Reconstructing the Phase Space With Fractional Derivatives

Fractional derivatives are applied in the reconstruction of the phase space of dynamical systems with a single observable. The fractional derivatives of time series data are obtained in the frequency domain. The method is applied to a Duffing oscillator, the Lorenz system, and data from a two-well experiment. The ability of the method to unfold the data is assessed by the method of global false nearest neighbors. The reconstructed data is used to compute recurrences and correlation dimensions. The reconstruction is compared to the method of delays in order to assess the choice of reconstruction parameters, and also the quality of results.

Phase-Space Reconstructions of Stick-Slip Systems

Nonsmooth processes such as stick-slip may introduce problems with phase-space reconstructions. We examine chaotic single-degree-of-freedom stick-slip friction models and use the method of delays to reconstruct the phase space. We illustrate that this reconstruction process can cause pseudo trajectories to collapse in a way that is unlike, yet related to, the dimensional collapse in the original phase-space. As a result, the reconstructed attractor is not topologically similar to the real attractor. Standard dimensioning tools are applied in effort to recognize this situation. The use of additional observables is examined as a possible remedy for the problem.

Experimental Study of Chaos in a Flexible Parametrically Excited Beam

Nonlinear vibrations in an axially driven limber cantilever beam are studied experimentally to determine whether the observed aperiodic motions are chaotic, and to find the attractor dimension. Using the method of delays and time series calculations, the largest Lyapunov exponent is found to be positive, indicating that the aperiodic motions are chaotic. The correlation and embedding dimensions are computed; a phase space dimension in the range of 5–7 is found for the chaotic attractor. The system is also studied analytically, using a truncated Galerkin reduction of the planar equation of motion. It is found that this analytical approach does model the near-harmonic periodic motions of this system; however, the chaotic motions and chaotic transitions are not predicted. The model does exhibit interesting chaotic responses for non-physical values of the driving parameters.