Efficient manifolds tracing for planar maps

Ciro, David; Caldas, Iberê L.; Viana, Ricardo L.; Evans, Todd E.

doi:10.1063/1.5027698

Title: Efficient manifolds tracing for planar maps

Abstract

Here, we present an exact calculation method and an approximation technique for tracing the invariant manifolds of unstable periodic orbits of planar maps. The exact approach relies on an adaptive refinement procedure that prevents redundant calculations occurring in other approaches, and the approximated method relies on a novel interpolation approach based on normal displacement functions. The resulting approximated manifold is precise when compared to the exact one, and its relative computational cost falls like the inverse of the manifold length. To present the tracing method, we obtain the invariant manifolds of the Chirikov-Taylor map, and as an application we illustrate the transition from homoclinic to heteroclinic chaos in the Duffing oscillator that leads from localized chaos to global chaotic motion.

Authors:

^[1];

^[2]; Viana, Ricardo L. ^[3];

^[4]

Federal Univ. of Paraná (Brazi); Univ. of São Paulo (Brazil)
Univ. of São Paulo (Brazil)
Federal Univ. of Paraná (Brazi)
General Atomics, San Diego, CA (United States)

Publication Date:: Tue Sep 11 00:00:00 EDT 2018

Research Org.:: General Atomics, San Diego, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC)

OSTI Identifier:: 1540200

Alternate Identifier(s):: OSTI ID: 1469428

Grant/Contract Number:: FC02-04ER54698; SC0012706; FC02-04ER54698, DE- SC0012706

Resource Type:: Accepted Manuscript

Journal Name:: Chaos: An Interdisciplinary Journal of Nonlinear Science

Additional Journal Information:: Journal Volume: 28; Journal Issue: 9; Journal ID: ISSN 1054-1500

Publisher:: American Institute of Physics (AIP)

Country of Publication:: United States

Language:: English

Subject:: 74 ATOMIC AND MOLECULAR PHYSICS

Citation Formats


                    Ciro, David, Caldas, Iberê L., Viana, Ricardo L., and Evans, Todd E. Efficient manifolds tracing for planar maps.  United States: N. p., 2018. 
Web.  doi:10.1063/1.5027698.

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                    Ciro, David, Caldas, Iberê L., Viana, Ricardo L., & Evans, Todd E. Efficient manifolds tracing for planar maps.  United States.  https://doi.org/10.1063/1.5027698

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                    Ciro, David, Caldas, Iberê L., Viana, Ricardo L., and Evans, Todd E. Tue .  
"Efficient manifolds tracing for planar maps".  United States.  https://doi.org/10.1063/1.5027698.  https://www.osti.gov/servlets/purl/1540200.

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@article{osti_1540200,

  title        = {Efficient manifolds tracing for planar maps},

  author       = {Ciro, David and Caldas, Iberê L. and Viana, Ricardo L. and Evans, Todd E.},

  abstractNote = {Here, we present an exact calculation method and an approximation technique for tracing the invariant manifolds of unstable periodic orbits of planar maps. The exact approach relies on an adaptive refinement procedure that prevents redundant calculations occurring in other approaches, and the approximated method relies on a novel interpolation approach based on normal displacement functions. The resulting approximated manifold is precise when compared to the exact one, and its relative computational cost falls like the inverse of the manifold length. To present the tracing method, we obtain the invariant manifolds of the Chirikov-Taylor map, and as an application we illustrate the transition from homoclinic to heteroclinic chaos in the Duffing oscillator that leads from localized chaos to global chaotic motion.},

  doi          = {10.1063/1.5027698},

  journal      = {Chaos: An Interdisciplinary Journal of Nonlinear Science},

  number       = 9,

  volume       = 28,

  place        = {United States},

  year         = {Tue Sep 11 00:00:00 EDT 2018},

  month        = {Tue Sep 11 00:00:00 EDT 2018}

}

Copy to clipboard

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Title: Efficient manifolds tracing for planar maps

Abstract

Citation Formats

An Algorithm for Least-Squares Estimation of Nonlinear Parameters journal, June 1963

High-Order Bisection Method for Computing Invariant Manifolds of Two-Dimensional maps journal, July 2011

A Survey of Methods for Computing (Un)Stable Manifolds of Vector Fields journal, March 2005

