Characterization of time-dependence for dissipative solitons stabilized by nonlinear gradient terms: Periodic and quasiperiodic vs chaotic behavior

Orazio Descalzi; M. Facão; Carlos Cartes; M. I. Carvalho; Helmut R. Brand

doi:10.1063/5.0156518

Characterization of time-dependence for dissipative solitons stabilized by nonlinear gradient terms: Periodic and quasiperiodic vs chaotic behavior

Orazio Descalzi^*, M. Facão, Carlos Cartes, M. I. Carvalho, Helmut R. Brand

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate the properties of time-dependent dissipative solitons for a cubic complex Ginzburg-Landau equation stabilized by nonlinear gradient terms. The separation of initially nearby trajectories in the asymptotic limit is predominantly used to distinguish qualitatively between time-periodic behavior and chaotic localized states. These results are further corroborated by Fourier transforms and time series. Quasiperiodic behavior is obtained as well, but typically over a fairly narrow range of parameter values. For illustration, two examples of nonlinear gradient terms are examined: the Raman term and combinations of the Raman term with dispersion of the nonlinear gain. For small quintic perturbations, it turns out that the chaotic localized states are showing a transition to periodic states, stationary states, or collapse already for a small magnitude of the quintic perturbations. This result indicates that the basin of attraction for chaotic localized states is rather shallow.

Original language	English
Article number	083151
Journal	Chaos
Volume	33
Issue number	8
DOIs	https://doi.org/10.1063/5.0156518
State	Published - 1 Aug 2023

Bibliographical note

Funding Information:
The work by M.I.C. and M.F. was financed by National Funds through the Portuguese funding agency, FCT—Fundação para a Ciência e a Tecnologia—within Project Nos. UIDB/50014/2020, UIDB/50025/2020, UIDP/50025/2020, and LA/P/0037/2020. O.D. and C.C. wish to acknowledge the support of FONDECYT (CL) (No. 1200357) and Universidad de los Andes through FAI initiatives. H.R.B. thanks the Deutsche Forschungsgemeinschaft (DE) for partial support of this work.

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© 2023 Author(s).

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abstract = "We investigate the properties of time-dependent dissipative solitons for a cubic complex Ginzburg-Landau equation stabilized by nonlinear gradient terms. The separation of initially nearby trajectories in the asymptotic limit is predominantly used to distinguish qualitatively between time-periodic behavior and chaotic localized states. These results are further corroborated by Fourier transforms and time series. Quasiperiodic behavior is obtained as well, but typically over a fairly narrow range of parameter values. For illustration, two examples of nonlinear gradient terms are examined: the Raman term and combinations of the Raman term with dispersion of the nonlinear gain. For small quintic perturbations, it turns out that the chaotic localized states are showing a transition to periodic states, stationary states, or collapse already for a small magnitude of the quintic perturbations. This result indicates that the basin of attraction for chaotic localized states is rather shallow.",

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AU - Facão, M.

AU - Cartes, Carlos

AU - Carvalho, M. I.

AU - Brand, Helmut R.

N1 - Funding Information: The work by M.I.C. and M.F. was financed by National Funds through the Portuguese funding agency, FCT—Fundação para a Ciência e a Tecnologia—within Project Nos. UIDB/50014/2020, UIDB/50025/2020, UIDP/50025/2020, and LA/P/0037/2020. O.D. and C.C. wish to acknowledge the support of FONDECYT (CL) (No. 1200357) and Universidad de los Andes through FAI initiatives. H.R.B. thanks the Deutsche Forschungsgemeinschaft (DE) for partial support of this work. Publisher Copyright: © 2023 Author(s).

PY - 2023/8/1

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N2 - We investigate the properties of time-dependent dissipative solitons for a cubic complex Ginzburg-Landau equation stabilized by nonlinear gradient terms. The separation of initially nearby trajectories in the asymptotic limit is predominantly used to distinguish qualitatively between time-periodic behavior and chaotic localized states. These results are further corroborated by Fourier transforms and time series. Quasiperiodic behavior is obtained as well, but typically over a fairly narrow range of parameter values. For illustration, two examples of nonlinear gradient terms are examined: the Raman term and combinations of the Raman term with dispersion of the nonlinear gain. For small quintic perturbations, it turns out that the chaotic localized states are showing a transition to periodic states, stationary states, or collapse already for a small magnitude of the quintic perturbations. This result indicates that the basin of attraction for chaotic localized states is rather shallow.

AB - We investigate the properties of time-dependent dissipative solitons for a cubic complex Ginzburg-Landau equation stabilized by nonlinear gradient terms. The separation of initially nearby trajectories in the asymptotic limit is predominantly used to distinguish qualitatively between time-periodic behavior and chaotic localized states. These results are further corroborated by Fourier transforms and time series. Quasiperiodic behavior is obtained as well, but typically over a fairly narrow range of parameter values. For illustration, two examples of nonlinear gradient terms are examined: the Raman term and combinations of the Raman term with dispersion of the nonlinear gain. For small quintic perturbations, it turns out that the chaotic localized states are showing a transition to periodic states, stationary states, or collapse already for a small magnitude of the quintic perturbations. This result indicates that the basin of attraction for chaotic localized states is rather shallow.

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Characterization of time-dependence for dissipative solitons stabilized by nonlinear gradient terms: Periodic and quasiperiodic vs chaotic behavior

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