Abstract
We extend the concept of linked twist maps to a 3D setting and develop a global geometrical method to detect the presence of complex dynamics. Our approach, which is based on a recent variant of the theory of topological horseshoes, provides an analytical proof of “chaos” which does not involve small/large parameter techniques and is robust with respect to small perturbations. An application is given to a predator-prey system in \({{\mathbb R}^3}\) with a Beddington-DeAngelis functional response.
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Arioli G., Zgliczyński P.: Symbolic dynamics for the Hénon-Heiles Hamiltonian on the critical level. J. Differ. Equ. 171, 173–202 (2001)
Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics. Dynamical Systems III, Encyclopaedia Math. Sci., vol. 3. Springer, Berlin (1993)
Aulbach B., Kieninger B.: On three definitions of chaos. Nonlinear Dyn. Syst. Theory 1, 23–37 (2001)
Battelli F., Palmer K.J.: Singular perturbations, transversality, and Sil’nikov saddle-focus homoclinic orbits. J. Dyn. Differ. Equ. 15, 357–425 (2003)
Beddington J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Animal Ecol. 44, 331–340 (1975)
Burns K., Weiss H.: A geometric criterion for positive topological entropy. Comm. Math. Phys. 172, 95–118 (1995)
Burton, R., Easton, R.W.: Ergodicity of linked twist maps. Global theory of dynamical systems. In: Proceedings of the International Conference, Northwestern University, Evanston, Ill., 1979. Lecture Notes in Math., vol. 819, pp. 35–49. Springer, Berlin (1980)
Capietto A., Dambrosio W., Papini D.: Superlinear indefinite equations on the real line and chaotic dynamics. J. Differ. Equ. 181, 419–438 (2002)
Carbinatto M., Kwapisz J., Mischaikow K.: Horseshoes and the Conley index spectrum. Ergod. Theory Dyn. Syst. 20, 365–377 (2000)
Cui J., Takeuchi Y.: Permanence, extinction and periodic solution of predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 317, 464–474 (2006)
DeAngelis D.L., Goldstein R.A., O’Neill R.V.: A model for trophic interaction. Ecology 56, 881–892 (1975)
Devaney R.L.: Subshifts of finite type in linked twist mappings. Proc. Am. Math. Soc. 71, 334–338 (1978)
DuBowy P.J.: Waterfowl communities and seasonal environments: temporal variability in interspecific competition. Ecology 69, 1439–1453 (1988)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Revised and Corrected Reprint of the 1983 Original. Applied Mathematical Sciences, vol. 42. Springer, New York, NY (1990)
Hsu S.-B., Zhao X.-Q.: A Lotka-Volterra competition model with seasonal succession. J. Math. Biol. 64, 109–130 (2012)
Hu S.S., Tessier A.J.: Seasonal succession and the strength of intra- and interspecific competition in a Daphnia assemblage. Ecology 76, 2278–2294 (1995)
Huppert A., Blasius B., Olinky R., Stone L.: A model for seasonal phytoplankton blooms. J. Theoret. Biol. 236, 276–290 (2005)
Keeling M., Rohani P., Grenfell B.T.: Seasonally forced disease dynamics explored as switching between attractors. Phys. D 148, 317–335 (2001)
Kennedy J., Koçak S., Yorke J.A.: A chaos lemma. Am. Math. Mon. 108, 411–423 (2001)
Kennedy J., Yorke J.A.: Topological horseshoes. Trans. Am. Math. Soc. 353, 2513–2530 (2001)
Kennedy J., Yorke J.A.: Generalized Hénon difference equations with delay. Univ. Iagel. Acta Math. 41, 9–28 (2003)
Kirchgraber U., Stoffer D.: On the definition of chaos. Z. Angew. Math. Mech. 69, 175–185 (1989)
Leray J., Schauder J.: Topologie et équations fonctionelles. Ann. Sci. École Norm. Sup. 51(3), 45–78 (1934)
Malik T., Smith H.L.: Does dormancy increase fitness of bacterial populations in time-varying environments?. Bull. Math. Biol. 70, 1140–1162 (2008)
