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This chapter is concerned with a problem to identify the structures of linear NW systems from input/output data, where the systems are supposed to possess a smaller number of actuators/sensors compared with the number of the nodes. In this chapter, two researches on such a NW structure identification are introduced with their comparison: One is a study proposing a characteristic-polynomial-based method using knock-out procedure, and the other is a study on the reconstruction of the dynamical structure function of the NW system.

Stabilization of unstable periodic orbits within a given chaotic hybrid dynamical system is realized by a variable threshold value. In the conventional chaos control methods, a control input is proportional to the difference between the target orbit and the current state and it is added to a specific system parameter or the state as a small perturbation. Thus the whole system consumes certain control energy even if small as the amount of such input values during the transition state. We propose a new control method that changing the threshold value dynamically to stabilize the chaotic orbit. No actual control input is added into the system unlike the OGY method and the delayed feedback control. When the orbit hits the threshold, the state-feedback only determines the next threshold value to convey the controlled orbit to the target unstable periodic orbit eventually. Thus the orbit starting from the current threshold value reaches the next controlled threshold value without any direct control energy. We obtain the variation of the threshold value from the composite Poincare map, and the controller is designed by the linear feedback theory with this variation. We demonstrate this method in simple hybrid chaotic systems and show its control performances with evaluating basins of attraction.

We propose a new concept of dynamic robust bifurcation analysis for uncertain dynamical systems. An uncertain system is described by a feedback form composed of a nonlinear dynamical system and a dynamic uncertainty that is defined by a set of differential equations. For such an uncertain system, a bifurcation point can be uncertain as well. Therefore, we formulate a dynamic robust bifurcation analysis problem of identifying the set of all potential bifurcation points. To this end, first, we study equilibrium analysis to evaluate the existence and location of equilibria. Next, we derive a condition for robust hyperbolicity of the evaluated set of potential equilibrium points. On the basis of the condition, we propose a method for identifying the set of potential bifurcation points. Finally, illustrative examples for robustness analysis of normal forms for various bifurcations are presented.

This chapter proposes a clustered model reduction method for interconnected linear systems evolving over bidirectional networks. This model reduction method belongs to a kind of structured model reduction methods, where network clustering, namely clustering of subsystems, is performed according to a notion of uncontrollability of local states. We refer to this notion of uncontrollability as cluster reducibility, which can be captured by a coordinate transformation called positive tridiagonal transformation in an algebraic manner. In this chapter, it is shown that the aggregation of the reducible clusters retains the stability of the original system as well as an interconnection topology among clustered subsystems. Furthermore, an \(\fancyscript{H}_{\infty }\)-error bound is derived for the state discrepancy due to the cluster aggregation. The efficiency of the clustered model reduction is shown through an example of large-scale complex networks.

In this chapter, robust bifurcation analysis for nonlinear parameterized dynamical systems is introduced. It provides a direct method for finding the values of the system parameters at which the system has a steady-state with a high degree of stability using an optimization method. The approach is based on the idea of characterising the degree of stability as a function of the parameters. As a result, we could design a system that was robust to unexpected factors such as environmental changes and major incidents in applications. The optimization method is verified by using numerical experiments. An example of numerical results obtained in a model of the ventricular muscle cell suggests that our method can suppress the alternans and reduce the risk of sudden death.