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References · Bibliography

Bibliography

Every built-in system and every analysis method in TSDynamics traces back to a primary source. This page collects them all — never a competitor library, always the original paper. It has two halves:

  • The systems bibliography is generated straight from the catalogue. Each built-in system carries a reference (and, where one exists, a doi) class attribute; this page walks the registry, deduplicates, and lists the citing systems under each paper. Add or edit a system and its citation lands here automatically.
  • The methods bibliography collects the papers behind the analysis toolkit — the same citations that close each page in the Analysis section.

Reproducing the numbers

The parameter defaults and initial conditions in each system's page match the cited paper wherever the source gives them. When a quantity is quoted in the docs (a Lyapunov spectrum, a fractal dimension) it is pinned to a fixed initial condition and, for stochastic systems, a fixed seed — so every number is reproducible from the snippet that produces it.

Systems

The 171 built-in systems cite 137 distinct sources, grouped below by family and ordered alphabetically by first author. A handful of textbook/folklore systems (the Lissajous figures, a couple of purely illustrative maps) carry no single primary source and are omitted from this list.

Ordinary differential equations

Delay differential equations

Stochastic differential equations

Discrete maps

Methods

The primary literature behind the analysis toolkit. These are the citations that close each page in the Analysis section, collected here by topic; every implementation names its source in its docstring and on its prose page.

Lyapunov exponents & tangent-space dynamics

  • G. Benettin, L. Galgani & J.-M. Strelcyn, "Kolmogorov entropy and numerical experiments", Phys. Rev. A 14, 2338 (1976). doi:10.1103/PhysRevA.14.2338
  • G. Benettin, L. Galgani, A. Giorgilli & J.-M. Strelcyn, “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them”, Meccanica 15, 9 & 21 (1980). doi:10.1007/BF02128236
  • H. Kantz, "A robust method to estimate the maximal Lyapunov exponent of a time series", Phys. Lett. A 185, 77 (1994). doi:10.1016/0375-9601(94)90991-1
  • J. L. Kaplan & J. A. Yorke, “Chaotic behavior of multidimensional difference equations”, in Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Mathematics 730, 204, Springer (1979). doi:10.1007/BFb0064319
  • M. T. Rosenstein, J. J. Collins & C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets”, Physica D 65, 117 (1993). doi:10.1016/0167-2789(93)90009-P

Chaos indicators

  • G. A. Gottwald & I. Melbourne, “A new test for chaos in deterministic systems”, Proc. R. Soc. Lond. A 460, 603 (2004). doi:10.1098/rspa.2003.1183
  • G. A. Gottwald & I. Melbourne, “On the implementation of the 0–1 test for chaos”, SIAM J. Appl. Dyn. Syst. 8, 129 (2009). doi:10.1137/080718851
  • B. R. Hunt & E. Ott, "Defining chaos", Chaos 25, 097618 (2015). doi:10.1063/1.4922973
  • Ch. Skokos, T. C. Bountis & Ch. Antonopoulos, “Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method”, Physica D 231, 30 (2007). doi:10.1016/j.physd.2007.04.004

Fractal dimensions

  • R. Badii & A. Politi, “Statistical description of chaotic attractors: the dimension function”, J. Stat. Phys. 40, 725 (1985). doi:10.1007/BF01009897
  • P. Grassberger & I. Procaccia, “Characterization of strange attractors”, Phys. Rev. Lett. 50, 346 (1983). doi:10.1103/PhysRevLett.50.346
  • P. Grassberger, “Generalizations of the Hausdorff dimension of fractal measures”, Phys. Lett. A 107, 101 (1985). doi:10.1016/0375-9601(85)90724-8
  • H. G. E. Hentschel & I. Procaccia, “The infinite number of generalized dimensions of fractals and strange attractors”, Physica D 8, 435 (1983). doi:10.1016/0167-2789(83)90235-X
  • J. Theiler, "Estimating fractal dimension", J. Opt. Soc. Am. A 7, 1055 (1990). doi:10.1364/JOSAA.7.001055

