# 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.

## Refs

ABDP12
Ay, N., Bernigau, H., Der, R., & Prokopenko, M. (2012) Information-driven self-organization: the dynamical system approach to autonomous robot behavior. Theory in Biosciences, 131(3), 161–179. DOI.
AyDP10
Ay, N., Der, R., & Prokopenko, M. (2010) Information Driven Self-Organization: The Dynamical System Approach to Autonomous Robot Behavior (No. 10-09-018). . Santa Fe Institute
AyWe03
Ay, N., & Wennekers, T. (2003) Dynamical properties of strongly interacting Markov chains. Neural Networks, 16(10), 1483–1497. DOI.
BaPo99
Badii, R., & Politi, A. (1999) Complexity: Hierarchical Structures and Scaling in Physics. . Cambridge University Press
Beal03
Beal, M. J.(2003) Variational algorithms for approximate Bayesian inference. . University of London
BoMc10
Bown, O., & McCormack, J. (2010) Taming nature: tapping the creative potential of ecosystem models in the arts. Digital Creativity, 21(4), 215–231. DOI.
CeLZ11
Ceguerra, R. V., Lizier, J. T., & Zomaya, A. Y.(2011) Information storage and transfer in the synchronization process in locally-connected networks. . Presented at the IEEE Symposium Series in Computational Intelligence (SSCI 2011) - IEEE Symposium on Artificial Life,
Chaz00
Chazottes, J.-R. (n.d.) An introduction to fluctuations of observables in chaotic dynamical systems.
Fras08
Fraser, A. M.(2008) Hidden Markov models and dynamical systems. . Philadelphia, PA: Society for Industrial and Applied Mathematics
GlLi88
Glazier, J. A., & Libchaber, A. (1988) Quasi-periodicity and dynamical systems: An experimentalist’s view. IEEE Transactions on Circuits and Systems, 35(7), 790–809. DOI.
Grassberger, P., Schreiber, T., & Schaffrath, C. (1991) Nonlinear time sequence analysis. International Journal of Bifurcation and Chaos, 1(3), 521–547. DOI.
HeDG15
Hefny, A., Downey, C., & Gordon, G. (2015) A New View of Predictive State Methods for Dynamical System Learning. arXiv:1505.05310 [cs, Stat].
HeBu14
Heinonen, M., & d’Alché-Buc, F. (2014) Learning nonparametric differential equations with operator-valued kernels and gradient matching. arXiv:1411.5172 [cs, Stat].
HPVB07
Hlaváčková-Schindler, K., Paluš, M., Vejmelka, M., & Bhattacharya, J. (2007) Causality detection based on information-theoretic approaches in time series analysis. Physics Reports, 441(1), 1–46. DOI.
IoBK06
Ionides, E. L., Bretó, C., & King, A. A.(2006) Inference for nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 103(49), 18438–18443. DOI.
KaST98
Kakizawa, Y., Shumway, R. H., & Taniguchi, M. (1998) Discrimination and clustering for multivariate time series. Journal of the American Statistical Association, 328–340.
Kels94
Kelso, J. A. S.(1994) The informational character of self-organized coordination dynamics. Human Movement Science, 13, 393–413. DOI.
Kels95
Kelso, J. A. S.(1995) Dynamic Patterns: The Self-Organization of Brain and Behavior (Complex Adaptive Systems). . The MIT Press
KEMW05
Kendall, B. E., Ellner, S. P., McCauley, E., Wood, S. N., Briggs, C. J., Murdoch, W. W., & Turchin, P. (2005) Population cycles in the pine looper moth: Dynamical tests of mechanistic hypotheses. Ecological Monographs, 75(2), 259–276.
MKWL11
Martini, M., Kranz, T. A., Wagner, T., & Lehnertz, K. (2011) Inferring directional interactions from transient signals with symbolic transfer entropy. Phys. Rev. E, 83(1), 011919. DOI.
Marw08
Marwan, N. (2008) A historical review of recurrence plots. The European Physical Journal Special Topics, 164(1), 3–12. DOI.
PCFS80
Packard, N. H., Crutchfield, J. P., Farmer, J. D., & Shaw, R. S.(1980) Geometry from a Time Series. Physical Review Letters, 45(9), 712–716. DOI.
Ragi11
Raginsky, M. (2011) Directed information and Pearl’s causal calculus. In 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton) (pp. 958–965). DOI.
RTKB04
Romano, M. C., Thiel, M., Kurths, J., & von Bloh, W. (2004) Multivariate recurrence plots. Physics Letters A, 330(3–4), 214–223. DOI.
RuSW05
Rudary, M., Singh, S., & Wingate, D. (2005) Predictive Linear-Gaussian Models of Stochastic Dynamical Systems. In arXiv:1207.1416 [cs].
Ruel98
Ruelle, D. (1998) Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics.
RyRy10
Ryabko, D., & Ryabko, B. (2010) Nonparametric Statistical Inference for Ergodic Processes. IEEE Transactions on Information Theory, 56(3), 1430–1435. DOI.
SaYC91
Sauer, T., Yorke, J. A., & Casdagli, M. (1991) Embedology. Journal of Statistical Physics, 65(3-4), 579–616. DOI.
Schö02
Schöner, G. (2002) Timing, Clocks, and Dynamical Systems. Brain and Cognition, 48(1), 31–51. DOI.
SHRK06
Shalizi, C. R., Haslinger, R., Rouquier, J.-B., Klinkner, K. L., & Moore, C. (2006) Automatic Filters for the Detection of Coherent Structure in Spatiotemporal Systems. Physical Review E, 73(3). DOI.
Smit00
Smith, L. A.(2000) Disentangling Uncertainty and Error: On the predictability of nonlinear systems. In Nonlinear Dynamics and Statistics.
Stro01
Strogatz, S. H.(2001) Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity). . Westview Press
Valp11
Valpine, P. de. (2011) Frequentist analysis of hierarchical models for population dynamics and demographic data. Journal of Ornithology, 152(2), 393–408. DOI.
BeDo95
van Beijeren, H., & Dorfman, J. R.(1995) Lyapunov Exponents and Kolmogorov-Sinai Entropy for the Lorentz Gas at Low Densities. Phys. Rev. Lett., 74(22), 4412–4415. DOI.
WoWT00
Wolpert, D. H., Wheeler, K. R., & Tumer, K. (2000) Collective intelligence for control of distributed dynamical systems. EPL (Europhysics Letters), 49, 708. DOI.