# B04Discretization as Perturbation: Qualitative and Quantitative Aspects

## Impact of Discretization for Integrable Hamiltonian and Nonholonomic Dynamics

Continuous time dynamical systems have to be discretized in order to find solutions numerically using a computer. This generally leads to a perturbation of the original dynamical behaviour. In particular, the long-term evolution might change drastically, e.g., regular behavior may turn into chaotic behavior. This raises the question of how to control the corresponding discrepancies. The research goal of B4 is to study this question for integrable and nonholonomic systems, both from quantitative and qualitative perspectives, exploiting the detour of embedding discrete dynamics into the dynamics of a non-autonomously perturbed continuous time system.

#### Mission-

Many processes in the real world may be described from the mathematical point of view as time continuous dynamical system. Unfortunately, their solutions are not always easy to find. In such cases, the application of numerical schemes, or in other words of discretization techniques, is useful to obtain at least approximate solutions and to gain insight into the qualitative and quantitative behavior of the actual dynamical system. When the original continuous dynamical system has particular geometric or structural features, it is a primary requirement that its
discretizations preserve those to a certain extend. However, it is in general impossible to achieve a complete correspondence, even if the discretization is performed in some kind of structure preserving way. This discrepancy can be rigorously established in some specific cases and is easily apprehensible in a more general perspective, since any practical discretization introduces a perturbation in the original time continuous dynamics. Thus, the question arises of how to control the discrepancies between the perturbed and unperturbed dynamics, in particular,
concerning the long-term behavior. Since this question is too general to admit a unified answer, the B4 project addresses it in two specific situations, namely symplectic integrators applied to integrable systems and nonholonomic integrators.

In particular, nonholonomic systems play a fundamental role in Mechanics. They include, for instance, rolling bodies on a surface which do not slide, such as wheeled vehicles or, as a very basic example, a rolling coin. Mathematically speaking, these systems are characterized by the so-called nonholonomic constraints: restrictions of the motion that involve the velocity variables. Moreover, as compared to holonomic systems, their equations of motion are not obtained by Hamilton’s but instead by the Lagrange-d’Alembert variational principle. The dynamical behavior of nonholonomic systems differs radically from holonomic ones. For instance, while the total energy is preserved as in the holonomic case, for nonholonomic systems with symmetry, the momentum is not necessarily preserved. Such facts make nonholonomic systems special so that the straightforward application of standard discretization procedures is not adequate. The same is true for the class of so-called integrable systems, where the special structure is due to the existence of sufficiently many conserved quantities.

The philosophy in research project B4 is to study discretizations of time continuous autonomous systems by a corresponding non-autonomous perturbation of those, i.e. by an appropriately modified time continuous system. This approach is not only used to control the discrepancies in the dynamical behavior, but also to develop new and more efficient discretization schemes for integrable and nonholonomic systems.

#### Scientific Details+

Within the general framework of Geometric Integration, a primary requirement to a discretized dynamical system is a correct correspondence of principal qualitative features of the dynamics, which is achieved by respecting geometric, structural properties of the original continuous system. However, it is in general impossible to achieve a complete qualitative and quantitative correspondence. A famous instance of formal statement expressing this impossibility is a theorem by Ge and Marsden, which says that, for a generic (non-integrable) Hamiltonian system, the only integrator which is symplectic and energy preserving is the shift along trajectories of the original system. Thus, any practical integrator disrespects some of the qualitative features of the original system. This raises the question about the unavoidable discrepancies introduced into the dynamics by discretization, or, in other words, about the perturbation of dynamics caused by discretization, even if the latter is performed in some kind of structure-preserving fashion. This is a central and fundamental question for all kinds of numerical investigation of long-term evolution of dynamical systems. However, this form of the question is obviously too general to admit a unifying general answer. Therefore, one is forced to restrict oneself to some more specific situations where more specific aspects can be studied. The present project addresses the above mentioned question in two specific situations: symplectic integrators applied to integrable systems, and nonholonomic integrators. In order to investigate these situations, we plan to invoke the so called method non-autonomous modified systems.

