Michele V. Bartuccelli

On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation

Abstract We present analytical methods which predict the occurrence of both soft (weak) and hard (strong) turbulence in the complex Ginzburg-Landau (CGL) equation: A t =RA+(1+iν)δA−(1+iμ)A¦A¦ 2 on a periodic domain [0,1] D in D spatial dimensions. Hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales. This form of hard turbulence appears to occur not in 1D but only in 2D and 3D in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the μ-ν planes where the system is modulationally unstable (ϵ=1+μν 2n: F n =∫(¦∇ n−1 A¦ 2 +α n ¦A¦ 2n )dx , for αn > 0. For large times and large R, upper bounds exist for the infinite set of Fns, constructed from the hierarchy of differential inequalities Fn≤(2nR+cn‖A‖2∞)Fn−bnF2n/Fn−1, for cn, bn > 0 (F0≡1). Estimates for the “bottom rung” F2 give upper bounds for the whole ladder. Long time upper bounds on F2 and ‖A‖2∞ (and hence all Fn) are well controlled in the soft region but become much larger in the hard region, whereas spati al and temporal averages remain comparatively small. When the nonlinearity is A¦A¦2q, the critical case qD=2 gives parallel results.

Length scales in solutions of the complex Ginzburg-Landau equation

Abstract We generalise and in certain cases improve on previous a priori estimates of Sobolev norms of solutions to the generalised complex Ginzburg-Landau equation. A set of dynamic length scales based on ratios of these norms is defined. We are able to derive lower bounds for time averages and long-time limits of these length scales. The bounds scale like the inverses of our L∞ bounds.

On the dynamics of a vertically driven damped planar pendulum

Results on the dynamics of the planar pendulum with parametric vertical timeperiodic 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 Poincaréesections. 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.

Length scales in solutions of the Navier-Stokes equations

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 a natural definition of a set of length scales. The authors discuss what happens to these scales if intermittent fluctuations in the vorticity field occur, and they consider how these scales compare to those derived from the attractor dimension and the number of determining modes. Their methods are based on estimates of ratios of norms which appear to play a natural role and which make many of the calculations comparatively easy. In 3d they cannot preclude length scales which are significantly shorter than the Kolmogorov length. In 2d their estimate for a length scale l turns out to be (l/L)-2<or=cG(1+logG)1/2 where G is the Grashof number. This estimate of l is shorter than that derived from the attractor dimension. The reason for this is discussed in detail.

Summation of divergent series and Borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms

We consider a class of second order ordinary differential equations describing one-dimensional systems with a quasiperiodic analytic forcing term and in the presence of damping. As a physical application one can think of a resistor–inductor–varactor circuit with a periodic (or quasiperiodic) forcing function, even if the range of applicability of the theory is much wider. In the limit of large damping we look for quasiperiodic solutions which have the same frequency vector of the forcing term, and we study their analyticity properties in the inverse of the damping coefficient. We find that even the case of periodic forcing terms is nontrivial, as the solution is not analytic in a neighborhood of the origin: it turns out to be Borel summable. In the case of quasiperiodic forcing terms we need renormalization group techniques in order to control the small divisors arising in the perturbation series. We show the existence of a summation criterion of the series in this case also; however, this cannot be interpreted as Borel summability.

Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems

We consider a class of ordinary differential equations describing one-dimensional analytic systems with a quasiperiodic forcing term and in the presence of damping. In the limit of large damping, under some generic nondegeneracy condition on the force, there are quasiperiodic solutions which have the same frequency vector as the forcing term. We prove that such solutions are Borel summable at the origin when the frequency vector is either any one-dimensional number or a two-dimensional vector such that the ratio of its components is an irrational number of constant type. In the first case the proof given simplifies that provided in a previous work of ours. We also show that in any dimension d, for the existence of a quasiperiodic solution with the same frequency vector as the forcing term, the standard Diophantine condition can be weakened into the Bryuno condition. In all cases, under a suitable positivity condition, the quasiperiodic solution is proved to describe a local attractor.

BIFURCATION CURVES OF SUBHARMONIC SOLUTIONS AND MELNIKOV THEORY UNDER DEGENERACIES

We 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.

Ideal orbits of toral automorphisms

Abstract Ideal orbits of toral automorphisms are the simplest (and arguably the most interesting) among the periodic orbits of fully chaotic maps. We show that for an ideal orbit the coordinate and momentum evolve independently, and that their dynamical evolution is determined by multiplication by a fixed integer modulo a prime. Their spectrum is found to be the product of a regular and an irregular function. The former is computed explicitly, while the latter is a sum of Dirichlet characters, whose irregular nature is the source of spectral fluctuations. We evaluate some statistical properties of these sums, and estimate the average spectral density of the phase coordinates. We show how to select toral automorphisms and initial conditions corresponding to ideal orbits of arbitrarily large period having certain prescribed properties. These results may be relevant to the problem of generating pseudorandom sequences.

LINDSTEDT SERIES FOR PERTURBATIONS OF ISOCHRONOUS SYSTEMS: A REVIEW OF THE GENERAL THEORY

We give a proof of the persistence of invariant tori for analytic perturbations of isochronous systems by using the Lindstedt series expansion for the solutions of the equations of motion. With respect to the case of anisochronous systems, there is the additional problem of finding the set of allowed rotation vectors, because they cannot be given a priori simply by looking at the unperturbed system. By considering the involved parameters (size of the perturbation, rotation vector and average action of a persisting invariant torus) as independent parameters we can introduce a function which is analytic in such parameters and only when the latter satisfy some constraint it becomes a solution: this can be regarded as a sort of singular implicit function problem. Therefore, although the dependence of the parameters, hence of the solution, upon the size of the perturbation is not smooth, in this way we construct explicitly the solution by using an absolutely convergent power series.

Frequency locking in an injection-locked frequency divider equation

We 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 devils 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.