Anti-control of Hopf bifurcations

Chen, D.S.; Wang, H.O.; Chen, G.

doi:10.1109/81.928149

Cited by 78 publications

(30 citation statements)

References 26 publications

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“…The created Hopf circle is stable if β (μ 0 , K 2 , K 3 ) < 0 and unstable if β (μ 0 , K 2 , K 3 ) > 0. From the formulas (13)-(26), we can see the stability condition (15) can be computed by the coefficients of the series expression of system (13). Therefore, after the analytical expression of β (μ 0 , K 2 , K 3 ) with respect to K 2 , K 3 is derived, the stability of the created Hopf bifurcation solutions can be desirably manipulated by choosing K 2 , K 3 to adjust the sign of β (μ 0 , K 2 , K 3 ).…”

Section: The Nonlinear Control Gains For the Type And Stability Of Bimentioning

confidence: 99%

“…In general, the objective of bifurcation control is to modify the bifurcation characteristics of a nonlinear system by delaying the onset of an inherent bifurcation [10] and modifying the amplitudes [11] and stability [12] of bifurcated solutions. Besides that, the inverse problem of conventional bifurcation analysis, i.e., the creation of a certain bifurcation with desired characters, has been reported [13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

“…In recent years, the attention has been paid to create Hopf bifurcation and design oscillatory solution of limit circle with specified properties by control. Chen et al [13] studied the problem of anti-control of Hopf bifurcation and suggested a new approach to design limit cycles with specified oscillatory behaviors via feedback control. Wen et al [17][18][19] studied the creation of Hopf bifurcation of discretetime systems in nonresonance and resonance cases and also investigated the control of degenerate Hopf Bifurcation in discrete-time systems.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Anti-controlling Hopf bifurcation in a type of centrifugal governor system

Wen

et al. 2015

Nonlinear Dyn

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Anti-controlling Hopf bifurcation is considered as one way to design Hopf limit circle into a dynamical system, and the oscillatory behavior of Hopf limit circle can be beneficial in many practical applications such as mixing, low-energy navigation control and fault diagnosis. In this paper, the feedback control problem of designing Hopf bifurcation in a centrifugal governor system is addressed. A feedback control method is proposed to achieve three aspects of controlling problem including existence, stability, and adjusting amplitude and frequency of the limit cycle to be designed. An explicit criterion of Hopf bifurcation including eigenvalue assignment and transversality conditions, without using eigenvalue computation, is utilized to derive the linear gains responsible for control of the bifurcation existence. The center manifold theory and normal form reduction is utilized to derive the nonlinear gains responsible for control of the stability of the created limit circle. The expressions of the approximate amplitude and frequency of the limit cycle are developed to derive the nonlinear gains responsible for controls of amplitude and frequency of the limit

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Section: The Nonlinear Control Gains For the Type And Stability Of Bimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Anti-controlling Hopf bifurcation in a type of centrifugal governor system

Wen

et al. 2015

Nonlinear Dyn

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“…H OPF bifurcation has been found or synthesized in numerous circuits and systems [2], [4], [10]- [12]. For the past two decades, stabilizing control of bifurcated systems has drawn a lot of attention from the control community [1], [6], [8], [13].…”

Section: Introductionmentioning

confidence: 99%

Local Robustness of Hopf Bifurcation Stabilization

Yang

Chen

2009

IEEE Trans. Circuits Syst. II

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Local robust analysis via 2 gain method is presented for a class of Hopf bifurcation stabilizing controllers. In particular, we first construct a family of Lyapunov functions for the corresponding critical system, then derive a sufficient condition to compute the 2 gain by solving the Hamilton-Jacobi-Bellman (HJB) inequalities. Local robust analysis can be conducted through computing the local 2 gain achieved by the stabilizing controllers at the critical situation. The theoretical results obtained in this brief provide useful guidance for selecting a robust controller from a given class of stabilizing controllers under Hopf bifurcation. As an example, application to a modified Van der Pol oscillator is discussed in details.

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“…In order to utilize the characteristics of limit circles, many researchers have been attracted to the inverse problem of the classical Hopf bifurcation control, i.e., anti-control of Hopf bifurcation that serves as a way to actively designing limit circles into a dynamical system. Over the past years, some methods [6][7][8] have been developed to create Hopf bifurcation at a desired parameter point but the amplitudes of the bifurcated limit circle are not operable in an explicit form. The research on control of limit circle amplitude [9] has attracted a great deal of attention.…”

mentioning

confidence: 99%

Designing Hopf limit circle to dynamical systems via modified projective synchronization

Wen

2010

Nonlinear Dyn

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In order to affirmatively utilize the characteristics of Hopf limit circle, a control method to design Hopf circle with proper characteristics into dynamical system is established based on the modified projective synchronization (MPS). The proposed method may serve as a complete solution to design a stable Hopf limit circle, which can simultaneously achieve the following three properties: with the desired amplitudes and shape changes, with the pre-specified location center, and at a pre-specified system parameter location. In contrast to the methods based on Hopf bifurcation theory, the new method is independent of the verbose procedures for the bifurcation critical conditions and the stability analysis. Numerical examples demonstrate the effectiveness of the proposed method.

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Anti-control of Hopf bifurcations

Cited by 78 publications

References 26 publications

Anti-controlling Hopf bifurcation in a type of centrifugal governor system

Anti-controlling Hopf bifurcation in a type of centrifugal governor system

Local Robustness of Hopf Bifurcation Stabilization

Designing Hopf limit circle to dynamical systems via modified projective synchronization

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