Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities

Fátima Drubi; Santiago Ibáñez; David Rivela

doi:10.3934/dcdsb.2019256

Article Contents

2020, Volume 25, Issue 2: 599-615. Doi: 10.3934/dcdsb.2019256

This issue Previous Article Well-posedness results for fractional semi-linear wave equations Next Article Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions

Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities

Department of Mathematics, University of Oviedo, Federico García Lorca 18, 33007 Oviedo, Spain

^* Corresponding author: mesa@uniovi.es
^* Corresponding author: mesa@uniovi.es

Received: January 31, 2019

Revised: April 30, 2019

Early access: November 2019

Published: February 2020

F. Drubi and S. Ibáñez are supported by the Spanish MICINN grant MTM2017-87697-P

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Abstract

A discussion on local bifurcations of codimension one and two is presented for generic unfoldings of Hopf-Bogdanov-Takens singularities of codimension three. Among all identified bifurcations, we focus on Hopf-Zero and Hopf-Hopf bifurcations, since, in certain cases, they can explain the emergence of chaotic dynamics. Moreover, numerical simulations are provided to illustrate that strange attractors appear at least when the second order normal form of the unfolding is considered.

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Mathematics Subject Classification: Primary: 58K45; Secondary: 37C10, 58K50.

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Figure 1. Primary bifurcations in the unfolding of a HBT singularity. A sphere $ \lambda_1^2+\lambda_2^2+\mu^2 = \delta $ is fixed and the front view corresponds to $ \lambda_1<0 $. Hence, points (respectively lines) correspond to bifurcation curves (respectively surfaces). We assume that $ \widehat C_{1, 1} <0 $

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Figure 3. Examples of period-doubling and chaotic dynamics close to a Hopf-Zero bifurcation. In all cases, we have $ \lambda_2 = -0.1 $, $ \mu = -0.3 $ and $ \widehat C_{1, 1} = -1 $. Initial conditions are $ x = y = u = v = 0.04 $

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Figure 5. Examples of period-doubling and chaotic dynamics close to a Hopf-Hopf bifurcation. In all cases, we have $ \lambda_2 = 0.21 $, $ \mu = -0.3 $ and $ \widehat C_{1, 1} = -1 $. Initial conditions are $ x = y = u = v = 0.04 $

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Figure 2. Green line corresponds to the segment of parameters where simulations in Figure 3 have been done. Note that we start after the birth of an invariant torus when parameters cross the secondary Hopf bifurcation. This bifurcation curve has been obtained using Matcont [13]

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Figure 4. Red line corresponds to the segment of parameters where simulations in Figure 5 have been done. Note that we start after the birth of an invariant torus when parameters cross the secondary Hopf bifurcation curve. This bifurcation curve has been obtained using Matcont [13]

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Table 1. Cases of HZ bifurcation unfolded by a HBT singularity. With the appropriate choice for coefficients in the normal form of the HBT singularity, any case can be obtained. In Case Ⅲ, it can be expected the emergence of chaotic behavior when higher order terms are considered

	$ \lambda_2\widehat C_{1, 1} <0 $	$ \lambda_2\widehat C_{1, 1}>0 $
$ \kappa=-1 $	Case II	Case I
$ \kappa=1 $	Case IV	Case III

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Table 2. Cases of HH bifurcation unfolded by a HBT singularity. In Case Ⅵa, it can be expected the emergence of chaotic behavior when higher order terms are considered

	$ \widehat C_{1, 1} <0 $	$ 0 <\widehat C_{1, 1} <\frac{1}{4} $	$ \frac{1}{4} <\widehat C_{1, 1} <\frac{1}{2} $	$ \frac{1}{2} <\widehat C_{1, 1} $
$ \kappa=-1 $	Case IVb	Case VIIa	Case VIIb	Case V
$ \kappa=1 $	Case VIa	Case Ib	Case Ia	Case III

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