Introduction to the perturbation theory of hamiltonian systems download

Please choose whether or not you want other users to be able to see on your profile that this library is a favorite of yours. Finding libraries that hold this item From the reviews:"This Springer monograph, based on lectures given by the first author at Moscow State University The style is concise and precise, and the book is suitable for graduate students and researchers.

Proofs are usually complete and, if not, references are given. In conclusion, the book constitutes a precious addition to the literature concerning the dynamics of perturbation theory of Hamiltonian systems. Also, special methods are used: asymptotical formulas describing quantitatively stochastic layers; averaging procedures. In conclusion, the book will be a very good reference for beginners. The appendix on diophantine properties, resonance, etc. Most results are given with complete proofs, so that the book may be of good service to researchers and graduate students with interest in mechanics.

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In this way the geometry of the integrable bundle can be carried over to the nearly-integrable one. The classical example of a Liouville integrable system with non-trivial monodromy [62, 58] is the spherical pendulum, which we now briefly revisit. From [24, ] it follows that the non-trival monodromy can be extended in the perturbed case. Also compare [41] and many of its references. For fully resonant tori the phenomenon of frequency locking leads to the destruction of the torus under sufficiently rich perturbations, and other resonant tori disintegrate as well.

In case of a single resonance between otherwise Diophantine frequencies the perturbation leads to quasi-periodic bifurcations, cf. It concerns all trajectories and states that they stay close to the unperturbed tori for long times that are exponential in the inverse of the perturbation strength. For trajectories starting close to surviving tori the diffusion is even superexponentially slow, cf.

Solenoids, which cannot be present in integrable systems, are constructed for generic Hamiltonian systems in [15, 97, ], yielding the simultaneous existence of representatives of all homeomorphy-classes of solenoids. Hyperbolic tori form the core of a construction proposed in [5] of trajectories that venture off to distant points of the phase space.

In the unperturbed system the union of a family of hyperbolic tori, parametrised by the actions conjugate to the toral angles, form a normally hyperbolic manifold. The latter is persistent under perturbations, cf.

The main difference between integrable and non-integrable systems already occurs for periodic orbits. For integrable systems these form together a pinched torus, but under generic perturbations the stable and unstable manifold of a hyperbolic periodic orbit intersect transversely.

In two degrees of freedom normalization leads to approximations that are integrable to all orders, which implies that the Melnikov integral is a flat function. In the real analytic case the Melnikov criterion is still decisive in many examples [66]. Genericity conditions are traditionally formulated in the universe of smooth vector fields, and this makes the whole class of analytic vector fields appear to be non-generic.

In this respect it is interesting that the generic properties may also be formulated in the universe of analytic vector fields, see [46] for more details. This still allows for a global understanding of a substantial part of the dynamics, but also leads to additional questions. Separatrices splitting yields the dividing surfaces in the sense of Wiggins et al. When passing through resonances 12 and 13 the lower-dimensional tori lose ellipticity and aquire hyperbolic Floquet exponents.

See [29] for a thorough treatment of the ensuing possibilities. The restriction to a single normal—internal resonance is dictated by our present possibil- ities. Indeed, already the bifurcation of equilibria with a fourfold zero eigenvalue leads to unfoldings that simultaneously contain all possible normal resonances. Thus, a satis- factory study of such tori which already may form one—parameter families in integrable Hamiltonian systems with five degrees of freedom has to await further progress in local bifurcation theory.

Similar remarks go for the onset of turbulence in fluid dynamics. Around this led to the scenario of Hopf-Landau-Lifschitz [78, 86, 87], which roughly amounts to the following. The first transition is a Hopf bifurcation [77, 67, 85], where a periodic solution branches off.

In a second transition of similar nature a quasi-periodic 2- torus branches off, then a quasi-periodic 3-torus, etc. The idea is that the motion picks up more and more frequencies and thus obtains an increasingly complicated power spectrum. By the quasi-periodic bifurcation theory [37, 36, 41] as sketched below these two approaches are unified in a generic way, keeping track of measure theoretic aspects.

For general background in dynamical systems theory we refer to [49, 81]. This transition consists of an infinite sequence of period doubling bifurcations ending up in chaos; it has several universal as- pects and occurs persistently in families of dynamical systems. In many of these cases also homoclinic bifurcations show up, where sometimes the transition to chaos is immediate when parameters cross a certain boundary, for general theory see [13, 14, 25, ]. The lower part of the figure corresponds to hyperbolic tori and the upper part to elliptic ones.

