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A kneading map of chaotic switching oscillations in a Kerr cavity with two interacting light fields

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Title: A kneading map of chaotic switching oscillations in a Kerr cavity with two interacting light fields
Authors: Bitha, Rodrigues D. Dikandé, Giraldo, Andrus, Broderick, Neil G. R., Krauskopf, Bernd
Publication Year: 2024
Collection: Mathematics
Mathematical Physics
Nonlinear Sciences
Physics (Other)
Subject Terms: Physics - Optics, Mathematical Physics, Mathematics - Dynamical Systems, Nonlinear Sciences - Chaotic Dynamics
More Details: Optical systems that combine nonlinearity with coupling between various subsystems offer a flexible platform for observing a diverse range of nonlinear dynamics. Furthermore, engineering tolerances are such that the subsystems can be identical to within a fraction of the wavelength of light; hence, such coupled systems inherently have a natural symmetry that can lead to either delocalization or symmetry breaking. We consider here an optical Kerr cavity that supports two interacting electric fields, generated by two symmetric input beams. Mathematically, this system is modeled by a four-dimensional $\mathbb{Z}_2$-equivariant vector field with the strength and detuning of the input light as control parameters. Previous research has shown that complex switching dynamics are observed both experimentally and numerically across a wide range of parameter values. Here, we show that particular switching patterns are created at specific global bifurcations through either delocalization or symmetry breaking of a chaotic attractor. We find that the system exhibits infinitely many of these global bifurcations, which are organized by $\mathbb{Z}_2$-equivariant codimension-two Belyakov transitions. We investigate these switching dynamics by means of the continuation of global bifurcations in combination with the computation of kneading invariants and Lyapunov exponents. In this way, we provide a comprehensive picture of the interplay between different switching patterns of periodic orbits and chaotic attractors.
Document Type: Working Paper
Access URL: http://arxiv.org/abs/2410.23588
Accession Number: edsarx.2410.23588
Database: arXiv
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  Data: A kneading map of chaotic switching oscillations in a Kerr cavity with two interacting light fields
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  Data: <searchLink fieldCode="AR" term="%22Bitha%2C+Rodrigues+D%2E+Dikandé%22">Bitha, Rodrigues D. Dikandé</searchLink><br /><searchLink fieldCode="AR" term="%22Giraldo%2C+Andrus%22">Giraldo, Andrus</searchLink><br /><searchLink fieldCode="AR" term="%22Broderick%2C+Neil+G%2E+R%2E%22">Broderick, Neil G. R.</searchLink><br /><searchLink fieldCode="AR" term="%22Krauskopf%2C+Bernd%22">Krauskopf, Bernd</searchLink>
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  Data: 2024
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  Data: Mathematics<br />Mathematical Physics<br />Nonlinear Sciences<br />Physics (Other)
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  Data: <searchLink fieldCode="DE" term="%22Physics+-+Optics%22">Physics - Optics</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Physics%22">Mathematical Physics</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+-+Dynamical+Systems%22">Mathematics - Dynamical Systems</searchLink><br /><searchLink fieldCode="DE" term="%22Nonlinear+Sciences+-+Chaotic+Dynamics%22">Nonlinear Sciences - Chaotic Dynamics</searchLink>
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  Data: Optical systems that combine nonlinearity with coupling between various subsystems offer a flexible platform for observing a diverse range of nonlinear dynamics. Furthermore, engineering tolerances are such that the subsystems can be identical to within a fraction of the wavelength of light; hence, such coupled systems inherently have a natural symmetry that can lead to either delocalization or symmetry breaking. We consider here an optical Kerr cavity that supports two interacting electric fields, generated by two symmetric input beams. Mathematically, this system is modeled by a four-dimensional $\mathbb{Z}_2$-equivariant vector field with the strength and detuning of the input light as control parameters. Previous research has shown that complex switching dynamics are observed both experimentally and numerically across a wide range of parameter values. Here, we show that particular switching patterns are created at specific global bifurcations through either delocalization or symmetry breaking of a chaotic attractor. We find that the system exhibits infinitely many of these global bifurcations, which are organized by $\mathbb{Z}_2$-equivariant codimension-two Belyakov transitions. We investigate these switching dynamics by means of the continuation of global bifurcations in combination with the computation of kneading invariants and Lyapunov exponents. In this way, we provide a comprehensive picture of the interplay between different switching patterns of periodic orbits and chaotic attractors.
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    Subjects:
      – SubjectFull: Physics - Optics
        Type: general
      – SubjectFull: Mathematical Physics
        Type: general
      – SubjectFull: Mathematics - Dynamical Systems
        Type: general
      – SubjectFull: Nonlinear Sciences - Chaotic Dynamics
        Type: general
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      – TitleFull: A kneading map of chaotic switching oscillations in a Kerr cavity with two interacting light fields
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            NameFull: Bitha, Rodrigues D. Dikandé
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            NameFull: Giraldo, Andrus
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            NameFull: Broderick, Neil G. R.
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            NameFull: Krauskopf, Bernd
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              Type: published
              Y: 2024
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