Little can be said that applies generally to nonlinear systems. We have access mainly to local results, such as the basic existence and uniqueness theorem, which establishes the possibility of rectifying vector fields. The conditions for the theorem exclude the neighbourhoods of equilibria, exactly the regions approximated by linear fields in chapter 1. This omission is filled in by the Hartman–Grobman theorem, which guarantees the existence of a coordinate transformation that linearizes a neighbourhood of generic points of equilibrium.

There follows a discussion of Poincaré sections. In principle they reduce the study of motion near a periodic orbit to the iterations of a map in the neighbourhood of a fixed point. The linear approximation near a fixed point was shown in chapter 1 to be equivalent to a linear flow. The nontrivial extension of the Hartman–Grobman theorem to maps thus provides information on the motion surrounding an important class of periodic orbits.

Poincaré sections reduce Hamiltonian systems to a map in a phase space with two fewer dimensions. That the map preserves symplectic area (just the area, in the case of the plane) is a consequence of the Poincaré–Cartan theorem, here derived from the variational principle. We will discuss the reduction of two dimensions for a Hamiltonian system, by making it periodic in time. In particular we will see how specific choices of coordinates may preserve some of the symmetry properties of the full system.

The motion along thin tori surrounding stable periodic orbits in the Birkhoff approximation was studied in section 1.3. The Poincaré maps on such tori are translations of the circle (rotations). According to the analysis in section 4.4, the periodic Birkhoff tori and their neighbours are broken up by the nearly resonant terms of the Hamiltonian, which generate small denominators in the normal-form transformation. Even so, the survival of nonperiodic tori is not excluded, as will be confirmed by the theorem of Kolmogorov, Arnold and Moser in chapter 6. Understanding the motion near stable periodic orbits thus requires a preliminary study of general motion on tori, or their Poincaré sections – maps on the circle.

There is a didactic incentive that perhaps even outweighs the abovementioned physical motivation for the study of circle maps. In the context of these maps, many of the mathematical difficulties that beset the analysis of stable motion manifest themselves in their clearest, simplest form. We will analyse the effect of the near commensurability of frequencies, that is, rational rotation numbers. The attempt to reduce general maps to rotations leads once again to small denominators. Convergence in their presence is possible, but this result is so surprising that considerable effort will be spent in sketching proof.

The quantization of classically integrable systems relies explicitly on the invariant tori that foliate phase space. No distinction is made in the traditional semiclassical theory as to whether the motion on a quantized torus is periodic or only quasi-periodic. A strong perturbation may break up tori with incommensurate frequencies as well as those with periodic orbits. The only surviving smooth invariant manifolds, beyond the energy shell itself, will then be isolated periodic orbits. The semiclassical limit of stationary states must therefore bear some relation to these invariant curves. The uncertainty principle prevents us from attempting to tighten too closely the association between a given orbit and a specific state. In the last resort a state has to rely on a collection of periodic orbits or at least on a periodic orbit dressed with the local motion around it.

The basic instrument for evoking the contribution of periodic orbits is the semiclassical propagator for points that return to their initial positions. The classical orbits determining this propagator are dominated by the periodic orbits, in a way that is explained in section 9.1. On the other hand, the propagator can be represented as a sum over the intensities of the stationary wave functions. A local analysis establishes that individual wave intensities may exhibit strong ‘scars’ along some periodic orbits. Taking the trace of the propagator, we obtain a formula for the density of states as a sum over all the periodic orbits.

The discovery of chaotic behaviour in deterministic dynamical systems has had a profound effect in many areas of physics. There is now a large literature on this subject, which includes some important surveys. However, a physicist just entering this beautiful field or an advanced graduate student still finds the need for a concise introduction to the basic concepts in unsophisticated mathematical language. What are the essential distinctions between Hamiltonian (conservative) systems and dissipative systems? In what way will the presence of chaos in the classical limit affect a system appropriately described by quantum mechanics?

There is much to be gained by studying the theory of Hamiltonian systems against the background of general systems, in order to emphasize both contrasts and similarities. Liouville's theorem provides a remarkable distinguishing feature of Hamiltonian systems. The preservation of volume in phase space prevents the asymptotic collapse of the motion onto equilibria, periodic orbits, or ‘strange attractors’. Even though period-doubling cascades do occur in Hamiltonian systems, the loss of stability of a periodic orbit is only a local occurrence, rather than the-apocalyptic event that can subvert an entire dissipative system. In contrast the motion generated by a given Hamiltonian may exhibit diverse chaotic and regular orbits interwoven into rich structures.

