On the existence of chaos in a class of two-degree-of-freedom, damped, strongly parametrically forced mechanical systems with broken \(O(2)\) symmetry
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.
modal interactions in parametrically excited surface waves
scientific article; zbMATH DE number 399117
scientific article