Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Predicting and estimating probability density functions of chaotic systems
Page number are going to be assigned later 2017

doi:10.3934/dcdsb.2017144      Abstract        References        Full text (463.5K)      

Noah H. Rhee - Dept. of mathematics & Statistics, University of Missouri-Kansas City, 5100 Rockhill Rd, Kansas City, MO 64110, United States (email)
Paweł Góra - Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada (email)
Majid Bani-Yaghoub - Dept. of Mathematics & Statistics, University of Missouri-Kansas City, 5100 Rockhill Rd, Kansas City, MO 64110, United States (email)

1 K. Alligood, T. Sauer and J. Yorke, Chaos: An Introduction to Dynamical Systems, Springer-Verlag, New York, 1996.       
2 M. Anazawa, Bottom-up derivation of discrete-time population models with the Allee effect, Theoretical Population Biology, 75 (2009), 56-67.
3 L. Becks, F. M. Hilker, H. Malchow, K. Jürgens and H. Arndt, Experimental demonstration of chaos in a microbial food web, Nature, 435 (2005), 1226-1229.
4 A. Boyarskyand and P. Góra, Laws of Chaos, Birkhauser, Boston, 1997.
5 F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics, 2012.
6 C. Castillo-Chavez and A. Yakubu, Epidemics on Attractors, Contemporary Mathematics, 284 (2001), 23-42.
7 B. Cipra, Beetlemania: Chaos in ecology, in What's happening in the mathematical sciences 1998-1999 (ed. P. Zorn), Volume 4, American Mathematical Society, Providence, Rhode Island, USA (1999), 72-81.
8 R. F. Costantino, R. A. Desharnais, J. M. Cushing and B. Dennis, Chaotic dynamics in an insect population, Science, 275 (1997), 389-391.
9 C. Dellacherie and P. A. Meyer, Probabilities and Potential, North-Holland Pub. Co., N.Y., 1978, (Chapter II, Theorem T25).
10 B. Dennis, R. A. Desharnais, J. M. Cushing, S. M. Henson and R. F. Costantino, Estimating chaos and complex dynamics in an insect population, Ecological Monographs, 71 (2001), 277-303.
11 R. A. Desharnais, R. F. Costantino, J. M. Cushing and B. Dennis, Estimating chaos in an insect population, Science, 276 (1997), 1881-1882.
12 W. Falck, O. N. Bjornstad and N. C. Stenseth, Voles and lemmings: Chaos and uncertainty in fluctuating populations, Proceedings of the Royal Society of London B, 262 (1995), 363-370.
13 H. C. J. Godfray and S. P. Blythe, Complex dynamics in multispecies communities, Philosophical Transactions of the Royal Society of London B, 330 (1990), 221-233.
14 C. Godfray and M. Hassell, Chaotic beetles, Science, 275 (1997), 323-324.
15 I. Hanski, P. Turchin, E. Korpimaki and H. Henttonen, Population oscillations of boreal rodents: Regulation by mustelid predators leads to chaos, Nature, 364 (1993), 232-235.
16 M. P. Hassell, H. N. Comins and R. M. May, Spatial structure and chaos in insect population dynamics, Nature, 353 (1991), 255-258.
17 H. Henttonen and I. Hanski, Population dynamics of small rodents in northern Fennoscandia, in Chaos in real data: The analysis of non-linear dynamics from short ecological time series (ed. J.N. Perry, R.H. Smith, I.P. Woiwod and D.R. Morse), Kluwer Academic, Dordrecht, The Netherlands, (2000), 73-96.       
18 M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Communications in Mathematical Physics, 81 (1981), 39-88.       
19 M. Jakobson, Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions, Proceedings of Symposia in Pure Mathematics, 69 (2001), 825-881.       
20 P. Kareiva, Predicting and producing chaos, Nature, 375 (1995), 189-190.       
21 G. Keller and T. Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Communications in Mathematical Physics, 149 (1992), 31-69.
22 B. E. Kendall, C. J. Briggs, W. W. Murdoch, P. Turchin, S. P. Ellner, E. McCauley, R. M. Nisbet and S. N. Wood, Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches, Ecology, 80 (1999), 1789-1805.
23 A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Springer-Verlag, New York, 1994, Chapter V, page 87.
24 T. Y. Li, Finite approximation for the Frobenius-Perron operator, a solution to Ulam's conjecture, Journal of Approximation Theory, 17 (1976), 177-186.
25 J. A. Logan and F. P. Hain (eds.), Chaos and Insect Ecology, Virginia Experiment Station Information Series, 91-3, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, 1991.
26 S. Luzzatto and H. Takahasi, Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps, Nonlinearity, 19 (2006), 1657-1695.
27 W. de Melo and S. van Strien, One-dimensional Dynamics, in Ergebnisse der Mathematik und ihrer Grenzgebiete Volume 3, Springer-Verlag, Berlin, 1993, Chapter V, section 6.
28 M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Institute Hautes Etudes Sci. Publ. Math., 53 (1981), 17-51.
29 R. M. Nisbet and W. S. C. Gurney, Modelling fluctuating populations, Wiley, 1982, 285-308.
30 T. Nowicki, Some dynamical properties of S-unimodal maps, Fundamenta. Mathematicae, 142 (1993), 45-57.
31 T. Nowicki and S. van Strien, Nonhyperbolicity and invariant measures for unimodal maps, Ann. Inst. H. Poincare Phys. Theor., 53 (1990), 427-429.
32 L. Oksanen and T. Oksanen, Long-term microtine dynamics in north Fennoscandian tundra: The vole cycle and lemming chaos, Ecography, 15 (1992), 226-236.
33 L. F. Olsen, G. L. Truty and W. M. Schaffer, Oscillations and chaos in epidemics: A nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark, Theoretical Population Biology, 33 (1988), 344-370.
34 L. F. Olsen and W. M. Schaffer, Chaos vs. noisy periodicity: Alternative hypotheses for childhood epidemics, Science, 249 (1990), 499-504.
35 E. Renshaw, Chaos in biometry, IMA Journal of Mathematics Applied in Medicine and Biology, 11 (1994), 17-44.
36 P. Rohani and D. J. D. Earn, Chaos in a cup of flour, Trends in Ecology and Evolution, 12 (1997), p171.
37 M. R. Rychlik, Another proof of Jakobson's theorem and related results, Ergodic Theory and Dynamical Systems, 8 (1988), 93-109.
38 S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology and Chemistry, Perseus Publishing, 2001.
39 P. Turchin, Chaos and stability in rodent population dynamics: evidence from nonlinear time-series analysis, Oikos, 68 (1993), 167-172.
40 P. Turchin, Chaos in microtine populations, Proceedings of the Royal Society of London B, 262 (1995), 357-361.
41 P. Turchin and S. P. Ellner, Living on the edge of chaos: Population dynamics of Fennoscandian voles, Ecology, 81 (2000), 3099-3116.
42 S. M. Ulam, Problems in Modern Mathematics, Interscience, New York, 1964.       
43 S. N. Wood, Statistical inference for noisy nonlinear ecological dynamic systems, Nature, 466 (2010), 1102-1104.       
44 J.-C. Yoccoz, A Proof of Jakobson's Theorem, https://matheuscmss.wordpress.com/category/mathematics/expository/.
45 C. Zimmer, Life after chaos, Science, 284 (1999), 83-86.       

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