By Kazuyuki Aihara, Jun-ichi Imura, Tetsushi Ueta

This booklet is the 1st to record on theoretical breakthroughs on keep watch over of advanced dynamical platforms constructed by means of collaborative researchers within the fields of dynamical structures conception and regulate thought. besides, its easy standpoint is of 3 varieties of complexity: bifurcation phenomena topic to version uncertainty, advanced habit together with periodic/quasi-periodic orbits in addition to chaotic orbits, and community complexity rising from dynamical interactions among subsystems. *Analysis and regulate of complicated Dynamical Systems* bargains a necessary source for mathematicians, physicists, and biophysicists, in addition to for researchers in nonlinear technology and keep watch over engineering, letting them increase a greater basic figuring out of the research and keep watch over synthesis of such complicated systems.

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**Additional info for Analysis and Control of Complex Dynamical Systems: Robust Bifurcation, Dynamic Attractors, and Network Complexity**

**Example text**

Aihara et al. 1007/978-4-431-55013-6_3 33 34 Y. Oishi et al. of Chap. 2. The obtained update rule is tested with a simple dynamical system to show its efficacy. Finally, generalization of this approach is considered for keeping the system from falling into chaos. A related result of the same authors can be found in [2]. The following notation is used. The transpose of a matrix or a vector is denoted by T . The trace of a matrix is expressed by tr and a diagonal matrix is expressed by diag. For a square matrix D, the symbol ρ(D) denotes the maximum of the absolute values of the eigenvalues of D.

Its key idea is to construct a penalty function that has a small value when the matrix inequality constraint is satisfied and a large value when it is not satisfied. Adding this penalty function to the objective function, we can perform minimization without considering the matrix inequality constraint explicitly. The desired penalty function can be constructed as follows. First, we define a function φ p (t) having a positive scheduling parameter p by φ p (t) := p 1 −1 . t/ p + 1 For t ≥ 0, the function φ p (t) approaches zero as p ↓ 0; for t < 0, it blows up to infinity as p ↓ −t.

Aihara et al. 1007/978-4-431-55013-6_5 49 50 K. Fujimoto et al. Bifurcations of stable periodic points occur when their degree of stability (stability index) defined in Chap. , their bifurcations can be avoided by suppressing the stability index below one. However, as described in Chap. 3, the optimization of the stability index has a difficulty because the stability index is not differentiable with respect to system parameters in general. In this chapter, by using the maximum Lyapunov exponent (MLE) [11, 13, 16] that is related to the stability index [4, 5, 11, 12], we present a parametric controller that can avoid bifurcations of stable periodic points for unexpected parameter variation [6].