Control of the Planar Takens–Bogdanov Bifurcation with Applications

Carrillo, Francisco A.; Verduzco, Fernando

doi:10.1007/s10440-008-9272-9

Cited by 10 publications

(9 citation statements)

References 10 publications

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“…It is interesting to remark that the relation θ(t) = θ(t) + 2πk (k = 0, ±1, ±2,..) must be taken into account in the numerical integration of Eqs (14), whereas Eqs (15) can be directly solved using the appropriate initial conditions. In accordance 8 with Eqs (13), Fig 2 shows …”

Section: Mathematical Modelsupporting

confidence: 72%

“…Eqs (8), (14) and (15) will be used in the analysis of the BT [4][5][6][7][8][9][10][11][12] and APH bifurcations [13][14][15][16][17][18][19][20]. It is interesting to remark that the relation θ(t) = θ(t) + 2πk (k = 0, ±1, ±2,..) must be taken into account in the numerical integration of Eqs (14), whereas Eqs (15) can be directly solved using the appropriate initial conditions.…”

Section: Mathematical Modelmentioning

confidence: 99%

“…To investigate this issue we apply the general Eqs (15), which account for a possible external disturbance T d (t) and an harmonic input θ r (t) defined by Eqs (8). In particular, taking into account the state variables defined in Eqs (16) , which leads to the appearance of chaotic behavior.…”

Section: Analysis Of the Chaotic Behaviormentioning

confidence: 99%

“…For example, the Bogdanov-Takens (BT) bifurcation (i.e. when the matrix of the linear part of the system has two null eigenvalues) [4][5][6][7][8][9][10][11][12] or the Andronov-Poincaré-Hopf (APH) bifurcation (which appears if the matrix of the linear part of the system has two pure imaginary eigenvalues) [13][14][15][16][17][18][19][20] has not been of common use in the resolution of problems associated to nonlinear oscillations in complex systems.…”

Section: Introductionmentioning

confidence: 99%

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Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

Pérez-Molina

Pérez-Polo

2014

Communications in Nonlinear Science and Numerical Simulation

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Please cite this article as: Pérez-Molina, M., F.Pérez-Polo, M., Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov-Takens and Andronov-Poincaré-Hopf bifurcations, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http://dx.doi.org/ 10. 1016/j.cnsns.2014.03.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractThis paper analyzes a controlled servomechanism with feedback and a cubic nonlinearity by means of the Bogdanov-Takens and Andronov-Poincaré-Hopf bifurcations, from which steady-state, self-oscillating and chaotic behaviors will be investigated using the center manifold theorem. The system controller is formed by a Proportional plus Integral plus Derivative action (PID) that allows to stabilize and drive to a prescribed set point a body connected to the shaft of a DC motor. The BogdanovTakens bifurcation is analyzed through the second Lyapunov stability method and the harmonic-balance method, whereas the first Lyapunov value is used for the AndronovPoincaré-Hopf bifurcation. On the basis of the results deduced from the bifurcation analysis, we show a procedure to select the parameters of the PID controller so that an arbitrary steady-state position of the servomechanism can be reached even in presence of noise. We also show how chaotic behavior can be obtained by applying a harmonical external torque to the device in self-oscillating regime. The advantage of achieving chaotic behavior is that it can be used so that the system reaches a set point inside a strange attractor with a small control effort. The analytical calculations have been verified through detailed numerical simulations.

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Section: Mathematical Modelsupporting

confidence: 72%

Section: Mathematical Modelmentioning

confidence: 99%

Section: Analysis Of the Chaotic Behaviormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

Pérez-Molina

Pérez-Polo

2014

Communications in Nonlinear Science and Numerical Simulation

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“…Different control systems were designed in order to create different types of bifurcation and to manipulate the bifurcation characteristics such as the stability and orientation of limit cycles or the stability of equilibria [2].…”

Section: Introductionmentioning

confidence: 99%

Control of planar Bautin bifurcation

Braga¹,

Mello²,

Rocşoreanu³

et al. 2010

Nonlinear Dyn

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On a versal deformation of the Bautin bifurcation it is possible to find dynamical systems that undergo Hopf or non-hyperbolic limit cycle bifurcations. Our paper concerns a nonlinear control system in the plane whose nominal vector field has a pair of purely imaginary eigenvalues. We find conditions to control the Hopf and Bautin bifurcation using the symmetric multilinear vector functions that appear in the Taylor expansion of the vector field around the equilibrium. The control law designed by us depends on two bifurcation parameters and four control parameters, which establish the stability of the equilibrium point and the orientation and stability of the limit cycles. Two examples are given.

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Bogdanov-Takens Bifurcation in a Leslie Type Tritrophic Model with General Functional Responses

Blé¹,

Dela–Rosa²

2019

Acta Appl Math

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Control of the Planar Takens–Bogdanov Bifurcation with Applications

Cited by 10 publications

References 10 publications

Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

Control of planar Bautin bifurcation

Bogdanov-Takens Bifurcation in a Leslie Type Tritrophic Model with General Functional Responses

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