refs.bib

@book {GuckenheimerHolmes,
author = { Guckenheimer, John and Holmes, Philip },
title = { Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields },
edition = { 3. print., rev. and corr. } ,
publisher = {Springer},
isbn = {3540908196},
isbn = {0387908196},
isbn = {9780387908199},
isbn = {9783540908197},
year = {1990},
booktitle = { Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields },
address = { New York },
doi = { 10.1007/978-1-4612-1140-2 },
url = { https://link.springer.com/book/10.1007/978-1-4612-1140-2 }
}

@book {Verhulst,
  title={Nonlinear Differential Equations and Dynamical Systems},
  author={Ferdinand Verhulst},
  publisher = {Springer},
  year={1989},
  isbn = {978-3-540-60934-6},
  doi = {10.1007/978-3-642-61453-8}
}

@book {Arnold,
	title= {Ordinary Differential Equations},
	author = {Vladimir Arnold},
	publisher = {Springer},
	isbn = {978-3-540-34563-3},
	url = {https://link.springer.com/book/9783540345633},
	year = {1992}
}

@book{Strogatz,
  title={Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering},
  author={Steven Strogatz},
  series={Studies in nonlinearity},
  year={2000},
  publisher={Westview}
}

@book{Chicone,
	title={Ordinary Differential Equations with Applications},
	author={Carmen Chicone},
	year={1999},
	publisher={Springer},
	doi={10.1007/b97645},
	issn = {2196-9949},
	isbn={978-0-387-22623-1}
}

@phdthesis{PalisPhd,
	title = {On Morse-Smale Diffeomorphisms},
	author= {J. Palis},
	school = {UC Berkely},
	year = {1967}
}

@article{Palis,
	title = {On Morse-Smale dynamical systems},
	author = {J. Palis},
	journal = {Topology},
	volume = {8},
	pages = {385-404},
	year = {1969}
}

@article{Haller2001,
title = {Distinguished material surfaces and coherent structures in three-dimensional fluid flows},
journal = {Physica D: Nonlinear Phenomena},
volume = {149},
number = {4},
pages = {248-277},
year = {2001},
issn = {0167-2789},
doi = {https://doi.org/10.1016/S0167-2789(00)00199-8},
url = {https://www.sciencedirect.com/science/article/pii/S0167278900001998},
author = {G. Haller},
keywords = {Fluid flows, Coherent structures, Invariant manifolds, Mixing},
abstract = {We prove analytic criteria for the existence of finite-time attracting and repelling material surfaces and lines in three-dimensional unsteady flows. The longest lived such structures define coherent structures in a Lagrangian sense. Our existence criteria involve the invariants of the velocity gradient tensor along fluid trajectories. An alternative approach to coherent structures is shown to lead to their characterization as local maximizers of the largest finite-time Lyapunov exponent field computed directly from particle paths. Both approaches provide effective tools for extracting distinguished Lagrangian structures from three-dimensional velocity data. We illustrate the results on steady and unsteady ABC-type flows.}
}

@article{Sun2016,
title = {Detection of Lagrangian Coherent Structures in the SPH framework},
journal = {Computer Methods in Applied Mechanics and Engineering},
volume = {305},
pages = {849-868},
year = {2016},
issn = {0045-7825},
doi = {https://doi.org/10.1016/j.cma.2016.03.027},
url = {https://www.sciencedirect.com/science/article/pii/S0045782516301128},
author = {P.N. Sun and A. Colagrossi and S. Marrone and A.M. Zhang},
keywords = {Smoothed Particle Hydrodynamics, Finite-Time Lyapunov Exponent, Lagrangian Coherent Structures, Vortex visualization, Breaking bow wave},
abstract = {The present work is dedicated to the detection of Lagrangian Coherent Structures (LCSs) in viscous flows through the Finite-Time Lyapunov Exponents (FTLEs) which have been addressed by several works in the recent literature. Here, a novel numerical technique is presented in the context of the Smoothed Particle Hydrodynamics (SPH) models. Thanks to the Lagrangian character of SPH, the trajectory of each fluid particle is explicitly tracked over the whole simulation. This allows for a direct evaluation of the FTLE field supplying a new way for the data analysis of complex flows. The evaluation of FTLE can be either implemented as a post-processing or nested into the SPH scheme conveniently. In the numerical results, three test-cases are presented giving a proof of concept for different conditions. The last test-case regards a naval engineering problem for which the present algorithm is successfully used to capture the submerged vortical tunnels caused by the splashing bow wave.}
}

