Manifold angles, the concept of self-similarity, and angle-enhanced bifurcation diagrams

Predicting regime changes and durations in Lorenz's atmospheric convection model

We show that a characteristic alignment between Lyapunov vectors can be used to predict regime changes as well as regime duration in the classical Lorenz model of atmospheric convection. By combining Lyapunov vector alignment with maxima in the local expansion of bred vectors, we obtain an effective and competitive method to significantly decrease errors in the prediction of regime durations.

Predictability of the onset of spiking and bursting in complex chemical reactions

For three complex chemical reactions displaying intricate dynamics, we assess the effectiveness of a recently proposed quantitative method to forecast bursting and large spikes, i.e. extreme events. Specifically, we consider predicting extreme events in (i) a copper dissolution model where Bassett and Hudson experimentally observed homoclinic (Shilnikov) chaos, (ii) a model derived from the mass action law of chemical kinetics, and (iii) an autocatalator model. For these systems, we describe how the alignment of Lyapunov vectors can be used to predict the imminence of large-amplitude events and the onset of complex dynamics in chaotic time-series of observables.

Exploring conservative islands using correlated and uncorrelated noise

In this work, noise is used to analyze the penetration of regular islands in conservative dynamical systems. For this purpose we use the standard map choosing nonlinearity parameters for which a mixed phase space is present. The random variable which simulates noise assumes three distributions, namely equally distributed, normal or Gaussian, and power law (obtained from the same standard map but for other parameters). To investigate the penetration process and explore distinct dynamical behaviors which may occur, we use recurrence time statistics (RTS), Lyapunov exponents and the occupation rate of the phase space. Our main findings are as follows: (i) the standard deviations of the distributions are the most relevant quantity to induce the penetration; (ii) the penetration of islands induce power-law decays in the RTS as a consequence of enhanced trapping; (iii) for the power-law correlated noise an algebraic decay of the RTS is observed, even though sticky motion is absent; and (iv) although strong noise intensities induce an ergodic-like behavior with exponential decays of RTS, the largest Lyapunov exponent is reminiscent of the regular islands.

Alignment of Lyapunov Vectors: A Quantitative Criterion to Predict Catastrophes?

We argue that the alignment of Lyapunov vectors provides a quantitative criterion to predict catastrophes, i.e. the imminence of large-amplitude events in chaotic time-series of observables generated by sets of ordinary differential equations. Explicit predictions are reported for a Rössler oscillator and for a semiconductor laser with optoelectronic feedback.