In this paper, I review recent progress in studies on mathematical modelling of complex systems from a general viewpoint. First, I explain our theoretical platform composed of (1) advanced control theory of complex systems,(2) complex networks theory, and (3) nonlinear data analysis and data-driven modelling, that has been developed for mathematical modelling of complex systems. Second, I introduce recent various applications based on the theoretical platform. Finally, I discuss possible future directions of this research on the mathematical modelling of complex systems.

Twenty years has passed since the book of “The Impact of Chaos on Science and Society,” edited by Prof. Celso Grebogi and Prof. James A. Yorke, has been published. This book had influenced the researches held during the FIRST program and have been influencing the current researches following them. Thus, I would like to summarize how the questions posed in the book have been answered partly by our current generation and what questions and the other emerging questions should be considered by the next generation.

In this paper, we analyze the relation between the stability of a noisy dynamical system based on linear approximation and the covariance matrix of its stationary distribution. We reformulate the theory of dynamical network biomarkers in terms of the covariance matrix and clarify the limiting behavior of the covariance matrix when a dynamical system approaches a bifurcation point. We also discuss the relation between the Jacobian matrix and principal component analysis. An application to a simple nonlinear network model is also demonstrated.

This paper addresses quantitative performance analysis of an interconnected passive system. The passivity property, which provides a unified and abstracted description of dynamical systems, plays an important role for qualitative stability analysis of interconnected large-scale systems. In this paper, quantitative performance is further evaluated for the interconnected passive system. To this end, a performance-characterizing parameter is integrated into the conventional passivity. Then, by using the parameter, the L₂-gain of the entire interconnected system is evaluated. Further assuming that the interconnection rule is described by a graph, more detailed performance analysis and its reinforcement via network expansion are studied.

Time delayed feedback can control chaotic motion to periodic motion by stabilizing an unstable periodic orbit that is embedded in a chaotic set. The time delayed feedback control method can be applied to a non-stationary stochastic process with Gaussian noise, and it can control the diffusion processes in a stochastic system. In this study, we apply time delayed feedback control to the diffusion processes in systems with noise that is more complicated than Gaussian noise, e.g., diffusions induced by chaotic noise generated by chaotic dynamical systems, a logistic map and a one-dimensional piecewise-linear map.

An exact linearization method using static input transformations is proposed for third-order continuous-time systems with pulse-width-modulation-type inputs whose eigenvalues are real and distinct. In this method, two rectangular waves are located in each control interval, and their rising/falling timings are treated as the control parameters. This paper demonstrates that the image of the nonlinear input map contains the image of a suitable linear map, which means that some input transformations exist for linearization.

Pole assignment control for stabilizing a quasi-periodic orbit in a discrete-time dynamical system has been previously proposed. In this paper, the pole assignment method is applied to a switched-capacitor chaotic neural network circuit. For circuit experiments in which there are unknown circuit characteristics and parameters, and inevitable noise, the control method is modified by introducing new control input signals. As a result, the quasi-periodic orbits are successfully stabilized through pole assignment control. In order to confirm the quasi-periodicity of the obtained orbits, bifurcation diagrams and phase-plane portraits are provided. In addition, a statistical test designed for noisy experimental data, in particular, further confirms the quasi-periodicity. Through circuit experiments, the feasibility, usefulness, efficacy, and robustness of pole assignment control for quasi-periodic orbits are verified.

We propose a statistical method to test whether time series data are quasiperiodic or periodic. A time series is defined to be quasiperiodic if its generating map is topologically conjugate to an irrational rotation. We present an algorithm to estimate the conjugacy map from time series data. We also show that a noisy irrational rotation is equivalent to a random walk model. Since there are general tests for a random walk, we can statistically evaluate quasiperiodicity of a time series by using the estimated conjugacy map. The proposed method is validated by asymmetric chaotic neural networks controlled to be quasiperiodic.

Using the theory of Fredholm determinants of Perron-Frobenius operators of piecewise-linear maps, we derive a mathematically rigorous upper bounds of mean squared errors of analog-to-digital converters based on β-expansions. We also explain the technique of calculating the upper bound numerically by means of the numerical verification method.

Bajwa et al. proposed a channel estimation method based on compressed sensing. This method is markedly superior to the conventional methods. However, there is a problem in the method that multi-path delays may not be resolved if they span between the grids. We study to overcome the drawback of the method. Firstly, we investigate upsampled codes so that we could more accurately estimate the channel. Secondly, we investigate Markov codes. It was shown that Spread Spectrum (SS) codes with negative autocorrelation reduces the Multiple Access Interference (MAI) as well as Bit Error Rate (BER) in chip-asynchronous Spread Spectrum Multiple Access (SSMA) systems. Such SS codes are generated from a Markov chain whose transition probability matrix has a negative eigenvalue λ = -2 + √3. The mean square error (MSE) of channel estimation using upsampled Markov code is shown to be better than the MSE of channel estimation using upsampled independent and identically distributed code at certain conditions.

We propose a coupled-map-lattice complementary-metal-oxide-semiconductor very-large-scale-integration (CMOS VLSI) circuit based on the threshold-coupled map (TCM) that has been proposed previously as a unidirectional connected network model exhibiting different spatiotemporal patterns according to its underlying nonlinear map and update scheme. We introduce mutual connections and arbitrarily valued connection weights into the TCM to realize cellular automata. In this study, we design, fabricate, and evaluate a CMOS integrated circuit with which to implement this extended TCM (ETCM). The ETCM is a universal Turing machine as confirmed in circuit experiments using the fabricated circuit, which can achieve Rule110 of a one-dimensional cellular automaton.

Reconstructing accurately the structure of neural networks from biological data is essential for the analysis of simultaneous recordings from many neurons, and, in turn, for the understanding of neural codes and the design of neural prostheses. Classical techniques are generally based on cross-correlations and cannot reconstruct unambiguously the network structure. Recently, we have proposed a method for which there is one-to-one correspondence between statistical properties of packets of spikes (or avalanches) and the network structure, but this mapping was only proven for simpler neuronal model. In the following, we show using numerical simulation of the Izhikevich model that the proposed method is general, and is particularly well-fitted for the analysis of neural activity recorded from cultured neuronal networks coupled to microelectrode arrays.

Despite of several complementary mechanisms that are inherently equipped with the immune system to eliminate tumor cells, some adaptive change of tumor cells such as down-regulating expression level of Major HistoCompatibility (MHC) class I to evade killing from T cells can be beneficial. Description of complex interactions among tumor and immune cells is useful to understand how some of adaptive change are effective for tumor cells to escape from immunosurveillance. In this paper, we formulate a mathematical model representing tumor killing by two different immune subsets to investigate under what conditions down-regulation of MHC class I expression can be favorable for tumor cells. Mathematical analysis and numerical computation results suggest that tumor cells prefer down-regulation of MHC class I if the killing of tumor cells by T cells is more potent than NK cells. This implication may support empirical observations that tumor escape via down-regulation of MHC class I expression can commonly occur.