Synchronization detection of biological CAM plants using instantaneous Lyapunov exponent

The Lyapunov exponent gives a measure of the mean decay/divergence rates of the flows of nonlinear systems. However, the Lyapunov exponent needs an infinite time interval of flows and the Jacobian matrix of system dynamics. In this paper, we propose an instantaneous decay that is a kind of generalized Lyapunov exponent and call the instantaneous Lyapunov exponent (ILE) with respect to a decay function. The instantaneous Lyapunov exponent is one of the measures that estimate the decay rates of flows of nonlinear systems by assigning a comparison function and can apply a stable system whose decay rate is slower than an exponential function. Finally, the proposed measures are applied to the synchronization detection of the endogenous circadian photosynthesis oscillations of plants performing crassulacean acid metabolism (CAM) and its adaptive estimator.