Appearance of chaos and hyperchaos in evolving pendulum network.

The study of deterministic chaos continues to be one of the important problems in the field of nonlinear dynamics. Interest in the study of chaos exists both in low-dimensional dynamical systems and in large ensembles of coupled oscillators. In this paper, we study the emergence of chaos in chains of locally coupled identical pendulums with constant torque. The study of the scenarios of the emergence (disappearance) and properties of chaos is done as a result of changes in (i) the individual properties of elements due to the influence of dissipation in this problem and (ii) the properties of the entire ensemble under consideration, determined by the number of interacting elements and the strength of the connection between them. It is shown that an increase of dissipation in an ensemble with a fixed coupling force and a number of elements can lead to the appearance of chaos as a result of a cascade of period-doubling bifurcations of periodic rotational motions or as a result of invariant tori destruction bifurcations. Chaos and hyperchaos can occur in an ensemble by adding or excluding one or more elements. Moreover, chaos arises hard since in this case, the control parameter is discrete. The influence of the coupling strength on the occurrence of chaos is specific. The appearance of chaos occurs with small and intermediate coupling and is caused by the overlap of the existence of various out-of-phase rotational mode regions. The boundaries of these areas are determined analytically and confirmed in a numerical experiment. Chaotic regimes in the chain do not exist if the coupling strength is strong enough. The dimension of an observed hyperchaotic regime strongly depends on the number of coupled elements.

Variety of rotation modes in a small chain of coupled pendulums.

This article studies the rotational dynamics of three identical coupled pendulums. There exist two parameter areas where the in-phase rotational motion is unstable and out-of-phase rotations are realized. Asymptotic theory is developed that allows us to analytically identify borders of instability areas of in-phase rotation motion. It is shown that out-of-phase rotations are the result of the parametric instability of in-phase motion. Complex out-of-phase rotations are numerically found and their stability and bifurcations are defined. It is demonstrated that the emergence of chaotic dynamics happens due to the period doubling bifurcation cascade. The detailed scenario of symmetry breaking is presented. The development of chaotic dynamics leads to the origin of two chaotic attractors of different types. The first one is characterized by the different phases of all pendulums. In the second case, the phases of the two pendulums are equal, and the phase of the third one is different. This regime can be interpreted as a drum-head mode in star-networks. It may also indicate the occurrence of chimera states in chains with a greater number of nearest-neighbour interacting elements and in analogical systems with global coupling.