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In this paper we propose a convex Sum-of-Squares optimization problem for finding outer approximations of forward reachable sets for nonlinear uncertain Ordinary Differential Equations (ODE’s) with either (or both) L2 or point-wise bounded input disturbances. To make our approximations tight we seek to minimize the volume of our approximation set. Our approach to volume minimization is based on the use of a convex determinant-like objective function.We provide several numerical examples including the Lorenz system and the Van der Pol oscillator.
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Geometric phase for nonlinear oscillators from perturbative renormalization group
We formulate a renormalization-group approach to a general nonlinear oscillator problem. The approach is based on the exact group law obeyed by solutions of the corresponding ordinary differential equation. We consider both the autonomous models with time-independent parameters, as well as nonautonomous models with slowly varying parameters. We show that the renormalization-group equations for the nonautonomous case can be used to determine the geometric phase acquired by the oscillator during the change of its parameters. We illustrate the obtained results by applying them to the Van der Pol and Van der Pol-Duffing models.
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- Award ID(s):
- 2138008
- PAR ID:
- 10472684
- Publisher / Repository:
- American Physical Society
- Date Published:
- Journal Name:
- Physical Review E
- Volume:
- 108
- Issue:
- 4
- ISSN:
- 2470-0045
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1103/PhysRevE.108.044215
