Reconstructing Bifurcation Diagrams of Dynamical Systems Using Measured Time Series

Reconstructing bifurcation diagrams only from time-series data generated by electronic circuits in discrete-time dynamical systems

Bifurcation-diagram reconstruction estimates various attractors of a system without observing all of them but only from observing several attractors with different parameter values. Therefore, the bifurcation-diagram reconstruction can be used to investigate how attractors change with the parameter values, especially for real-world engineering and physical systems for which only a limited number of attractors can be observed. Although bifurcation diagrams of various systems have been reconstructed from time-series data generated in numerical experiments, the systems that have been targeted for reconstructing bifurcation diagrams from time series measured from physical phenomena so far have only been continuous-time dynamical systems. In this paper, we reconstruct bifurcation diagrams only from time-series data generated by electronic circuits in discrete-time dynamical systems with different parameter values. The generated time-series datasets are perturbed by dynamical noise and contaminated by observational noise. To reconstruct the bifurcation diagrams only from the time-series datasets, we use an extreme learning machine as a time-series predictor because it has a good generalization property. Hereby, we expect that the bifurcation-diagram reconstruction with the extreme learning machine is robust against dynamical noise and observational noise. For quantitatively verifying the robustness, the Lyapunov exponents of the reconstructed bifurcation diagrams are compared with those of the bifurcation diagrams generated in numerical experiments and by the electronic circuits.

Bifurcation diagrams in estimated parameter space using a pruned extreme learning machine

We propose an algorithm to estimate parameter spaces by using a pruned extreme learning machine, but without using principal component analysis, and we plot bifurcation diagrams in the estimated parameter spaces to visualize changes in system patterns. The estimation of parameter spaces can predict changes in the behavior of a system when its parameters are changed. It can be very helpful to adjust the optimal parameters of unknown systems. In this paper, we estimate the dimension of parameter spaces using the singular values of trained synaptic weights and the parameter spaces based on the method proposed by Bagarinao et al., using a pruned extreme learning machine. We motivate this use of a pruned extreme learning machine through numerical experiments, with the estimation of the dimension of parameter spaces for various systems, and we show that the proposed method can successfully plot bifurcation diagrams in the estimated parameter spaces. In addition, we show the results of a bifurcation diagram in the estimated parameter space for maps that are nonlinear-in-parameters, since Bagarinao et al. limited their method to linear-in-parameters maps. In particular, we estimate the parameter spaces for a mathematical model of induction motor drives and for a model of vegetation biomass in ecosystems, which are both nonlinear-in-parameter maps corresponding to real-world systems.

Nanoshells for in vivo imaging using two-photon excitation microscopy

Gold nanoshells have been intensively investigated and applied to various biomedical fields because of their flexible optical tunability and biological compatibility. They hold great potential to serve as luminescent contrast agents excitable with near-infrared (NIR) lasers. In this paper, we describe the development of nanoshells with a peak of plasmon resonance at 800 nm and their subsequent use for in vivo blood vessel imaging using two-photon excitation microscopy at an excitation wavelength of 750 nm. We were able to image single nanoshell particles in blood vessels and generate optical contrast for blood vessel structure using luminescent signals. These results confirm the feasibility of engineering nanoshells with controlled optical properties for single-particle-based in vivo imaging.