#### Publications+

##### Papers
• M. Vermeeren.
A dynamical solution to the Basel problem.
preprint, 2015.
arXiv:1506.05288.
• Fernando Jiménez and Hiroaki Yoshimura.
Dirac Structures in Vakonomic Mechanics.
J. Geom. Phys., 2015. accepted.
arXiv:1405.5394.
• F. Jiménez.
Hamilton-Dirac systems for charged particles in gauge fields.
J. Geom. Phys., 94:35-49, 2015.
arXiv:1410.3249.
• Jürgen Scheurle and Sebastian Walcher.
Minima of invariant functions: The inverse problem.
Acta Applicandae Mathematicae, 137(1):233-252, 2015. accepted for publication by Acta Applicandae Mathematicae.
doi:10.1007/s10440-014-9997-6.
• Sebastián Ferraro, Fernando Jiménez, and David Martín de Diego.
New developments on the Geometric Nonholonomic Integrator.
Nonlinearity, 28:871-900, 2015.
arXiv:1312.1587.
• Fernando Jimenez and Juergen Scheurle.
On the discretization of nonholonomic dynamics in $\mathbb R^n$.
J. Geom. Mechanics, 7(1):43-80, 2015.
arXiv:1407.2116.
• F. Jiménez and J. Scheurle.
On the discretization of the Euler-Poincaré-Suslov equations in $SO(3)$.
preprint, 2015.
arXiv:1506.01289.
• A. Delshams, M. Gonchenko, and P. Gutiérrez.
A methodology for obtaining asymptotic estimates for the exponentially small splitting of separatrices to whiskered tori with quadratic frequencies.
To appear in Research Perspectives CRM Barcelona, July 2014.
arXiv:1407.6524.
• A. Delshams, M. Gonchenko, and S. Gonchenko.
On dynamics and bifurcations of area-preserving maps with homoclinic tangencies.
Submitted to Nonlinearity, July 2014.
arXiv:1407.5473.
• Amadeu Delshams, Marina Gonchenko, and Pere Gutiérrez.
Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio.
Regul. Chaotic Dyn., 19(6):663–680, 2014.
arXiv:1409.4944, doi:10.1134/S1560354714060057.
• Amadeu Delshams, Marina Gonchenko, and Pere Gutiérrez.
Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies.
Electron. Res. Announc. Math. Sci., 21:41–61, 2014.
arXiv:1306.0728, doi:10.3934/era.2014.21.41.
• Amadeu Delshams, Marina Gonchenko, and Pere Gutiérrez.
Exponentially Small Lower Bounds for the Splitting of Separatrices to Whiskered Tori with Frequencies of Constant Type.
International Journal of Bifurcation and Chaos, 24(08):1440011, 2014.
arXiv:1402.1654, doi:10.1142/S0218127414400112.
• Amadeu Delshams, Marina Gonchenko, and Sergey V. Gonchenko.
On bifurcations of area-preserving and nonorientable maps with quadratic homoclinic tangencies.
Regul. Chaotic Dyn., 19(6):702–717, 2014.
arXiv:1410.5704, doi:10.1134/S1560354714060082.
• Manuele Santoprete, Jürgen Scheurle, and Sebastian Walcher.
Motion in a Symmetric Potential on the Hyperbolic Plane.
Canadian J. of Mathematics, August 2013. 27 pages.
arXiv:1305.3788, doi:10.4153/CJM-2013-026-2.
• Leonardo Colombo, Fernando Jiménez, and David Martín de Diego.
Variational integrators for underactuated mechanical control systems with symmetries.
submitted, September 2012.
arXiv:1209.6315.

#### Prof. Dr. Yuri Suris   +

Projects: B02, B10
University: TU Berlin
E-Mail: suris[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~suris/