Robust Bifurcation Analysis Based on Degree of Stability

Kohno, Hiroyuki; Yoshinaga, Tetsuya; Imura, Jun‐ichi; Aihara, Kazuyuki

doi:10.1007/978-4-431-55013-6_2

Cited by 3 publications

(9 citation statements)

References 10 publications

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“…For those systems and other time-delay systems with many factors (arbitrary, restricted switchings [16]; nonlinearities [24,28]; network-induced phenomena [5,17,28], 165 etc. ), future work will focus on such topics as bifurcation analysis [10] and exponential synchronization [19] in addition to stability. …”

Section: Remarkmentioning

confidence: 99%

“…It frequently occurs 5 in many practical systems, such as networked control systems [5,17,28], genetic regulatory networks [18,19], and power systems [29]. So, the stability analysis of time-delay systems has aroused a great deal of interest, and various approaches A C C E P T E D M A N U S C R I P T have been devised in the past few decades, such as model transformation [2], Park's inequality [12], Moon's inequality [11], Jensen's inequality [8,9,14], and 10 the free-weighting matrix technique [6,7].…”

Section: Introductionmentioning

confidence: 99%

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Stability analysis for discrete time-delay systems based on new finite-sum inequalities

Wan

et al. 2016

Information Sciences

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Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stability analysis for discrete time-delay systems based on new finite-sum inequalities

Wan

et al. 2016

Information Sciences

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“…Thus, local bifurcations of stable fixed and periodic points occur when the characteristic multiplier of the maximum modulus is on the circumference of the unit circle. Kitajima et al [18] defined the stability index on fixed and periodic points using the characteristic multiplier of the maximum modulus.…”

Section: Bifurcation Of Fixed and Periodic Pointsmentioning

confidence: 99%

“…Local bifurcations of stable, hyperbolic fixed and periodic points occur when the maximum modulus of characteristic multipliers becomes one. Kitajima et al [18] defined the degree of stability (the stability index) on hyperbolic fixed and periodic points using the characteristic multiplier of the maximum modulus and proposed updating parameter values to optimize the stability index. It is expected that this method can be used to construct robust dynamical systems that are controlled so as not to make bifurcations.…”

Section: Introductionmentioning

confidence: 99%

“…the maximum characteristic multiplier) is not differentiable with respect to system parameters in general, simple gradient methods cannot be used to optimize it directly. In addition, this method [18] needs off-line calculation with a high computational cost to find the exact positions of fixed and periodic points and their characteristic multipliers whenever parameter values change. This means that the parametric control cannot be realized along the passage of time.…”

Section: Introductionmentioning

confidence: 99%

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Bifurcation avoidance control of stable periodic points using the maximum local Lyapunov exponent

Fujimoto

Aihara

2015

NOLTA

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We propose a novel controlling methodology for avoiding local bifurcations of stable, hyperbolic fixed and periodic points in nonlinear discrete-time dynamical systems with parameter variation. Dynamical systems may not work as expected owing to bifurcations of desired behavior caused by parameter variation. Controlling parameters to avoid bifurcations enables us to construct robust dynamical systems against unexpected parameter variation. Assuming that desired behavior is a hyperbolic fixed or periodic point, as a control strategy, optimizing the degree of its stability (i.e. the maximum modulus of its characteristic multipliers) is considerable. However, it cannot be optimized by simple gradient methods, and off-line calculation to find the exact positions of fixed and periodic points and their characteristic multipliers is also needed. In contrast, the method we propose is to control the maximum local Lyapunov exponent (MLLE) that is defined in finite time and is closely related to the degree of stability on hyperbolic fixed and periodic points. This method can predict occurrence of local bifurcations for parameter variation by monitoring the MLLE along the passage of time and adjust parameter values on line to control the MLLE. This leads that occurrence of local bifurcations can be avoided without directly analyzing bifurcations in advance. The parameter regulation to avoid local bifurcations is theoretically derived from the minimization of an objective function with respect to the MLLE. Our experimental results applied to the Kawakami and Hénon maps demonstrate that the proposed controller could be used to avoid local bifurcations.

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Robustification of a Nonlinear Dynamical System with a Stability Index and a Matrix Inequality

Oishi

Kobayashi

Yoshinaga

2015

SICE Journal of Control, Measurement, and System Integration

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A problem in the field of nonlinear dynamics is considered with a technique of control theory. In particular, for preventing a nonlinear dynamical system from making undesirable bifurcation, it is proposed to control its parameter with a stability index and a matrix inequality. A basic idea is to update the parameter of the system so as to minimize the stability index. Since the stability index is not differentiable as a function of the parameter, its minimization is carried out through equivalent transformation to a smooth minimization problem, where a matrix inequality is used. The proposed method is applied to a simple dynamical system and shown to be effective. Its generalization is also considered for driving a system to avoid chaos.

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Robust Bifurcation Analysis Based on Degree of Stability

Cited by 3 publications

References 10 publications

Stability analysis for discrete time-delay systems based on new finite-sum inequalities

Stability analysis for discrete time-delay systems based on new finite-sum inequalities

Bifurcation avoidance control of stable periodic points using the maximum local Lyapunov exponent

Robustification of a Nonlinear Dynamical System with a Stability Index and a Matrix Inequality

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