See the text for further interpretations. Quasi-periodic bifurcation theory concerns the extension of these bifurcations to invariant tori in nearly-integrable systems, e. Broadly speaking one could say that in these cases the Preparation The- orem [] is partly replaced by kam Theory. This results in the existence of two invariant 2-tori, one elliptic and the other hyperbolic. As before, cf.

The gaps at the border furthermore lead to the phenomenon of parabolic resonance, cf. Similar programs exist for all cuspoid and umbilic catastrophes [30, 31, 69] as well as for the Hamiltonian Hopf bifurcation [27, 28].

For applications of this approach see [29]. For a reversible analogue see [22]. As so often within the gaps generically there is an infinite regress of smaller gaps [11, 29]. For theoretical background we refer to [, 37, 33], for more references also see [41]. Given the standard bifurcations of equilibria and periodic orbits, we get more complex situations when invari- ant tori are involved as well.

The simplest examples are the quasi-periodic saddle-node and quasi-periodic period doubling [37] also see [36, 41]. To illustrate the whole approach let us start from the Hopf bifurcation of an equilibrium point of a vector field [77, 67, , 85] where a hyperbolic point attractor loses stability and branches off a periodic solution, cf. Figure 7 shows an amplitude response diagram often called bifurcation diagram. Let us briefly consider the Hopf bifurcation for fixed points of diffeomorphisms.

Here, due to the invariance of the rotation numbers of the invariant circles, no topological stability can be obtained []. Still this bifurcation can be characterized by many persistent properties. If the diffeomorphism is the return map of a periodic orbit for flows, this bifurcation produces an invariant 2-torus. Quasi-periodic versions exist of the saddle-node, the period doubling and the Hopf bi- furcation. We shall illustrate our results on the latter of these cases, compare with [36, 20].

For earlier results in this direction see [53]. The present interest is with small non-integrable perturbations of such integrable models. We now discuss the quasi-periodic Hopf bifurcation [16, 37], largely following [57]. Observe that if the nonlinearity g satisfies the well-known Hopf nondegeneracy conditions, e.

The story runs much like before. The contact between the disc boundaries and H is infinitely flat [16, 37]. For earlier results in the same spirit in a case study of the quasi-periodic saddle-node bifurcation see [50, 51, 52], also compare with [11].

As said earlier, period doubling sequences and homoclinic bifurcations may accompany this. As an example consider a family of maps that undergoes a generic quasi-periodic Hopf bifurcation from circle to 2-torus. It turns out that here the Cantorized fold of Figure 6 is relevant, where now the vertical coordinate is a bifurcation parameter.

A fattening process as explained above, also can be carried out here. This is a tough problem as can already be seen when considering 2-dimensional diffeomorphisms. In higher dimension this problem is even harder to handle, e. In the conservative case a related problem concerns a better understanding of Arnold diffusion.

Somewhat related to this is the analysis of dynamical systems without an explicit per- turbation setting. Here numerical and symbolic tools are expected to become useful to develop computer assisted proofs in extended perturbation settings, diagrams of Lyapunov exponents, symbolic dynamcics, etc. Compare with []. Also see [44, 45] for applications and further reference. Regarding nearly-integrable Hamiltonian systems, several problems are in order. Contin- uing the above line of thought, one interest is the development of Hamiltonian bifurcation theory without integrable normal form and, likewise, of kam theory without action an- gle coordinates [90].

A first step has been made in [89] where internally resonant parabolic tori involved in a quasi-periodic Hamiltonian pitchfork bifurcation are consid- ered. The resulting large dynamical instabilities may be further amplified for tangent or flat parabolic resonances, which fail to satisfy the iso-energetic non-degeneracy condition. The construction of solenoids in [15, 97] uses elliptic periodic orbits as starting points, the simplest example being the result of a period-doubling sequence.

In this way one might be able to construct solenoid-type invariant sets that are limits of tori with varying dimension. Concerning the global theory of nearly-integrable torus bundles [24], it is of interest to understand the effects of quasi-periodic bifurcations on the geometry and its invariants.

Also it is of interest to extend the results of [] when passing to semi-classical approxi- mations. References [1] R. Abraham and J. Marsden, Foundations of Mechanics, 2nd ed. Arnold, On the classical perturbation theory and the stability problem of the planetary system, Dokl. Nauk SSSR , Arnold, Proof of a theorem by A. Kolmogorov on the persistence of conditionally periodic motions under a small change of the Hamilton function, Russian Math. Surveys 18 5 , English; Russian original.

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