Alternative Methods of the Largest Lyapunov Exponent Estimation with Applications to the Stability Analyses Based on the Dynamical Maps—Introduction to the Method

Controlling stability of dynamical systems is one of the most important challenges in science and engineering. Hence, there appears to be continuous need to study and develop numerical algorithms of control methods. One of the most frequently applied invariants characterizing systems’ stability are Lyapunov exponents (LE). When information about the stability of a system is demanded, it can be determined based on the value of the largest Lyapunov exponent (LLE). Recently, we have shown that LLE can be estimated from the vector field properties by means of the most basic mathematical operations. The present article introduces new methods of LLE estimation for continuous systems and maps. We have shown that application of our approaches will introduce significant improvement of the efficiency. We have also proved that our approach is simpler and more efficient than commonly applied algorithms. Moreover, as our approach works in the case of dynamical maps, it also enables an easy application of this method in noncontinuous systems. We show comparisons of efficiencies of algorithms based our approach. In the last paragraph, we discuss a possibility of the estimation of LLE from maps and for noncontinuous systems and present results of our initial investigations.

A toy model to investigate stability of AI-based dynamical systems

<p>The development of atmospheric parameterizations based on neural networks is often hampered by numerical instability issues. Previous attempts to replicate these issues in a toy model have proven ineffective. We introduce a new toy model for atmospheric dynamics, which consists in an extension of the Lorenz'63 model to a higher dimension. While neural networks trained on a single orbit can easily reproduce the dynamics of the Lorenz'63 model, they fail to reproduce the dynamics of the new toy model, leading to unstable trajectories. Instabilities become more frequent as the dimension of the new model increases, but are found to occur even in very low dimension. Training the neural network on a different learning sample, based on Latin Hypercube Sampling, solves the instability issue. Our results suggest that the design of the learning sample can significantly influence the stability of dynamical systems driven by neural networks.</p>

Phase transitions and stability of dynamical processes on hypergraphs

Hypergraphs naturally represent higher-order interactions, which persistently appear in social interactions, neural networks, and other natural systems. Although their importance is well recognized, a theoretical framework to describe general dynamical processes on hypergraphs is not available yet. In this paper, we derive expressions for the stability of dynamical systems defined on an arbitrary hypergraph. The framework allows us to reveal that, near the fixed point, the relevant structure is a weighted graph-projection of the hypergraph and that it is possible to identify the role of each structural order for a given process. We analytically solve two dynamics of general interest, namely, social contagion and diffusion processes, and show that the stability conditions can be decoupled in structural and dynamical components. Our results show that in social contagion process, only pairwise interactions play a role in the stability of the absorbing state, while for the diffusion dynamics, the order of the interactions plays a differential role. Our work provides a general framework for further exploration of dynamical processes on hypergraphs.

Deep Reinforcement Learning Controller for 3D Path Following and Collision Avoidance by Autonomous Underwater Vehicles

Control theory provides engineers with a multitude of tools to design controllers that manipulate the closed-loop behavior and stability of dynamical systems. These methods rely heavily on insights into the mathematical model governing the physical system. However, in complex systems, such as autonomous underwater vehicles performing the dual objective of path following and collision avoidance, decision making becomes nontrivial. We propose a solution using state-of-the-art Deep Reinforcement Learning (DRL) techniques to develop autonomous agents capable of achieving this hybrid objective without having a priori knowledge about the goal or the environment. Our results demonstrate the viability of DRL in path following and avoiding collisions towards achieving human-level decision making in autonomous vehicle systems within extreme obstacle configurations.

Multi-machine system "Synchronous generator - steam turbine"

In this article, we will consider the problems of controlling multidimensional phase systems described by nonlinear differential equations. These mathematical models describe processes in complex systems consisting of many turbines and generators and are used for their analysis. The relevance of these models lies in the fact that they allow simulating different pre-emergency, emergency, and post-emergency situations. The controllability of the model under consideration is determined by studying the global asymptotic stability of dynamical systems in cylindrical phase systems. The results obtained are demonstrated by a numerical example.

The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems

The article is devoted to the analysis of partial stability of nonlinear systems of ordinary differential equations using Lie algebras and groups. It is shown that the existence of a group of transformations invariant under partial stability in the system under study makes it possible to simplify the analysis of the partial stability of the initial system. For this it is necessary that the associated linear differential operator Lie in the enveloping Lie algebra of the original system, and the operator defined by the one-parameter Lie group is commutative with this operator. In this case, if the found group has invariance with respect to partial stability, then the resulting transformation performs to the decomposition of the system under study, and the partial stability problem reduces to the investigation of the selected subsystem. Finding the desired transformation uses the first integrals of the original system. Examples illustrating the proposed approach are given.

The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems

The article is devoted to the analysis of partial stability of nonlinear systems of ordinary differential equations using Lie algebras and groups. It is shown that the existence of a group of transformations invariant under partial stability in the system under study makes it possible to simplify the analysis of the partial stability of the initial system. For this it is necessary that the associated linear differential operator Lie in the enveloping Lie algebra of the original system, and the operator defined by the one-parameter Lie group is commutative with this operator. In this case, if the found group has invariance with respect to partial stability, then the resulting transformation performs to the decomposition of the system under study, and the partial stability problem reduces to the investigation of the selected subsystem. Finding the desired transformation uses the first integrals of the original system. Examples illustrating the proposed approach are given.

Mechanical Representation and Stability of Dynamical Systems Containing Fractional Springpot Elements

In this article we consider the Lyapunov stability of mechanical systems containing fractional springpot elements. We obtain the potential energy of a springpot by an infinite dimensional mechanical analogue model. Furthermore, we consider a simple dynamical system containing a springpot as a functional differential equation and use the potential energy of the springpot in a Lyapunov functional to prove uniform stability and discuss asymptotic stability of the equilibrium with the help of an invariance theorem.