded exist.