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2007) 061208) have detected the presence of fractal escape basins in Hénon-Heiles potentials in the unbounded range. Upon fixing the energy value, these basins are detected on the (x, y) and (y, ẏ) planes. In this paper, we explore the appearance of different kinds of fractal structures. We present an analysis of the fractal structures on the escape basins of the (x, y) and (y, E) planes (allowing the energy value E to change and studying the fat-fractal exponent); later, we present these structures on the KAM tori for low energy values, on small regular islands inside the chaotic sea close to the critical energy level on the (y, E)-plane, and most interestingly, on small regular regions inside the escape region. These small regions of bounded motion and regular behavior appear after the critical escape energy, when most of the orbits are escape orbits.

We study a plethora of chaotic phenomena in the Hindmarsh-Rose neuron model with the use of several computational techniques including the bifurcation parameter continuation, spike-quantification, and evaluation of Lyapunov exponents in bi-parameter diagrams. Such an aggregated approach allows for detecting regions of simple and chaotic dynamics, and demarcating borderlines-exact bifurcation curves. We demonstrate how the organizing centers-points corresponding to codimension-two homoclinic bifurcations-along with fold and period-doubling bifurcation curves structure the biparametric plane, thus forming macro-chaotic regions of onion bulb shapes and revealing spike-adding cascades that generate micro-chaotic structures due to the hysteresis.

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