This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.

Regular and Chaotic Dynamics

THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

We characterize the polynomial decay of orbits of Hilbert space C
 0-semigroups in resolvent terms. We also show that results of the same type for general Banach space semigroups and functions obtained recently in Batty and Duyckaerts (J Evol Eq 8:765–780, 2008) are sharp. This settles a conjecture posed in Batty and Duyckaerts (2008).

Optimal polynomial decay of functions and operator semigroups

Foundations Of Time Frequency Analysis

We characterize the polynomial decay of orbits of Hilbert space $C_0$-semigroups in resolvent terms. We also show that results of the same type for general Banach space semigroups and functions obtained recently in the paper by C.J.K.Batty and T.Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces (J. Evol. Eq. 2008), are sharp. This settles a conjecture posed in the paper by C.J.K.Batty and T.Duyckaerts.

We consider the wave equation with a damping term on a partially rec\-tan\-gular planar domain, assuming that the damping is concentrated close to the non-rectangular part of the domain. Polynomial decay estimates for the ener\-gy of the solution are established.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2007/0014/0001/MRL-2007-0014-0001-a003.pdf

Energy decay for damped wave equations on partially rectangular domains

We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space intrinsically associated to their Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity. We also describe the large time behavior of contraction semigroups generated by these operators.

/pdf/spectra-and-semigroup-smoothing-for-non-elliptic-quadratic-3dwqfemegu.pdf

Spectra and semigroup smoothing for non-elliptic quadratic operators

For the class of super-exponentially decaying potentials on the real line sharp upper bounds on the counting function of the poles in discs are derived and the density of the poles in strips is estimated. In the case of nonnegative potentials, explicit estimates for the width of a pole-free strip are obtained.

/pdf/bounds-on-scattering-poles-in-one-dimension-275jes59q7.pdf

Bounds on Scattering Poles in One Dimension

This is the first in a series of works devoted to small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength $ \epsilon $ of the perturbation is $ \gg h $ (or sometimes only $ \gg h^2 $ ) and bounded from above by $ h^\delta $ for some $ \delta > 0 $ . We get a complete asymptotic description of all eigenvalues in certain rectangles $ [-1/C, 1/C] + i\epsilon[F_{0} - 1/C, F_{0} + 1/C] $ .

/pdf/non-selfadjoint-perturbations-of-selfadjoint-operators-in-2-qszleq0as3.pdf

Non-Selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I

This is the first in a series of works devoted to small non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength $\epsilon$ of the perturbation is $\gg h$ (or sometimes only $\gg h^2$) and bounded from above by $h^{\delta}$ for some $\delta>0$. We get a complete asymptotic description of all eigenvalues in certain rectangles $[-1/C, 1/C]+ i\epsilon [F_0-1/C,F_0+1/C]$.

Michael Hitrik

Papers

Energy decay for damped wave equations on partially rectangular domains

Spectra and semigroup smoothing for non-elliptic quadratic operators

Bounds on Scattering Poles in One Dimension

Non-Selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I

Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions I