Modal Series Method for Nonautonomous Nonlinear Systems

Stochastic bifurcation

Two-dimensional nonlinear dynamic system model of interspecific interaction and numerical simulation research on it

IntroductionThe dynamic behavior analysis of nonlinear physical systems plays a critical role in understanding complex processes across various domains, including education, where interactive simulations of such systems can enhance conceptual learning. Traditional modeling techniques for nonlinear systems often fail to capture their high-dimensional, multi-scale, and chaotic nature due to oversimplified assumptions or reliance on linear approximations.MethodsIn this study, we present a novel framework leveraging computer vision and advanced neural architectures to analyze the dynamic behaviors of nonlinear physical systems. The proposed Physics-Informed Nonlinear Dynamics Network (PNDN) integrates data-driven embeddings with physics-based constraints, offering a robust solution for capturing intricate dynamics and ensuring adherence to physical principles.ResultsExperimental results highlight the model’s superior performance in reconstructing and predicting nonlinear system behaviors under diverse conditions, establishing its utility for real-time educational simulations.DiscussionThis approach bridges the gap between computational modeling and educational innovation, providing learners with interactive tools to explore complex physical phenomena.

Among the irregular responses of nonlinear dynamic systems, chaotic responses of nonlinear systems are probably the most attractive phenomena along with the new observations in the last decades. A nonlinear deterministic system may behavior chaotically under regular such as periodic excitations. Regular motion of a system subjected to periodic exertions is usually periodic. In contrast with regular motions, final states of chaotic vibrations are extremely nonperiodic. This research is to analyzing the irregular behavior of dynamic systems with implementation of a newly developed criterion named Periodicity-Ratio. The development of a methodology for diagnosing the irregular motions from the regular motions of a dynamic system is presented. The Periodicity-Ratio describes the degree of periodicity of motion and can be conveniently used to distinguish a nonperiodic motion from a regular vibration or oscillation and to diagnose whether or not a motion is chaotic and the other irregular responses of the nonlinear dynamic systems, without plotting any figures. The analyses on the irregular behavior of nonlinear dynamic systems with the implementation of the Periodicity-Ratio will be demonstrated.

In previous studies, information dynamics methods such as Von Neumann entropy and Rényi entropy played an important role in many fields, covering both macroscopic and microscopic studies. They have a solid theoretical foundation, but there are few reports in the field of mechanical nonlinear systems. So, can we apply Von Neumann entropy and Rényi entropy to study and analyze the dynamic behavior of macroscopic nonlinear systems? In view of the current lack of suitable methods to characterize the dynamics behavior of mechanical systems from the perspective of nonlinear system correlation, we propose a new method to describe the nonlinear features and coupling relationship of mechanical systems. This manuscript verifies the above hypothesis by using a typical chaotic system and a real macroscopic physical nonlinear system through theory and practical methods. The nonlinear vibration correlation in multi-body mechanical systems is very complex. We propose a full-vector multi-scale Rényi entropy for exploring the chaos and correlation between the dynamic behaviors of mechanical nonlinear systems. The research results prove the effectiveness of the proposed method in modal identification, system dynamics evolution and fault diagnosis of nonlinear systems. It is of great significance to extend these studies to the field of mechanical nonlinear system dynamics.

Introducing some classes of stable systems without any smooth Lyapunov functions

In this paper a new balanced realization method for nonlinear systems is proposed which is based on singular value analysis of Hankel operators. The proposed method balances the relationship between the input-to-state behavior and the state-to-output behavior of nonlinear dynamical systems, whereas the existing results only balance the relationship among the coordinate axes of the state-space. This result is expected to be a basis for new model reduction and system identification of nonlinear systems.

This paper deals with the stability problems of nonautonomous impulsive differential systems with time-delay. By utilizing the method of mathematic induction, proofs by contradiction and Lyapunov-Krasovskii functional, we obtain several uniformly stable criteria about the linear and nonlinear nonautonomous impulsive systems with time delay. One numerical example and its simulations are given to illustrate the effectiveness of the theoretical result.

International audience

Extinction in two species nonautonomous nonlinear competitive system

This paper presents a method to determine a nonlinear state-space model from a finite number of measurements of the inputs and outputs. The method is based on embedding theory for nonlinear systems, and can be viewed as an extension of the subspace identification method for linear systems. The paper describes the underlying theory and provides some guidelines for using the method in practice. To illustrate the use of the identification method, it was applied to a second-order nonlinear system.

On stability of nonlinear nonautonomous discrete fractional Caputo systems

In many domains, including biology, engineering, physics, and finance, the ability to forecast chaotic behavior is of the utmost importance. Predicting nonlinear systems accurately is a critical task due to their intrinsic sensitivity to initial conditions and lack of apparent patterns. One potential way to address this difficulty is by utilizing machine learning techniques. These methods can help us better understand and manage complex systems that display chaotic dynamics. Complex nonlinear systems, with their great dimensionality, temporal interdependence, and sensitivity to initial conditions, make chaotic behavior prediction difficult. The development of sophisticated tools that can detect patterns in chaotic dynamics is often necessary because traditional methods fail to capture these subtleties. The present research presents Chaotic Dynamics Prediction through Deep Ensemble Learning (CDP-DEL), an approach to chaotic dynamics prediction (CDP) that combines deep learning architectures with ensemble learning methodologies. Ensemble learning improves generalizability and reduces overfitting, whereas deep neural networks are made to capture complicated nonlinear interactions. CDP-DEL is an all-inclusive method for predicting chaotic behavior in nonlinear systems since it uses feature engineering, integrates system factors, and handles temporal dependencies. Among the many fields that have found use for CDP-DEL are control systems, ecological research, financial modeling, and weather forecasting. When applied to real-world scenarios with substantial chaotic dynamics repercussions, CDP-DEL's improved prediction capabilities may help optimize decision-making processes. Extensive simulation evaluations are performed utilizing benchmark chaotic systems to validate the efficacy of CDP-DEL. Research comparing CDP-DEL to more conventional methods show that it is more effective at predicting chaotic behavior and overall outperforms the competition.

Applications of deterministic chaos theory to corrosion

The proportional-integrative-derivative (PID) controller provides effective control in linear systems. However, performance is poor in nonlinear systems unless combined with a fuzzy logic controller (FLC) that modifies the PID controller gains as needed. System behavior can adapt to operational environment changes by switching different FLCs online. But that capability requires accurate operational environment identification. In autonomous systems identification must be done without significant human presence. This paper describes how Discrete Fourier Transforms can identify operational environments in an autonomous nonlinear system, which in this work is a DC motor. Simulation results shows the proposed method accurately identifies the physical environment and altering the FLC online correctly adapts the system's behavior.