#### Title

Noise-Induced Stabilization of Hamiltonian Systems

#### Faculty Sponsor

Dr. Tiffany Kolba

#### College

Arts and Sciences

#### Department/Program

Mathematics and Statistics

#### ORCID Identifier(s)

0000-0002-0457-5077 0000-0001-6604-2367 0000-0003-1986-7336

#### Document Type

Poster Presentation

#### Symposium Date

Summer 7-31-2017

#### Abstract

Noise-induced stabilization is the phenomenon in which the addition of randomness to an unstable deterministic system of ordinary differential equations (ODEs) results in a stable system of stochastic differential equations (SDEs). A Hamiltonian system is a two-dimensional system of ODEs defined by a Hamiltonian function, which is constant along each solution curve. With stability defined as global stochastic boundedness, Hamiltonian systems cannot be stabilized by the addition of noise that is constant in space. Therefore we seek to deterministically perturb the Hamiltonian systems in such a way that the qualitative behavior of solutions is preserved, but noise-induced stabilization becomes possible. Our goal is to provide a systematic framework for methods of perturbing the systems and proving noise-induced stabilization.

#### Recommended Citation

Coniglio, Anthony; Sparks, Sarah; and Weithers, Daniel, "Noise-Induced Stabilization of Hamiltonian Systems" (2017). *Summer Interdisciplinary Research Symposium*. 18.

https://scholar.valpo.edu/sires/18

## Biographical Information about Author(s)

Anthony Coniglio is an incoming junior at Indiana University Bloomington majoring in mathematics, physics, and music. Some of his favorite areas of mathematics include differential equations, complex variables, and abstract algebra. Upon completion of his undergraduate degree, he hopes to attend graduate school and obtain a PhD in Mathematics.

Sarah Sparks is a senior undergraduate student at Frostburg State University in Maryland, majoring in mathematics. She has experience working with differential equations and mathematical modeling. Upon graduation she hopes to attend a graduate program in mathematics.

Daniel Weithers is a rising senior math major at Carleton College. Some of his favorite areas in math include stochastic processes, mathematical models of decision-making, and applications of math to the natural sciences. Next year he hopes to attend a graduate program specializing in applied math.