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Nonlinear dynamical systems such as coupled oscillators are being actively investigated as Ising machines for solving computationally hard problems in combinatorial optimization. Prior works have established the equivalence between the global minima of the cost function describing the coupled oscillator system and the ground state of the Ising Hamiltonian. However, the properties of the oscillator Ising machine (OIM) from a nonlinear control viewpoint, such as the stability of the OIM solutions, remain unexplored. Therefore, in this work, using nonlinear control-theoretic analysis, we (i) identify the conditions required to ensure the functionality of the coupled oscillators as an Ising machine, (ii) show that all globally optimal phase configurations may not always be stable, resulting in some configurations being more favored over others and, thus, creating a biased OIM, and (iii) elucidate the impact of the stability of locally optimal phase configurations on the quality of the solution computed by the system. Our work, fostered through the unique convergence between nonlinear control theory and analog systems for computing, provides a new toolbox for the design and implementation of dynamical system-based computing platforms.
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Most probable escape paths in periodically driven nonlinear oscillators
The dynamics of mechanical systems, such as turbomachinery with multiple blades, are often modeled by arrays of periodically driven coupled nonlinear oscillators. It is known that such systems may have multiple stable vibrational modes, and transitions between them may occur under the influence of random factors. A methodology for finding most probable escape paths and estimating the transition rates in the small noise limit is developed and applied to a collection of arrays of coupled monostable oscillators with cubic nonlinearity, small damping, and harmonic external forcing. The methodology is built upon the action plot method [Beri et al., Phys. Rev. E 72, 036131 (2005)] and relies on the large deviation theory, the optimal control theory, and the Floquet theory. The action plot method is promoted to non-autonomous high-dimensional systems, and a method for solving the arising optimization problem with a discontinuous objective function restricted to a certain manifold is proposed. The most probable escape paths between stable vibrational modes in arrays of up to five oscillators and the corresponding quasipotential barriers are computed and visualized. The dependence of the quasipotential barrier on the parameters of the system is discussed.
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- Award ID(s):
- 1760366
- PAR ID:
- 10439845
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- Chaos: An Interdisciplinary Journal of Nonlinear Science
- Volume:
- 32
- Issue:
- 8
- ISSN:
- 1054-1500
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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