Mathematical Problems in Engineering

Volume 2015, Article ID 367036, 9 pages

http://dx.doi.org/10.1155/2015/367036

## FFT Bifurcation Analysis of Routes to Chaos via Quasiperiodic Solutions

Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland

Received 5 August 2015; Revised 25 October 2015; Accepted 1 December 2015

Academic Editor: Jonathan N. Blakely

Copyright © 2015 L. Borkowski and A. Stefanski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The dynamics of a ring of seven unidirectionally coupled nonlinear Duffing oscillators is studied. We show that the FFT analysis presented in form of a bifurcation graph, that is, frequency distribution versus a control parameter, can provide a valuable and helpful complement to the corresponding typical bifurcation diagram and the course of Lyapunov exponents, especially in context of detailed identification of the observed attractors. As an example, bifurcation analysis of routes to chaos via 2-frequency and 3-frequency quasiperiodicity is demonstrated.

#### 1. Introduction

The spectral analysis of a signal using the fast Fourier transform (FFT) is a widespread method for investigation and diagnostics of dynamical systems in science and engineering. The bibliography concerning the FFT algorithms and their application is very huge. Therefore, this paper concentrates on selected application associated with the researched problem only. The FFT is an algorithm for computing the discrete Fourier transform (DFT) and its inverse [1]. For , which are complex numbers, the DFT is defined by the following formula: where . Using the FFT analysis, the frequency components included in the time waveform can be presented. Calculation of the sum by formula (1) would take operations. Using the Cooley-Tukey algorithm [2], which is based on the divide and conquer algorithm, the fast Fourier transform is calculated recursively dividing the transform of size into transform of size and with the use of multiplications. The computational complexity of the fast Fourier transform is , instead of algorithm which follows from the formula determining the DFT. There are other algorithms for calculating the DFT, for example, the Prime-factor algorithm also called the Good-Thomas algorithm [3, 4], Bruun’s algorithm [5], Rader’s algorithm [6], and Bluestein’s algorithm [7]. As a result of the existence of the above-mentioned algorithms, it became possible to apply digital signal processing (DSP) [8, 9] and the use of discrete cosine transform (DCT) to data compression [10, 11] (e.g., JPEG or MP3 files).

Nowadays, in many areas of science and technology, we can observe the use of the fast Fourier transform in order to present the results of research and calculations. The use of the FFT analysis to study nonlinear dynamical systems is present in works of many scientists and researchers. A few selected examples of such applications are mentioned in [12–14]. In a series of three articles, Krysko et al. used the FFT analysis to study(i)dynamics of continuous dynamical systems such as flexible plate and shallow shells [15],(ii)classical and novel scenarios of transition from periodic to chaotic solutions of dissipative continuous mechanical systems [16],(iii)dynamic loss of stability and different routes of transition to chaos of flexible curvilinear beam using Lyapunov exponents [17].

One can also mention a few examples of the FFT application in cases similar to the system analyzed in this paper. In 2006, Sánchez et al. [18] studied in their works a ring of unidirectionally coupled Lorenz oscillators. They observed occurrence of so-called rotating wave between oscillators and the transition from periodic rotating wave through quasiperiodic solutions to chaotic rotating wave. Numerical investigations were confirmed by experimental research. They used the FFT analysis as a tool for the presentation of results. Also in the electrical systems the FFT analysis is widely used. For example, Hajimiri and Lee used the FFT analysis to study phase noise in nonlinear electrical oscillators [19, 20]. Also in the article of Razavi we can observe the use of the FFT analysis test phase noise in a ring of CMOS oscillators [21].

In this paper, the FFT analysis is applied to study dynamics and bifurcations of the ring of unidirectionally coupled nonlinear Duffing oscillators. In this system a route to chaos via 2-frequency and 3-frequency quasiperiodicity can be observed. The FFT investigation accompanies classical qualitative and quantitative tools for dynamical systems research as Poincaré maps, bifurcation diagrams, and Lyapunov exponents. The paper is organized as follows. Section 2 contains a brief description of analyzed ring of Duffing oscillators. In Section 3, the results of numerical investigation of the system under consideration are demonstrated. Classical bifurcation diagrams and values of Lyapunov exponents are summarized with results of the FFT analysis. Finally, Section 4 presents a discussion of our results and conclusions.

#### 2. Analyzed System

The system under consideration is a closed ring of unidirectionally coupled identical oscillators shown in Figure 1. As a node system we took autonomous single-well Duffing oscillator given by where , , and are real positive parameters. Introducing the substitution and assuming diffusive coupling between the oscillators, we can describe the dynamics of each th ring node by the following pair of 1st-order ODEs: where and is an overall coupling coefficient [22].