@article{BreunningHaller,
	author = {T. Breunung and G. Haller},
	title = {When does a periodic response exist in a periodic forced multi-degree-of-freedom mechanical system?},
	journal = {Nonlinear Dynamics},
	volume = {98},
	issue = {3},
	url = {https://doi.org/10.1007/s11071-019-05284-z},
	doi = {10.1007/s11071-019-05284-z},
	year = {2019},
	pages = {1761-1780}
}

@article{Schmid2010,
  title={Dynamic mode decomposition of numerical and experimental data},
  author={Peter Schmid},
  journal={Journal of Fluid Mechanics},
  year={2010},
  volume={656},
  pages={5 - 28}
}

@article{Kutz2016,
	journal={SIAM Journal on Applied Dynamical Systems},
  doi = {10.48550/ARXIV.1506.00564},
  url = {https://arxiv.org/abs/1506.00564},
  author = {Kutz, J. Nathan and Fu, Xing and Brunton, Steven L.},
  keywords = {Dynamical Systems (math.DS), FOS: Mathematics, FOS: Mathematics},
  title = {Multi-Resolution Dynamic Mode Decomposition},
  year = {2015},
}

@article{Williams2015,
	title={A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition},
	author = {M. Williams and I. Kevrekidis and C. Rowley},
	journal={Journal of Nonlinear Science},
	issue={25},
	year={2015},
	pages={1307-1346},
	url={https://doi.org/10.1007/s00332-015-9258-5}
}

@article{SaariUrenko,
 ISSN = {00029890, 19300972},
 URL = {http://www.jstor.org/stable/2322163},
 author = {Donald G. Saari and John B. Urenko},
 journal = {The American Mathematical Monthly},
 number = {1},
 pages = {3--17},
 publisher = {Mathematical Association of America},
 title = {Newton's Method, Circle Maps, and Chaotic Motion},
 urldate = {2022-11-01},
 volume = {91},
 year = {1984}
}

@article{Volterra1926,
author={V. Volterra},
title={Variazioni e fluttuazioni del numero d'individui in specie animali conviventi},
journal={Memor. Accad. Lincei.},
year={1926},
volume={6},
pages={31-113},
url={https://cir.nii.ac.jp/crid/1574231874580495104}
}

@book{Lotka1925,
	year={1925},
	author={Alfred Lotka},
title={Elements of Physical Biology},
publisher={Williams and Wilkins Company},
pages={495},
abstract={FOR a long time methods analogous to those of statistical mechanics have been applied to animal and plant communities. But just as in physical chemistry these methods soon become intolerably cumbrous and may generally be replaced by thermodynamical calculations which ignore the individual molecule, so in the book before us the author has applied to biological problems a treatment of the type familiar in chemical statics and kinetics. These methods are applied to the growth of populations, whether of bacteria, insects, men,. or railway engines, and to relationships between different species, for example, between a parasite and one or more hosts. The author has the problem of evolution always before him, and considers analytically the effect on population of a change in the behaviour of individuals.},
doi={10.1038/116461b0}
}

@book{Milnor1965,
	year={1965},
	title={Topology from the differentiable viewpoint},
	author={John Milnor},
	publisher={University Press of Virginia},
	pages={64}
}

@book{LiapunovDirect,
	year={1977},
	title={Stability Theory by Liapunov’s Direct Method},
	author={N. Rouche and P. Habets and M. Laloy},
	publisher={Springer New York, NY},
	doi={10.1007/978-1-4684-9362-7},
	pages={396},
	issn={0066-5452},
	isbn={978-0-387-90258-6}
}

@book{Haken1977,
  title={Synergetics: an Introduction, Nonequilibrium Phase Transitions and Self-organization in Physics, Chemistry, and Biology},
  author={Hermann Haken},
  publisher={Springer Berline, Heidelberg},
  doi={10.1007/978-3-642-96469-5},
  issn={0172-7389},
  url={https://link.springer.com/book/10.1007/978-3-642-96469-5},
  year={1977}
}

