Route to Hyperchaos in Quadratic Optomechanics

Hyperchaos is a qualitatively stronger form of chaos in which several degrees of freedom contribute simultaneously to exponential divergence of small changes. A hyperchaotic dynamical system is, therefore, even more unpredictable than a chaotic one, and it has a higher fractal dimension. While hyperchaos has been studied extensively over the last decades, only a few experimental systems are known to exhibit hyperchaotic dynamics. Here, we introduce hyperchaos in the context of cavity optomechanics, in which light inside an optical resonator interacts with a suspended oscillating mass. We show that hyperchaos can arise in optomechanical systems with quadratic coupling and is well within reach of current experiments. We compute the two positive Lyapunov exponents characteristic of hyperchaos and independently verify the correlation dimension. We also identify a possible mechanism for the emergence of hyperchaos. As systems designed for high-precision measurements, optomechanical systems enable direct measurement of all four dynamical variables and, therefore, the full reconstruction of the hyperchaotic attractor. Our results may contribute to better understanding of nonlinear systems and the chaos-hyperchaos transition.