Delay as an energy regulator of the generation of deterministic chaos in hydrodynamic systems with limited excitation

TL;DR

This work investigates how finite excitation-delay $\delta$ influences the nonlinear dynamics of a tank partially filled with liquid driven by a limited-power electric motor, using a Miles–Krasnopolskaya framework. A delayed five-dimensional system is formulated and then approximated to an ordinary differential system by expanding $\beta(\tau-\delta)$ for small $\delta$, enabling detailed numerical bifurcation analysis. The key finding is that delay can both induce and annihilate deterministic chaos, producing period-doubling cascades and generalized intermittency, and even triggering transitions between two distinct chaotic attractor types as $\delta$ varies. These results have practical implications for energy-efficient excitation control, showing that precise timing of the excitation can qualitatively alter system behavior in real tank-liquid machines.

Abstract

The Miles-Krasnopolskaya system is considered, which is used to study the nonlinear interaction of a tank with a liquid and the source of excitation of its oscillations. Additionally, delay time of impulse from the source of excitation of oscillations on the dynamics of the aggregate system "tank with liquid - source of excitation" is taken into account. A technique for studying the attractors of such systems is proposed. It is shown that delay plays a key role in the emergence (disappearance) of deterministic chaos in the Miles-Krasnopolskaya system. Quantitative changes in the value of the delay can lead to qualitative changes in the types of attractors of the system. So, regular attractors can turn into chaotic ones and vice versa. Also, a change in the delay value can lead to the implementation of new scenarios, both transitions from regular attractors to chaotic ones and transitions from a chaotic attractor of one type to a chaotic attractor of another type.

Figures (7)

• Figure 1: Scheme of the system "tank with liquid--electric motor"
• Figure 2: Phase-parametric characteristic for first set of parameters
• Figure 3: Limit cycles at $\delta =0$ (a) and at $\delta =0.0075$ (b); Chaotic atttractors at $\delta =0.0099$ (c), (d)
• Figure 4: Fragment of phase-parametric characteristic
• Figure 5: Chaotic attractor at $\delta=0.01479$$(a) ,(b)$; chaotic attractor at $\delta=0.0148$$(c) ,(d)$;