Classical method of Lyapunov exponents spectrum estimation for a n-th-order continuous-time, smooth dynamical system involves Gram–Schmidt orthonormalization and calculations of perturbations lengths logarithms. In this paper, we have shown that using a new, simplified method, it is possible to estimate full spectrum of n Lyapunov exponents by integration of \((n-1)\) perturbations only. In particular, it is enough to integrate just one perturbation to obtain two largest Lyapunov exponents, which enables to search for hyperchaos. Moreover, in the presented algorithm, only very basic mathematical operations such as summation, multiplication or division are applied, which boost the efficiency of computations. All these features together make the new method faster than any other known by the authors if the order of the system under consideration is low. Correctness the method has been tested for three examples: Lorenz system, Duffing oscillator and three Duffing oscillators coupled in the ring scheme. Moreover, efficiency of the method has been confirmed by two practical tests. It has been revealed that for low-order systems, the presented method is faster than any other known by authors.

A discontinuous transition to hyperchaos is observed at discrete critical parameters in the Zeeman laser model for three well known nonlinear sources of instabilities, namely, quasiperiodic breakdown to chaos followed by interior crisis, quasiperiodic intermittency, and Pomeau–Manneville intermittency. Hyperchaos appears with a sudden expansion of the attractor of the system at a critical parameter for each case and it coincides with triggering of occasional and recurrent large-intensity pulses. The transition to hyperchaos from a periodic orbit via Pomeau–Manneville intermittency shows hysteresis at the critical point, while no hysteresis is recorded during the other two processes. The recurrent large-intensity pulses show characteristic features of extremes with their height larger than a threshold and the probability of a rare occurrence. The phenomenon is robust to weak noise although the critical parameter of transition to hyperchaos shifts with noise strength. This phenomenon appears as common in many low dimensional systems as reported earlier by Chowdhury et al. [Phys. Rep. 966, 1–52 (2022)], there the emergent large-intensity events or extreme events dynamics have been recognized simply as chaotic in nature although the temporal dynamics shows occasional large deviations from the original chaotic state in many examples. We need a new metric, in the future, that would be able to classify such significantly different dynamics and distinguish from chaos.

We report intermittent large-intensity pulses that originate in Zeeman laser due to instabilities in quasiperiodic motion, one route follows torus-doubling to chaos and another goes via quasiperiodic intermittency in response to variation in system parameters. The quasiperiodic breakdown route to chaos via torus-doubling is well known; however, the laser model shows intermittent large-intensity pulses for parameter variation beyond the chaotic regime. During quasiperiodic intermittency, the temporal evolution of the laser shows intermittent chaotic bursting episodes intermediate to the quasiperiodic motion instead of periodic motion as usually seen during the Pomeau-Manneville intermittency. The intermittent bursting appears as occasional large-intensity events. In particular, this quasiperiodic intermittency has not been given much attention so far from the dynamical system perspective, in general. In both cases, the infrequent and recurrent large events show non-Gaussian probability distribution of event height extended beyond a significant threshold with a decaying probability confirming rare occurrence of large-intensity pulses.

In this paper the construction of a neural-network based closed-loop control of a discontinuous capsule drive is analyzed. The foundation of the designed controller is an optimized open-loop control function. A neural network is used to determine the dependence between the output of the open-loop controller and the state of the system. Robustness of the neural controller with respect to variation of parameters of the controlled system is analyzed and compared with the original optimized open-loop control. It is expected that the presented method can facilitate the construction of closed-loop controllers for which alternative methods are not effective, such as non-smooth or discontinuous ones.