Traveling waves propagation on networks of dynamical systems (original) (raw)

A travelling wave approach to a multi-agent system with a path-graph topology

Systems & Control Letters, 2017

The paper presents a novel approach for the analysis and control of a multi-agent system with non-identical agents and a pathgraph topology. With the help of irrational wave transfer functions, the approach describes the interaction among the agents from the 'local' perspective and identify the travelling waves in the multi-agent system. The local treatment of the multi-agent system is complementary to the traditional 'overall' approach. It is shown that the different dynamics of the agents creates a virtual boundary that causes a partial reflection of the travelling waves. Undesired effects due to the reflection of the waves, such as amplification/attenuation, long transient or string instability, can be compensated by the feedback controllers introduced in this paper. A set of functions in MATLAB, which allows numerical simulation of the proposed approach, have been made available on MATLAB Central.

Zero dynamics for networks of waves

Automatica, 2019

The zero dynamics of infinite-dimensional systems can be difficult to characterize. The zero dynamics of boundary control systems are particularly problematic. In this paper the zero dynamics of port-Hamiltonian systems are studied. A complete characterization of the zero dynamics for port-Hamiltonian systems with invertible feedthrough as another port-Hamiltonian system on the same state space is given. It is shown that the zero dynamics for any port-Hamiltonian system with commensurate wave speeds are a well-posed system, and are also a port-Hamiltonian system. Examples include wave equations with uniform wave speed on a network. A constructive procedure for calculation of the zero dynamics that can be used for very large system order is provided.

Network models for the numerical solution of coupled ordinary non-lineal differential equations

COUPLED V : proceedings of the V International Conference on Computational Methods for Coupled Problems in Science and Engineering :, 2013

Many apparently simple problems in mechanics or mechanical engineering, particularly problems related to chaotic systems, are governing by coupled differential equations, generally non-lineal, that have to be solved numerically by specialists in this field. The network model, a tool very used in the last decades for numerical problems in different fields of science and engineering, allows that non-specialists, and even students familiarized with circuits theory, to design networks whose governing equations are just those of the engineering phenomenon, assuming a suitable or formal equivalence between electrical and physical variables. The design of the model, which is composed of a principal network, which implements a balance between the addends of the differential equations, and auxiliary networks to implement the derivative terms, follows a standard procedure. Non-lineal terms of the differential equations are implemented by a controlled source, a kind of device whose operation is quite intuitive. In this communication the models of two characteristic non-lineal mechanical problems are designed step by step with a detailed explanation: the elastic pendulum and the chaotic double pendulum: Solutions are presented graphically by using MATLAB.

Periodic orbits in nonlinear wave equations on networks

Journal of Physics A: Mathematical and Theoretical

We consider a cubic nonlinear wave equation on a network and show that inspecting the normal modes of the graph, we can immediately identify which ones extend into nonlinear periodic orbits. Two main classes of nonlinear periodic orbits exist: modes without soft nodes and others. For the former which are the Goldstone and the bivalent modes, the linearized equations decouple. A Floquet analysis was conducted systematically for chains; it indicates that the Goldstone mode is usually stable and the bivalent mode is always unstable. The linearized equations for the second type of modes are coupled, they indicate which modes will be excited when the orbit destabilizes. Numerical results for the second class show that modes with a single eigenvalue are unstable below a treshold amplitude. Conversely, modes with multiple eigenvalues seem always unstable. This study could be applied to coupled mechanical systems.

Preservation of relevant properties of interconnected dynamical systems over complex networks

Journal of the Franklin Institute, 2013

The inference and analysis of properties of solutions of differential equations is at the core of qualitative theory of dynamical systems. Understanding the conditions under which these properties are conserved under interconnections and coordinate transformations is a fundamental paradigm in the qualitative systems theory. Particularly, the concepts of passivity and dissipativity, which are embedded in the systems theory decades ago, are increasingly relevant in nowadays technological developments in the energy sector. Another property of special interest is the capacity of dynamical systems to synchronize or to desynchronize through weak interaction. Indeed, the study of synchronization is ubiquitous in basic sciences, such as physics, biology, medicine, as well as in disciplines linked to engineering and applied mathematics, such as computer science and control theory. On one hand, the study of dissipativity and energy exchange, as well as understanding the capacity of dynamical systems to (de)synchronize due to their interaction over networks (possibly with changing topology), has a major impact in a number of scientific and technological areas. On the other hand, qualitative systems theory contributes with original and efficient analytical methods. Thus, driven by authentic technological and scientific paradigms.

Localized solutions of nonlinear network wave equations

Journal of Physics A: Mathematical and Theoretical, 2018

We study localized solutions for the nonlinear graph wave equation on finite arbitrary networks. Assuming a large amplitude localized initial condition on one node of the graph, we approximate its evolution by the Duffing equation. The rest of the network satisfies a linear system forced by the excited node. This approximation is validated by reducing the nonlinear graph wave equation to the discrete nonlinear Schrödinger equation and by Fourier analysis. Finally, we examine numerically the condition for localization in the parameter plane, coupling versus amplitude and show that the localization amplitude depends on the maximal normal eigenfrequency.

P S ] 1 6 N ov 2 01 8 Localized solutions of nonlinear network wave equations

2018

Analysis of nonlinear oscillatory network dynamics via time-varying amplitude and phase variables

International Journal of Circuit Theory and Applications, 2007

The goal of this manuscript is to propose a method for investigating the global dynamics of nonlinear oscillatory networks, with arbitrary couplings. The procedure is mainly based on the assumption that the dynamics of each oscillator is accurately described by a couple of variables, that is, the oscillator periodic orbits are represented through time-varying amplitude and phase variables. The proposed method allows one to derive a set of coupled nonlinear ordinary differential equations governing the time-varying amplitude and phase variables. By exploiting these nonlinear ordinary differential equations, the prediction of the total number of periodic oscillations and their bifurcations is more accurate and simpler with respect to the one given by the latest available methodologies. Furthermore, it is proved that this technique also works for weakly connected oscillatory networks. Finally, the method is applied to a chain of third-order oscillators (Chua's circuits) and the results are compared with those obtained via a numerical technique, based on the harmonic balance approach.

Traveling waves propagation on networks of dynamical systems (original) (raw)

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