(DMAT, IMTech, CRM). Received 23 July, 2022.
The paper [7] (see also [8]) solves a version of the Arnold diffusion problem for Hamiltonian systems that remained open since the 1960s.
Hamiltonian systems appear naturally as models of many systems with negligible friction. We could mention, for instance, the models in Celestial Mechanics or the models for motion of charged particles in magnetic fields. Among Hamiltonian systems, integrable systems (systems with enough constants of motion) are characterized because all their orbits lie in maximal invariant tori. Therefore, the motion is stable.
One of the problems that appears naturally in the applications and which has attracted the attention for a long time is whether the effect of periodic small perturbations on integrable systems accumulate over time and lead to large effect (instability) or whether these effects average out (stability).
It is often the case that Hamiltonian systems exhibit both stable and unstable regimes. In fact, beginning in the 60s, there exist rigorous results proving that for the mentioned systems most of the invariant tori persist and therefore most of the trajectories are stable for all the time: this is the well known KAM Theorem (Kolmogorov-Arnol’d-Moser), or they are stable for very long times (Nekhoroshev).
The orbits that do not lie on KAM invariant tori and evolve over time scales where Nekhoroschev theory breaks up may possibly drift arbitrarily far. Indeed, in [1] Arnold conjectured that this is a phenomenon that happens in rather general systems. In that celebrated paper he constructed an example of a nearly integrable Hamiltonian system for which he proved the existence of trajectories that avoided the KAM tori and that performed long excursions.
The example provided by Arnold has the characteristic that the unperturbed system is integrable but presents hyperbolicity (the system has conserved quantities but the motion is not foliated by maximal invariant tori). In fact, it has a manifold foliated by whiskered tori whose stable and unstable manifolds coincide. When the perturbation is applied, the mechanism of diffusion is based on the existence of chains of whiskered tori such that the unstable manifold of one torus intersects the stable manifold of the next one (transition chains).
A characteristic of Arnold’s example is that the perturbation has been chosen carefully so that it does not affect the foliation of invariant tori present in the unperturbed system, but it causes the stable and unstable manifolds to intersect transversally. But general perturbations destroy the foliation of persisting whiskered KAM tori creating gaps of size bigger than the splitting of the stable and unstable manifolds. Therefore, the transition chains proposed by Arnold can not exist. This is known in the literature as the large gap problem.
Generalizations of Arnold’s model are the so-called a priori unstable systems and there has been a very active area of research to prove the existence of Arnold diffusion for such systems.
Overcoming the large gap problem and identifying new mechanisms for diffusion has become an important direction of study itself. This problem has attracted the attention of both mathematicians and physicists due to its practical importance and mathematical depth. Some works use variational methods to obtain the diffusing orbits in the case of convex Hamiltonians [9].
Other works use geometric methods and follow a different idea: to look at the Normally Hyperbolic Invariant Manifold (NHIM) formed by these whiskered tori. This object is preserved after the perturbation is applied and gives the opportunity for looking at other invariant objects that can play the role of the disappeared invariant tori in the transition chain: the so-called secondary tori, or tori of lower dimension [3], [5].
An important tool developed in [4] and used in lots of works was the use of the so-called scattering map: a map defined in the NHIM which labels the heteroclinic excusions. Understanding this map and finding perturbative formulas for it has been crucial to prove that objects of different dimension or topology can participate in the chains proposed by Arnold.
Using the scattering map several works obtained results on diffusion in models of Celestial mechanics, see for instance [2], [6].
The disadvantage of previous results is that they needed a deep knowledge of the so-called inner dynamics along the NHIM. The work [7] (and also [8]) follows a different approach. It presents a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. The approach relies on successive applications of the scattering map along homoclinic orbits to a normally hyperbolic invariant manifold without any knowledge of the invariant objects in the NHIM. We find pseudo-orbits of the scattering map that keep advancing in some privileged direction. Analogously, we find heteroclinic orbits between points in the NHIM. Then we use the recurrence property of the ‘inner dynamics’, restricted to the normally hyperbolic invariant manifold, to return to those pseudo-orbits. Finally, we apply topological methods to show the existence of true orbits that follow the successive applications of the two dynamics.
This method differs, in several crucial aspects, from earlier works. Unlike the well known ‘two-dynamics’ approach, the method we present relies on the outer dynamics by the scattering map alone. There are virtually no assumptions on the inner dynamics, as its invariant objects (e.g., primary and secondary tori, lower dimensional hyperbolic tori and their stable/unstable manifolds, Aubry-Mather sets) are not used at all. The method applies to unperturbed Hamiltonians of arbitrary degrees of freedom that are not necessarily convex (a strong hypothesis needed in the variational approach) avoiding several other technical assumptions needed in all the previous works. In addition, this mechanism is easy to verify (analytically or numerically) in concrete examples, as well as to establish diffusion in generic systems.
References
[1] V.I. Arnold. Instability of dynamical systems with several degrees of freedom. Sov. Math. Doklady 5:581–585, 1964.
[2] M. J. Capinski, M. Gidea. Arnold Diffusion, Quantitative Estimates, and Stochastic Behavior in the Three-Body Problem. Communications on Pure and Applied Mathematics, Vol. LXXIV, 0001–0064 (2021). DOI: 10.1002/cpa.22014
[3] A. Delshams, R. de la Llave, T.M. Seara. A Geometric Mechanism for Diffusion in Hamiltonian Systems Overcoming the large Gap Problem: Heuristics and Rigorous Verification on a Model. Mem. Amer. Math. Soc. 179(844): 1-141, 2006. DOI:
10.1090/memo/0844
[4] A. Delshams, R. de la Llave, T.M. Seara. Geometric properties of the scattering map of a normally hyperbolic invariant manifold. Adv. Math. 217(3): 1096-1153, 2008. DOI: 10.1016/j.aim.2007.08.014
[5] A. Delshams, R. de la Llave, T.M. Seara. Instability of high dimensional Hamiltonian Systems: Multiple resonances do not impede diffusion. Adv. Math. 294, 689-755, 2016.
DOI:10.1016/j.aim.2015.11.010
[6] A. Deslhams, V. Kaloshin, A. de la Rosa, T. M-Seara. Global Instability in the Restricted Planar Elliptic Three Body Problem. Communications in Mathematical Physics 366(3):1173-1228, 2019. DOI:10.1007/s00220-018-3248-z
[7] Marian Gidea, Rafael de la Llave, T. M-Seara. A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results. Communications on pure and applied mathematics 73 (1): 110-149, 2020. DOI: 10.1002/cpa.21856
[8] Marian Gidea, Rafael de la Llave, T. M-Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete and continuous dynamical systems 40(12): 6795-6813, 2020. DOI: 10.3934/dcds.2020166
[9] V. Kaloshin, K. Zhang. Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom. Annals of Mathematics Studies (AMS-208), 2020.