# Power Spectrum of a Noisy System Close to a Heteroclinic Orbit

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## Abstract

We consider a two-dimensional dynamical system that possesses a heteroclinic orbit connecting four saddle points. This system is not able to show self-sustained oscillations on its own. If endowed with white Gaussian noise it displays stochastic oscillations, the frequency and quality factor of which are controlled by the noise intensity. This stochastic oscillation of a nonlinear system with noise is conveniently characterized by the power spectrum of suitable observables. In this paper we explore different analytical and semianalytical ways to compute such power spectra. Besides a number of explicit expressions for the power spectrum, we find scaling relations for the frequency, spectral width, and quality factor of the stochastic heteroclinic oscillator in the limit of weak noise. In particular, the quality factor shows a slow logarithmic increase with decreasing noise of the form \(Q\sim [\ln (1/D)]^2\). Our results are compared to numerical simulations of the respective Langevin equations.

## Keywords

Power Spectrum Saddle Point Noise Intensity Planck Equation Heteroclinic Orbit## Notes

### Acknowledgements

JGB, PJT, and BL would like to acknowledge funding by La Caixa and DAAD (program 50015239), the National Science Foundation (Grant DMS-1413770), and BMBF (FKZ: 486 01GQ1001A) respectively.

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