On the existence of chaos in a class of two-degree-of-freedom, damped, strongly parametrically forced mechanical systems with broken \(O(2)\) symmetry
type: Item
instance of: scholarly article
title: On the existence of chaos in a class of two-degree-of-freedom, damped, strongly parametrically forced mechanical systems with broken \(O(2)\) symmetry
zbMATH Open document ID: 0786.34050
DOI: 10.1007/BF00914282
author: Zaichun Feng
author: Stephen Wiggins
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1993-08-25T00:00:00Z
review text: The homoclinic orbits of the ordinary differential equation system of the type \[ \begin{aligned} \dot z_ 1 &= -az_ 1+ i\bigl[-(\sigma-\beta)z_ 1+ \pi_ 1 z^ 2_ 1\bar z_ 1+ \Delta\bar z_ 1+ \pi_ 2 z_ 1 z_ 2\bar z_ 2+\pi_ 3\bar z_ 1 z^ 2_ 2\bigr]\\ \dot z_ 2 &= - az_ 2+ i\bigl[-(\sigma+ \beta)z_ 2+ \Delta\bar z_ 2+ \pi_ 1 z^ 2_ 2\bar z_ 2+ \pi_ 2 z_ 1 z_ 2\bar z_ 1+\pi_ 3\bar z_ 2 z^ 2_ 1\bigr]\tag{1}\end{aligned} \] are studied. Here \(z_ 1\) and \(z_ 2\) are two complex variables, \(\sigma\), \(\beta\), \(\Delta\), \(\pi_ 1\), \(\pi_ 2\) are all real parameters. The system (1) is Hamiltonian if \(a=0\). This set of equations arises as a normal form in different applied problems. In particular, it is as the amplitude equations that govern the motion of two Fourier modes that are in resonance with an external parametric sinusoidal exitation. For example, the system (1) is applied for the description of modal interactions in parametrically excited surface waves, nonlinear elastic plates and elastic beams. The case \(a\neq 0\) and \(\Delta= 0(1)\) is studied in the paper in contrast to the previous works. The main result is a proof of existence of a ``Shilnilov type'' homoclinic orbit to a fixed point of saddle focus type which is created through the internal resonance of the system.
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 70K50
zbMATH DE Number: 399117
zbMATH Keywords: Shilnikov type homoclinic orbit
zbMATH Keywords: normal form
zbMATH Keywords: modal interactions in parametrically excited surface waves
zbMATH Keywords: nonlinear elastic plates
zbMATH Keywords: elastic beams
zbMATH Keywords: internal resonance
reviewed by: Yu. P. Virchenko
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00914282
OpenAlex ID: W1991729125
description: scientific article; zbMATH DE number 399117