# Dynamical systems

Remember linear time invariant systems, as made famous by signal processing? Now relax the assumption that the model is linear, or even that its state space is in $$\mathbb{R}^n$$. Maybe its state is a measure, or a symbol, or whatever. Now say the word “chaos!”. Pronounce the exclamation mark. Maybe it’s a random system, a stochastic process, or a deterministic process representing the evolution of the measure of stochastic process or whatever.

(Regarding that, one day I should try to understand how Talagrand uses isoperimetric inequalities to derive concentration inequalities.)

Topics that I should connect to this one: the weird end: “nonlinear time series wizardy”, Also “sync”. And “ergodic theory”.

To wish I understood: Takens embedding, and whether it is any statistical use at all.

There is too much to do here, and it’s done better elsewhere. therefore: Idiosyncratic notes only.

## References

Ay, Nihat, Holger Bernigau, Ralf Der, and Mikhail Prokopenko. 2012. “Information-Driven Self-Organization: The Dynamical System Approach to Autonomous Robot Behavior.” Theory in Biosciences 131 (3): 161–79. https://doi.org/10.1007/s12064-011-0137-9.
Ay, Nihat, and Thomas Wennekers. 2003. “Dynamical Properties of Strongly Interacting Markov Chains.” Neural Networks 16 (10): 1483–97. https://doi.org/10.1016/S0893-6080(03)00190-4.
Badii, Remo, and Antonio Politi. 1999. Complexity: Hierarchical Structures and Scaling in Physics. Cambridge Nonlinear Science Series. Cambridge University Press.
Fraser, Andrew M. 2008. Hidden Markov Models and Dynamical Systems. Philadelphia, PA: Society for Industrial and Applied Mathematics.
Grassberger, Peter, Thomas Schreiber, and C Schaffrath. 1991. “Nonlinear Time Sequence Analysis.” International Journal of Bifurcation and Chaos 1 (3): 521–47. https://doi.org/10.1142/S0218127491000403.
Hefny, Ahmed, Carlton Downey, and Geoffrey Gordon. 2015. “A New View of Predictive State Methods for Dynamical System Learning.” May 20, 2015. http://arxiv.org/abs/1505.05310.
Heinonen, Markus, and Florence d’Alché-Buc. 2014. “Learning Nonparametric Differential Equations with Operator-Valued Kernels and Gradient Matching.” November 19, 2014. http://arxiv.org/abs/1411.5172.
Ionides, E. L., C. Bretó, and A. A. King. 2006. “Inference for Nonlinear Dynamical Systems.” Proceedings of the National Academy of Sciences 103 (49): 18438–43. https://doi.org/10.1073/pnas.0603181103.
Kantz, Holger, and Thomas Schreiber. 2004. Nonlinear Time Series Analysis. 2nd ed. Cambridge, UK ; New York: Cambridge University Press.
Kelso, J A Scott. 1995. Dynamic Patterns: The Self-Organization of Brain and Behavior (Complex Adaptive Systems). The MIT Press.
Kendall, Bruce E., Stephen P. Ellner, Edward McCauley, Simon N. Wood, Cheryl J. Briggs, William W. Murdoch, and Peter Turchin. 2005. “Population Cycles in the Pine Looper Moth: Dynamical Tests of Mechanistic Hypotheses.” Ecological Monographs 75 (2): 259–76. http://www.sysecol2.ethz.ch/Refs/EntClim/K/Ke169.pdf.
Marwan, N. 2008. “A Historical Review of Recurrence Plots.” The European Physical Journal Special Topics 164 (1): 3–12. https://doi.org/10.1140/epjst/e2008-00829-1.
Mordvintsev, Alexander, Ettore Randazzo, Eyvind Niklasson, and Michael Levin. 2020. “Growing Neural Cellular Automata.” Distill 5 (2): e23. https://doi.org/10.23915/distill.00023.
Packard, Norman H, James P Crutchfield, J Doyne Farmer, and R S Shaw. 1980. “Geometry from a Time Series.” Physical Review Letters 45 (9): 712–16. https://doi.org/10.1103/PhysRevLett.45.712.
Raginsky, M. 2011. “Directed Information and Pearl’s Causal Calculus.” In 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 958–65. https://doi.org/10.1109/Allerton.2011.6120270.
Ruelle, David. 1998. “Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics.” http://www.arxiv.org/abs/chao-dyn/9812032.
Ryabko, Daniil, and Boris Ryabko. 2010. “Nonparametric Statistical Inference for Ergodic Processes.” IEEE Transactions on Information Theory 56 (3): 1430–35. https://doi.org/10.1109/TIT.2009.2039169.
Schöner, Gregor. 2002. “Timing, Clocks, and Dynamical Systems.” Brain and Cognition 48 (1): 31–51. https://doi.org/10.1006/brcg.2001.1302.
Shalizi, Cosma Rohilla, Robert Haslinger, Jean-Baptiste Rouquier, Kristina L Klinkner, and Cristopher Moore. 2006. “Automatic Filters for the Detection of Coherent Structure in Spatiotemporal Systems.” Physical Review E 73 (3). https://doi.org/10.1103/PhysRevE.73.036104.
Smith, Leonard A. 2000. “Disentangling Uncertainty and Error: On the Predictability of Nonlinear Systems.” In Nonlinear Dynamics and Statistics.
Strogatz, Steven H. 2001. Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity). Westview Press.
Valpine, Perry de. 2011. “Frequentist Analysis of Hierarchical Models for Population Dynamics and Demographic Data.” Journal of Ornithology 152 (2): 393–408. https://doi.org/10.1007/s10336-010-0642-5.
Wolpert, David H, Kevin R Wheeler, and Kagan Tumer. 2000. “Collective Intelligence for Control of Distributed Dynamical Systems.” EPL (Europhysics Letters) 49: 708. https://doi.org/10.1209/epl/i2000-00208-x.

Warning! Experimental comments system! If is does not work for you, let me know via the contact form.