Delay embedding & state-space reconstruction

  • L. Cao, “Practical method for determining the minimum embedding dimension of a scalar time series”, Physica D 110, 43 (1997). doi:10.1016/S0167-2789(97)00118-8
  • A. M. Fraser & H. L. Swinney, “Independent coordinates for strange attractors from mutual information”, Phys. Rev. A 33, 1134 (1986). doi:10.1103/PhysRevA.33.1134
  • M. B. Kennel, R. Brown & H. D. I. Abarbanel, “Determining embedding dimension for phase-space reconstruction using a geometrical construction”, Phys. Rev. A 45, 3403 (1992). doi:10.1103/PhysRevA.45.3403
  • F. Takens, "Detecting strange attractors in turbulence", in Dynamical Systems and Turbulence, Lecture Notes in Mathematics 898, 366, Springer (1981). doi:10.1007/BFb0091924

Entropy & complexity

  • C. Bandt & B. Pompe, “Permutation entropy: a natural complexity measure for time series”, Phys. Rev. Lett. 88, 174102 (2002). doi:10.1103/PhysRevLett.88.174102
  • M. Costa, A. L. Goldberger & C.-K. Peng, “Multiscale entropy analysis of complex physiologic time series”, Phys. Rev. Lett. 89, 068102 (2002). doi:10.1103/PhysRevLett.89.068102
  • B. Fadlallah, B. Chen, A. Keil & J. Príncipe, “Weighted-permutation entropy: a complexity measure for time series incorporating amplitude information”, Phys. Rev. E 87, 022911 (2013). doi:10.1103/PhysRevE.87.022911
  • F. Kaspar & H. G. Schuster, “Easily calculable measure for the complexity of spatiotemporal patterns”, Phys. Rev. A 36, 842 (1987). doi:10.1103/PhysRevA.36.842
  • A. Lempel & J. Ziv, "On the complexity of finite sequences", IEEE Trans. Inf. Theory 22, 75 (1976). doi:10.1109/TIT.1976.1055501
  • S. M. Pincus, “Approximate entropy as a measure of system complexity”, Proc. Natl. Acad. Sci. USA 88, 2297 (1991). doi:10.1073/pnas.88.6.2297
  • J. S. Richman & J. R. Moorman, “Physiological time-series analysis using approximate entropy and sample entropy”, Am. J. Physiol. Heart Circ. Physiol. 278, H2039 (2000). doi:10.1152/ajpheart.2000.278.6.H2039
  • M. Rostaghi & H. Azami, “Dispersion entropy: a measure for time-series analysis”, IEEE Signal Process. Lett. 23, 610 (2016). doi:10.1109/LSP.2016.2542881

Recurrence & RQA

  • J.-P. Eckmann, S. O. Kamphorst & D. Ruelle, “Recurrence plots of dynamical systems”, Europhys. Lett. 4, 973 (1987). doi:10.1209/0295-5075/4/9/004
  • N. Marwan, M. C. Romano, M. Thiel & J. Kurths, “Recurrence plots for the analysis of complex systems”, Phys. Rep. 438, 237 (2007). doi:10.1016/j.physrep.2006.11.001
  • L. L. Trulla, A. Giuliani, J. P. Zbilut & C. L. Webber, “Recurrence quantification analysis of the logistic equation with transients”, Phys. Lett. A 223, 255 (1996). doi:10.1016/S0375-9601(96)00741-4
  • J. P. Zbilut & C. L. Webber, “Embeddings and delays as derived from quantification of recurrence plots”, Phys. Lett. A 171, 199 (1992). doi:10.1016/0375-9601(92)90426-M

