# Unstable Dynamics in Hamiltonian Systems

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

The key challenge of the modern theory of Hamiltonian Dynamical Systems is to provide adequate mathematical tools for describing chaotic dynamics in a system which combines both regular and chaotic components as an overwhelming majority of models relevant for applications of the theory fell into this category. In this area two major problems have resisted the efforts of scientists for decades. The first one is called "Arnold Diffusion" and is related to instability of action variables on long time scales. The second one is known as "Positive Metric Entropy Conjecture" and states that chaotic motions are physically relevant, i.e., they occupy a subset of positive Lebesgue measure.

In the period of the fellowship I will concentrate on the study of Hamiltonian systems with multiple time scales, as this property is often present in the equations either explicitly or implicitly. The aim of this study is to develop mathematical tools for studying instabilities of dynamics. Preliminary results show that we are able to prove existence of normally hyperbolic invariant objects with different dynamical behaviour of slow components. It is probable that on longer time scales the restriction of the dynamical system on this family can be approximated by a stochastic ordinary differential equation in the slow variables. If confirmed, it will establish an important connection between two different fields of Mathematics: the theory of deterministic Hamiltonian systems and stochastic differential equations, which are considered mostly unrelated at the present.

An extension of these results should provide a new insight on the theory of the Fermi acceleration.

A comparison with variational approach to Arnold Diffusion announced by Mather (Princeton) suggests that our mechanism could be used to solve the long-standing problem of genericity of Arnold Diffusion in near-integrable Hamiltonian systems. The progress in this direction should require radical improvements of methods for detection of transversal homoclinic trajectories associated with various invariant objects, i.e., in the area where I have a substantial technical expertise.

As a summary, the following list of technical topics will be addressed initially: exponentially small splitting of invariant manifolds in higher dimension, stochastic description of slow dynamics in slow-fast systems with chaotic fast component, Fermi acceleration, Arnold Diffusion, positive metric entropy conjecture.

I expect that as the fellowship advances new research directions will arise partially motivated by the development of the theory and partially by questions coming from its applications.

The research will be curried out at the Mathematics Institute, University of Warwick, and will involve collaboration with several groups in the UK and oversees.

In the period of the fellowship I will concentrate on the study of Hamiltonian systems with multiple time scales, as this property is often present in the equations either explicitly or implicitly. The aim of this study is to develop mathematical tools for studying instabilities of dynamics. Preliminary results show that we are able to prove existence of normally hyperbolic invariant objects with different dynamical behaviour of slow components. It is probable that on longer time scales the restriction of the dynamical system on this family can be approximated by a stochastic ordinary differential equation in the slow variables. If confirmed, it will establish an important connection between two different fields of Mathematics: the theory of deterministic Hamiltonian systems and stochastic differential equations, which are considered mostly unrelated at the present.

An extension of these results should provide a new insight on the theory of the Fermi acceleration.

A comparison with variational approach to Arnold Diffusion announced by Mather (Princeton) suggests that our mechanism could be used to solve the long-standing problem of genericity of Arnold Diffusion in near-integrable Hamiltonian systems. The progress in this direction should require radical improvements of methods for detection of transversal homoclinic trajectories associated with various invariant objects, i.e., in the area where I have a substantial technical expertise.

As a summary, the following list of technical topics will be addressed initially: exponentially small splitting of invariant manifolds in higher dimension, stochastic description of slow dynamics in slow-fast systems with chaotic fast component, Fermi acceleration, Arnold Diffusion, positive metric entropy conjecture.

I expect that as the fellowship advances new research directions will arise partially motivated by the development of the theory and partially by questions coming from its applications.

The research will be curried out at the Mathematics Institute, University of Warwick, and will involve collaboration with several groups in the UK and oversees.

### Planned Impact

The primary beneficiaries will be other researchers in related areas. The results of the proposed research will potentially lead to better understanding of celestial mechanics, some processes in plasma physics, and other areas of mathematical and theoretical physics which use Hamiltonian equations for modelling, and also affect understanding of mathematical background of Statistical Physics. Although the main goal of the proposed research is related to understanding the fundamental principles and direct technical applications do not form a part of the proposals, it is feasible that, in longer run, the theory may have practical applications, e.g., it can be useful for development of devices capable of fine tuning a laser frequency or for conducting an analysis of dynamics of small bodies in the Solar System. The project will also contribute to development of the human resources by providing training to a PhD student and a young postdoctoral scientist in the advanced theory of Hamiltonian systems.

### Organisations

- University of Warwick (Fellow, Lead Research Organisation)
- Weizmann Institute of Science (Collaboration, Project Partner)
- University of Barcelona (Collaboration, Project Partner)
- IMPERIAL COLLEGE LONDON (Collaboration)
- University of Bristol (Project Partner)
- Imperial College London (Project Partner)

## People |
## ORCID iD |

Vasily Gelfreykh (Principal Investigator / Fellow) |

### Publications

Gelfreich V
(2012)

*Fermi acceleration and adiabatic invariants for non-autonomous billiards.*in Chaos (Woodbury, N.Y.)
Gelfreich V
(2014)

*Separatrix splitting at a Hamiltonian 02 i? bifurcation*in Regular and Chaotic Dynamics
Gelfreich V
(2014)