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Heat flux modeling using ion drift effects in DIII-D H-mode plasmas with resonant magnetic perturbations journal, January 2014

Stagger-and-Step Method: Detecting and Computing Chaotic Saddles in Higher Dimensions journal, March 2001

Growing 1D and Quasi-2D Unstable Manifolds of Maps journal, October 1998

Modeling non-stationary, non-axisymmetric heat patterns in DIII-D tokamak journal, October 2016

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Field-induced barrier penetration in the quartic potential journal, April 1984

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields book, January 1983

A procedure for finding numerical trajectories on chaotic saddles journal, June 1989

Escape patterns, magnetic footprints, and homoclinic tangles due to ergodic magnetic limiters journal, December 2002

Modeling of stochastic magnetic flux loss from the edge of a poloidally diverted tokamak journal, December 2002

Fractal structures in nonlinear dynamics journal, March 2009

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Symplectic maps, variational principles, and transport journal, July 1992

Traces of stable and unstable manifolds in heat flux patterns journal, April 2007

A universal instability of many-dimensional oscillator systems journal, May 1979

Numerical calculation of invariant manifolds for maps journal, March 1994

The PIM-simplex method: an extension of the PIM-triple method to saddles with an arbitrary number of expanding directions journal, February 1999

Magnetic field lines, Hamiltonian dynamics, and nontwist systems journal, June 2000

Escape patterns of chaotic magnetic field lines in a tokamak with reversed magnetic shear and an ergodic limiter journal, September 2008

An Efficient Method for Computing Invariant Manifolds of Planar Maps journal, January 1993

Experimental signatures of homoclinic tangles in poloidally diverted tokamaks journal, January 2005

Regular and Chaotic Dynamics book, January 1992

Quantum dynamical tunneling in bound states journal, July 1981

Repellers, semi-attractors, and long-lived chaotic transients journal, August 1985

The Plasma Boundary of Magnetic Fusion Devices book, January 2000

An efficient method for computing invariant manifolds of planar maps journal, August 1992

Datasets of seed mucilage traits for Arabidopsis thaliana natural accessions with atypical outer mucilage journal, March 2021

The Plasma Boundary of Magnetic Fusion Devices journal, January 2001

An Algorithm for Least-Squares Estimation of Nonlinear Parameters
journal, June 1963

High-Order Bisection Method for Computing Invariant Manifolds of Two-Dimensional maps
journal, July 2011

A Survey of Methods for Computing (Un)Stable Manifolds of Vector Fields
journal, March 2005

The parameterization method for invariant manifolds III: overview and applications
journal, November 2005

Heat flux modeling using ion drift effects in DIII-D H-mode plasmas with resonant magnetic perturbations
journal, January 2014

Stagger-and-Step Method: Detecting and Computing Chaotic Saddles in Higher Dimensions
journal, March 2001

Growing 1D and Quasi-2D Unstable Manifolds of Maps
journal, October 1998

Modeling non-stationary, non-axisymmetric heat patterns in DIII-D tokamak
journal, October 2016

A two-dimensional mapping with a strange attractor
journal, February 1976

Field-induced barrier penetration in the quartic potential
journal, April 1984

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
book, January 1983

A procedure for finding numerical trajectories on chaotic saddles
journal, June 1989

Escape patterns, magnetic footprints, and homoclinic tangles due to ergodic magnetic limiters
journal, December 2002

Modeling of stochastic magnetic flux loss from the edge of a poloidally diverted tokamak
journal, December 2002

Fractal structures in nonlinear dynamics
journal, March 2009

Computation of maximal local (un)stable manifold patches by the parameterization method
journal, January 2016

Symplectic maps, variational principles, and transport
journal, July 1992

Traces of stable and unstable manifolds in heat flux patterns
journal, April 2007

A universal instability of many-dimensional oscillator systems
journal, May 1979

Numerical calculation of invariant manifolds for maps
journal, March 1994

The PIM-simplex method: an extension of the PIM-triple method to saddles with an arbitrary number of expanding directions
journal, February 1999

Magnetic field lines, Hamiltonian dynamics, and nontwist systems
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Escape patterns of chaotic magnetic field lines in a tokamak with reversed magnetic shear and an ergodic limiter
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Experimental signatures of homoclinic tangles in poloidally diverted tokamaks
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Regular and Chaotic Dynamics
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Quantum dynamical tunneling in bound states
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