Surrogates & nonlinearity tests

  • C. Diks, J. C. van Houwelingen, F. Takens & J. DeGoede, “Reversibility as a criterion for discriminating time series”, Phys. Lett. A 201, 221 (1995). doi:10.1016/0375-9601(95)00239-Y
  • H. Kantz & T. Schreiber, Nonlinear Time Series Analysis, 2nd ed., Cambridge University Press (2004). doi:10.1017/CBO9780511755798
  • T. Schreiber & A. Schmitz, “Improved surrogate data for nonlinearity tests”, Phys. Rev. Lett. 77, 635 (1996). doi:10.1103/PhysRevLett.77.635
  • G. Sugihara & R. M. May, “Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series”, Nature 344, 734 (1990). doi:10.1038/344734a0
  • J. Theiler, "Spurious dimension from correlation algorithms applied to limited time-series data", Phys. Rev. A 34, 2427 (1986). doi:10.1103/PhysRevA.34.2427
  • J. Theiler, S. Eubank, A. Longtin, B. Galdrikian & J. D. Farmer, “Testing for nonlinearity in time series: the method of surrogate data”, Physica D 58, 77 (1992). doi:10.1016/0167-2789(92)90102-S

Fixed points, periodic orbits & interval methods

  • R. L. Davidchack & Y.-C. Lai, “Efficient algorithm for detecting unstable periodic orbits in chaotic systems”, Phys. Rev. E 60, 6172 (1999). doi:10.1103/PhysRevE.60.6172
  • R. Krawczyk, “Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken”, Computing 4, 187 (1969). doi:10.1007/BF02234767
  • A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press (1990).
  • P. Schmelcher & F. K. Diakonos, “Detecting unstable periodic orbits of chaotic dynamical systems”, Phys. Rev. Lett. 78, 4733 (1997). doi:10.1103/PhysRevLett.78.4733

Attractors, basins & global stability

  • G. Datseris & A. Wagemakers, “Effortless estimation of basins of attraction”, Chaos 32, 023104 (2022). doi:10.1063/5.0076568
  • G. Datseris, K. L. Rossi & A. Wagemakers, “Framework for global stability analysis of dynamical systems”, Chaos 33, 073151 (2023). doi:10.1063/5.0159675
  • A. Daza, A. Wagemakers, M. A. F. Sanjuán & J. A. Yorke, “Testing for basins of Wada”, Sci. Rep. 5, 16579 (2015). doi:10.1038/srep16579
  • A. Daza, A. Wagemakers, B. Georgeot, D. Guéry-Odelin & M. A. F. Sanjuán, “Basin entropy: a new tool to analyze uncertainty in dynamical systems”, Sci. Rep. 6, 31416 (2016). doi:10.1038/srep31416
  • C. Grebogi, S. W. McDonald, E. Ott & J. A. Yorke, “Final state sensitivity: an obstruction to predictability”, Phys. Lett. A 99, 415 (1983). doi:10.1016/0375-9601(83)90945-3
  • L. Halekotte & U. Feudel, “Minimal fatal shocks in multistable complex networks”, Sci. Rep. 10, 11783 (2020). doi:10.1038/s41598-020-68805-6
  • P. J. Menck, J. Heitzig, N. Marwan & J. Kurths, “How basin stability complements the linear-stability paradigm”, Nat. Phys. 9, 89 (2013). doi:10.1038/nphys2516

Orbit diagrams, Poincaré maps & the classics

  • M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations”, J. Stat. Phys. 19, 25 (1978). doi:10.1007/BF01020332
  • M. Hénon, “On the numerical computation of Poincaré maps”, Physica D 5, 412 (1982). doi:10.1016/0167-2789(82)90034-3
  • R. M. May, “Simple mathematical models with very complicated dynamics”, Nature 261, 459 (1976). doi:10.1038/261459a0
  • H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars (1892–1899).

The systems bibliography is generated from the registry by docs/_tooling/make_bibliography.py; re-run it after adding or editing a system. If a DOI resolves to the wrong paper, fix the doi class attribute on the system, not this page.