Recurrence and Joint Recurrence Analysis of Multiple Attractors Energy Harvesting System

Abstract

The method of recurrence plots and joint recurrence plots are considered as tools for the nonlinear analysis of a dimensionless model of magnetoelastric piezoelectric energy harvester under wind flow excitation with low Reynolds number. The dynamics of the system is investigated by considering the bifurcation of the recurrence rate, the laminarity and the determinism and illustrations of system response are presented though the recurrence plots and phase diagrams. In order to enhance the efficiency of the system, a second degree of freedom is added to the mechanical part. The method of joint recurrence plot is used to analyze the global synchronization of the system. In this spirit, a feedback Master-Slave configuration is adopted to ensure optimal synchronized mechanical excursion and thus maximal electric voltage harvested in the electric load. Throughout the paper, attention is focussed on the effects of feedback coupling and mistuning parameter, as well as the relevance of the method of recurrence plots and joint recurrence plots in the analysis of such system. Specifically, it is shown that the joint recurrence plot synchronization parameter effectively detects domain of maximal output electric power as well as domain of out-of-phase motion leading to minimal output power.

Appendix: Definition of Recurrence Quantification Analysis Parameters

• 1. RR which defines the percentage of recurring points in the whole matrix. The RR is higher for periodic dynamics and smaller for chaotic or random dynamics. By definition, one has

  $$\begin{aligned} RR=\frac{1}{N^2}\sum _{i,j=1}^N\mathbf {R}_{i,j}(\varepsilon )\cdot \end{aligned}$$

  (39)

• 2. The percentage of recurrent points that form diagonal lines (of at least length $\ell _{min}$) parallel to the main diagonal DET gives information on the deterministic nature of the system. A chaotic system tends to have none or very short diagonals in opposite to periodic or quasi-periodic dynamics which tend to form regular diagonals parallel to the central diagonal along with mixture of short and long diagonals. The DET is defined as

  $$\begin{aligned} DET=\frac{\sum _{\ell =\ell _{min}}^N\ell P(\ell )}{\sum _{\ell =1}^N\ell P(\ell )}, \end{aligned}$$

  (40)

  where $\ell$ is the length of the diagonal line and P(x) is the histogram of x for a given threshold $\varepsilon$. If v is the length of the vertical line, one has

  $$\begin{aligned} LAM=\frac{\sum _{v=v_{min}}^Nv P(v)}{\sum _{v=1}^Nv P(v)} \end{aligned}$$

  (41)

  LAM decreases if the RP consists of more single recurrence points than vertical structures. This is related to the existence of intermittency in the system response [33].

In obtaining the RP of the system, we used the fourth order Runge Kutta algorithm to obtain sets of N = 8000 points for time series. The first 1000 values were ignored (transient time) and the time step was kept constant $\varDelta t=0.01$. The RR and DET are evaluated using the above definitions in a self made codes. However, various numerical codes are available online.

The bifurcation diagrams were obtained by increasing adiabatically (constant initial conditions) the bifurcation parameter and used the above procedure to generate the RR, LAM and DET.