@book{Wiggins1994,
	title={Normally Hyperbolic Invariant Manifolds in Dynamical Systems},
	year={1994},
	author = {Stephen Wiggins},
	publisher={Springer New York, NY},
	doi = {10.1007/978-1-4612-4312-0},
	url = {https://link.springer.com/book/10.1007/978-1-4612-4312-0},
	isbn = {978-0-387-94205-6},
	pages={194}
}

@article{Fenichel1979,
  title={Geometric singular perturbation theory for ordinary differential equations},
  author={Neil Fenichel},
  journal={Journal of Differential Equations},
  year={1979},
  volume={31},
  pages={53-98}
}

@article{Burns1999,
  title={A perturbation study of particle dynamics in a plane wake flow},
  author={Thomas J. Burns and R. W. Davis and Elizabeth F. Moore},
  journal={Journal of Fluid Mechanics},
  year={1999},
  volume={384},
  pages={1 - 26}
}

@article{Maxey1983,
author = {Maxey,Martin R.  and Riley,James J. },
title = {Equation of motion for a small rigid sphere in a nonuniform flow},
journal = {The Physics of Fluids},
volume = {26},
number = {4},
pages = {883-889},
year = {1983},
doi = {10.1063/1.864230},
URL = {https://aip.scitation.org/doi/abs/10.1063/1.864230},
eprint = {https://aip.scitation.org/doi/pdf/10.1063/1.864230}
}

@article{Sapsis2008,
title = {Where do inertial particles go in fluid flows?},
journal = {Physica D: Nonlinear Phenomena},
volume = {237},
number = {5},
pages = {573-583},
year = {2008},
issn = {0167-2789},
doi = {https://doi.org/10.1016/j.physd.2007.09.027},
url = {https://www.sciencedirect.com/science/article/pii/S0167278907003557},
author = {George Haller and Themistoklis Sapsis},
keywords = {Inertial particles, Slow manifolds, Singular perturbation theory, Nonautonomous systems},
abstract = {We derive a general reduced-order equation for the asymptotic motion of finite-size particles in unsteady fluid flows. Our inertial equation is a small perturbation of passive fluid advection on a globally attracting slow manifold. Among other things, the inertial equation implies that particle clustering locations in two-dimensional steady flows can be described rigorously by the Q parameter, i.e., by one-half of the squared difference of the vorticity and the rate of strain. Use of the inertial equation also enables us to solve the numerically ill-posed problem of source inversion, i.e., locating initial positions from a current particle distribution. We illustrate these results on inertial particle motion in the Jung–Tél–Ziemniak model of vortex shedding behind a cylinder in crossflow.}
}

@article{Sapsis2010,
author = {Sapsis,Themistoklis  and Haller,George },
title = {Clustering criterion for inertial particles in two-dimensional time-periodic and three-dimensional steady flows},
journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
volume = {20},
number = {1},
pages = {017515},
year = {2010},
doi = {10.1063/1.3272711},
URL = {https://doi.org/10.1063/1.3272711},
eprint = {https://doi.org/10.1063/1.3272711}
}

@article{Ponsioen2016,
author={Haller, George and Ponsioen, Sten},
year={2016},
title ={ Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction},
journal  ={ Nonlinear Dynamics},
pages ={ 1493- 1534},
volume  ={ 86},
issue  ={ 3},
abstract  ={ We propose a unified approach to nonlinear modal analysis in dissipative oscillatory systems. This approach eliminates conflicting definitions, covers both autonomous and time-dependent systems and provides exact mathematical existence, uniqueness and robustness results. In this setting, a nonlinear normal mode (NNM) is a set filled with small-amplitude recurrent motions: a fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a NNM, serving as the smoothest nonlinear continuation of a spectral subspace of the linearized system along the NNM. The existence and uniqueness of SSMs turns out to depend on a spectral quotient computed from the real part of the spectrum of the linearized system. This quotient may well be large even for small dissipation; thus, the inclusion of damping is essential for firm conclusions about NNMs, SSMs and the reduced-order models they yield.},
issn  ={ 1573-269X},
url  ={ https://doi.org/10.1007/s11071-016-2974-z},
doi ={ 10.1007/s11071-016-2974-z},
}