*Oscillating mushrooms: adiabatic theory for a non-ergodic system*in Journal of Physics A: Mathematical and Theoretical
Gelfreich V
(2018)

*Interpolating vector fields for near identity maps and averaging*in Nonlinearity
Gelfreich V
(2013)

*Dynamics of symplectic maps near a double resonance*in Physica D: Nonlinear Phenomena
Gelfreich V
(2014)

*Unique normal forms near a degenerate elliptic fixed point in two-parametric families of area-preserving maps*in Nonlinearity
Gelfreich V
(2017)

*Arnold Diffusion in A Priori Chaotic Symplectic Maps*in Communications in Mathematical Physics
Gelfreich V.
(2012)

*Splitting of separatrices near resonance periodic trajectories*in Analysis and Singularities
Gelfreich V.
(2014)

*Adiabatic theory for a non-ergodic system: an oscillating mushroom*in Proceedings of 8th European Nonlinear Dynamics
Gelfreich Vassily
(2014)

*Arnold Diffusion in a priory chaotic Hamiltonian systems*in arXiv e-printsDescription | We studied various mathematical models and developed new tools for studying unstable dynamics in Hamiltonian systems. One of the application of this theory is the study of energy growth in non-autonomous billiard systems, the process known as Fermi acceleration. We explored the influence of the billiard shape on the energy growth rate and established that it is much higher compare to the ergodic or integrable cases. We studied model systems for Arnold diffusion in order to improve the understanding of the underlying mathematical mechanisms of instability. We developed a theory for a drift in the phase space of a-priory chaotic Hamiltonian systems. We developed new analytical tools, including new normal form technics, and provided new technics for analysis and visualisation of the four dimensional dynamics. |

Exploitation Route | Our results are of interest to other researchers in the field of dynamical systems, applied mathematics and mathematical physics. For example, the Web of Science reports that our paper on Fermi acceleration (published in 2012) was cited 34 times in other research papers. Our work leads to development of new and improvement of existing mathematical tools. In particular, our simplified normal forms are used for other research purposes, as they represent an important technical tool which can be of help academics from various adjacent fields of research. |

Sectors | Other |

Description | The primary beneficiaries of this project are researchers in related areas. The results of our research have been made available to wider research community via publication of research papers, presentation at relevant conferences and seminars. The results of the project lead to better understanding of instabilities determined by Hamiltonian equations. In particular, we applied our methods to the study of Fermi acceleration and studied energy transfer from heavy slowly moving objects to fast and light ones. The impact of these results on other researchers is clearly visible through the citations of our paper by other researchers. We also applied our methods to testing mathematical backgrounds of Statistical Physics, the results being published in PNAS, a prestigious interdisciplinary research journal. The project also contributed to the development of the human resources by providing training to a PhD student and a young postdoctoral scientist in the advanced theory of Hamiltonian systems. The project also lead to creation of new visualisation technics which can be used to explain difficult mathematical concepts related to Hamiltonian dynamics in a 4 dimensional space. An animation which illustrates Arnold diffusion in a four-dimensional space is available from the PI web page at Warwick. |

Sector | Education |

Impact Types | Cultural |

Description | C. Simo, A. Vieiro, Arnold Diffusion in 4d symplectic maps |

Organisation | University of Barcelona |

Country | Spain |

Sector | Academic/University |

PI Contribution | Analytical and numerical studies of the dynamics of 4d symplectic maps near double resonances. |

Collaborator Contribution | Development of numerical algorithms and implementation of those in software. Development of analytical tools. |

Impact | Publication of a research paper (Physica D (2013)), presentation of the results at appropriate conferences and seminars. |

Start Year | 2011 |

Description | D. Turaev, Arnold Diffusion in Hamiltonian Systems |

Organisation | Imperial College London |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Developments of the theory of instability (Arnold Diffusion) in a-priori chaotic symplectic maps |

Collaborator Contribution | Developments of the theory of instability (Arnold Diffusion) in a-priori chaotic symplectic maps |

Impact | Preliminary version of our paper is available at arxiv (2014), results of the study were presented at appropriate conferences and seminars. |

Start Year | 2011 |

Description | V. Rom-Kedar, D. Turaev, K. Shah, Fermi acceleration in time-dependent billiards |

Organisation | Imperial College London |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Development of analytical and numerical tools for studying Fermi acceleration in non-autonomous billiard dynamical systems. |

Collaborator Contribution | Development of analytical tools for studying Fermi acceleration in non-autonomous billiard dynamical systems. |

Impact | publication of research papers (Chaos (2012), J.Phys A (2014)) and an extended conference abstract (ENOC 2014), presentation of the results of the research at appropriate conferences and seminars. |

Start Year | 2011 |

Description | V. Rom-Kedar, D. Turaev, K. Shah, Fermi acceleration in time-dependent billiards |

Organisation | Weizmann Institute of Science |

Country | Israel |

Sector | Academic/University |

PI Contribution | Development of analytical and numerical tools for studying Fermi acceleration in non-autonomous billiard dynamical systems. |

Collaborator Contribution | Development of analytical tools for studying Fermi acceleration in non-autonomous billiard dynamical systems. |

Impact | publication of research papers (Chaos (2012), J.Phys A (2014)) and an extended conference abstract (ENOC 2014), presentation of the results of the research at appropriate conferences and seminars. |

Start Year | 2011 |