In the previous chapter we attained global understanding of a very simple kind of system. We now return to full Hamiltonian systems, but these will be restricted in such a way as to allow a global knowledge of their main characteristics. This class of integrable systems contains all the solved problems in classical mechanics, as well as the truncations of the Birkhoff normal forms studied in chapter 4. We will find that no limitation need be made on the number of freedoms of an integrable system. In fact, their definition can be extended up to an infinite number of freedoms, leading to the solution of important partial differential equations, but this subject lies beyond the scope of this book.

We start by defining integrable systems and by studying the geometry of their invariant surfaces. This leads to the definition of general action-angle variables. The consideration of a few simple examples elicits the concept of caustics, that is, singularities of the projections of invariant surfaces. We will discuss briefly Thorn and Arnold's classification of some of the simpler generic caustics.

In conclusion we will study perturbations of integrable systems. In section 6.5 we will discuss the averaging principle, which is then related to a stationary perturbation theory, reminiscent of the resonant normal forms encountered in section 4.4. This evokes the question of survival of the invariant surfaces of an integrable system after a perturbation, to be answered by the Kolmogorov, Arnold and Moser (KAM theorem).

The remarkable self-consistency of the semiclassical theory of integrable systems prevents its natural generalization in the absence of invariant tori. The two ways we may attempt to tackle the problem are by studying the quantization of quasi-integrable systems or, at the other extreme, by plunging straight into ergodic systems. The first option has so far been confined to the quantization of the nearest integrable system that can be found. If the perturbations from integrability are sufficiently weak, this procedure can be justified by the analysis presented in section 9.4.

The present chapter starts with the investigation of the Berry–Voros hypothesis that the classical limit of the Wigner function for an ergodic system is uniform on the energy shell. This leads to some interesting consequences for the smoothed probability density and spatial correlation function. The classical limit of the Wigner function can be refined through multiple iterations of the pure-state condition. In section 8.2 it is shown that the torus Wigner function satisfies the pure-state condition, whereas the energy shell Wigner function is brought into a form that depends on individual orbits. A very important feature is that for an ergodic system we can identify the Wigner caustic with the energy shell itself – the semiclassical Wigner function oscillates inside and decays outside the shell.

The flow in an autonomous Hamiltonian system with one freedom is even more restricted than the generic motion categorized by Peixoto's theorem. The constancy of the Hamiltonian holds all the orbits to its level curves. For a bound system most of these will be closed loops, the exception being the levels of saddle points, that is, unstable equilibria of the Hamiltonian. The corresponding orbits either start out at an unstable point as t → - ∞ and return to the same point as t → + ∞, or go on to another unstable point. These two cases, known respectively as homoclinic and heteroclinic orbits, are shown in figs. 3.1a and b [under the potential curves V(q) that generate them, if the Hamiltonian has the form p2 + V(q)]. The homoclinic orbit of the pendulum (fig. 3.1b) is known as the separatrix between the small librations of the system and the full rotations arising at higher energies.

Consider making a small perturbation to this system, that is, coupling to another degree of freedom or, equivalently, adding an oscillatory forcing term to the Hamiltonian. Far from the unstable equilibrium, an orbit close to the separatrix will feel mainly the unperturbed Hamiltonian. But this gives a zero force near the saddle, so the perturbation becomes dominant near the unstable point: It can switch librations into rotations and back again. Successive switches will be uncorrelated, because the periods of rotations and librations lying close to the separatrix vary infinitely.

Having established the ubiquity of periodic orbits in dynamical systems, we now return to the study of the motion near a given periodic orbit, fixed point or point of equilibrium. The Hartman–Grobman theorem of section 2.2 guarantees the existence of a continuous coordinate transformation that linearizes the vector field near a hyperbolic fixed point, but no indication is given as to how to construct this transformation. The method of normal forms, invented by Poincaré, consists of eliminating nonlinear terms of the vector field by successive polynomial transformations. If this process can be carried out to all orders, the resulting compound transformation can be shown to be convergent in some cases, and an analytic reduction of the nonlinear vector field to a linear one is thus achieved. This transformation can be approximated to arbitrary accuracy.

One of the cases in which this process can never be carried out is that of Hamiltonian systems. The Hamiltonian cannot generally be made quadratic by a canonical transformation, though Birkhoff showed that it can be simplified into a form that shares some of the important features of quadratic Hamiltonians. For hyperbolic points this transformation is analytic in a narrow neighbourhood of the separatrices, allowing us to calculate precisely some homoclinic orbits and the periodic orbit families that accumulate on them.