  Optical non-linearities, such as thermo-optic effects and
free-carrier-dispersion, are often considered as undesired effects in
silicon-based resonators and, more specifically, optomechanical (OM) cavities,
affecting the relative detuning between an optical resonance and the excitation
laser. However, the interplay between such mechanisms could also enable
unexpected physical phenomena to be used in new applications. In the present
work, we exploit those non-linearities and their intercoupling with the
mechanical degrees of freedom of a silicon OM nanobeam to unveil a rich set of
fundamentally different complex dynamics. By smoothly changing the parameters
of the excitation laser, namely its power and wavelength, we demonstrate
accurate control for activating bi-dimensional and tetra-dimensional
limit-cycles, a period doubling route and chaos. In addition, by scanning the
laser parameters in opposite senses we demonstrate bistability and hysteresis
between bi-dimensional and tetra-dimensional limit-cycles, between different
coherent mechanical states and between tetra-dimensional limit-cycles and
chaos. As a result of implementing the Rosenstein algorithm to the experimental
time series we have extracted a Largest Lyapunov Exponent of LLE=1.3x10^5 s-1,
which is between one and two orders of magnitude lower than the typical
oscillating frequencies of the system. Most of the experimental features can be
well reproduced with a model of first-order non-linear differential equations
coupled through the number of intracavity photons. Our findings open several
routes towards exploiting silicon-based OM photonic crystals for systems with
memory and, more specifically, for chaos based applications such as secure
information transfer or sensing.