Margheri A., Rebelo C., Zanolin F.: Chaos in periodically perturbed planar Hamiltonian systems using linked twist maps. J. Differ. Equ. 249, 3233–3257 (2010)
Mawhin J.: Leray-Schauder degree: a half century of extensions and applications. Topol. Methods Nonlinear Anal. 14, 195–228 (1999)
Medio A., Pireddu M., Zanolin F.: Chaotic dynamics for maps in one and two dimensions: a geometrical method and applications to economics. Int. J. Bifur. Chaos Appl. Sci. Eng. 19, 3283–3309 (2009)
Moser, J.: Stable and random motions in dynamical systems. With special emphasis on celestial mechanics, Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, NJ. Annals of Mathematics Studies, no. 77. Princeton University Press, Princeton, NJ (1973)
Naji R.K., Balasim A.T.: On the dynamical behavior of three species food web model. Chaos Solitons Fractals 34, 1636–1648 (2007)
Palmer, K.J.: Exponential Dichotomies, the Shadowing Lemma and Transversal Homoclinic Points. Dynamics reported, vol. 1, pp. 265–306, Dynam. Report. Ser. Dynam. Systems Appl., 1. Wiley, Chichester (1988)
Pang P.Y.H., Wang M.: Strategy and stationary pattern in a three-species predator-prey model. J. Differ. Equ. 200, 245–273 (2004)
Papini D., Zanolin F.: On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill’s equations. Adv. Nonlinear Stud. 4, 71–91 (2004)
Pireddu, M., Zanolin, F.: Cutting surfaces and applications to periodic points and chaotic-like dynamics. Topol. Methods Nonlinear Anal. 30, 279–319 (2007). Correction in Topol. Methods Nonlinear Anal. 33, 395 (2009)
Przytycki F.: Ergodicity of toral linked twist mappings. Ann. Sci. École Norm. Sup. 16(4), 345–354 (1983)
Robinson, C.: Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, Second edition. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1999)
Schreiber S.: Coexistence for species sharing a predator. J. Differ. Equ. 196, 209–225 (2004)
Smale, S.: Diffeomorphisms with many periodic points. In: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) pp. 63–80. Princeton University Press, Princeton, NJ (1965)
Smale S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)
Springham, J.: Ergodic properties of linked-twist maps. PhD thesis, University of Bristol. Available at ArXiv:0812.0899v1. http://arxiv.org/abs/0812.0899v1
Springham J., Wiggins S.: Bernoulli linked-twist maps in the plane. Dyn. Syst. 25, 483–499 (2010)
Srzednicki, R., Wójcik, K., Zgliczyński, P.: Fixed point results based on the Ważewski method. In: Handbook of topological fixed point theory, pp. 905–943. Springer, Dordrecht (2005)
Sturman R., Meier S.W., Ottino J.M., Wiggins S.: Linked twist map formalism in two and three dimensions applied to mixing in tumbled granular flows. J. Fluid Mech. 602, 129–174 (2008)
Sturman, R., Ottino, J., Wiggins, S.: The Mathematical Foundations of Mixing. The Linked Twist Map as a Paradigm in Applications: Micro to Macro, Fluids to Solids, Cambridge Monographs on Applied and Computational Mathematics, vol. 22. Cambridge University Press, Cambridge (2006)
Waldvogel J.: The period in the Lotka-Volterra system is monotonic. J. Math. Anal. Appl. 114, 178–184 (1986)
Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79. Springer, New York, NY, Berlin (1982)
Wiggins, S.: Global Bifurcations and Chaos. Analytical Methods. Applied Mathematical Sciences, vol. 73. Springer, New York, NY (1988)
Wiggins S., Ottino J.M.: Foundations of chaotic mixing. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362, 937–970 (2004)
Wójcik K.: Remark on complicated dynamics of some planar system. J. Math. Anal. Appl. 271, 257–266 (2002)
Wojtkowski, M.: Linked twist mappings have the K-property. In: Nonlinear Dynamics (International Conference, New York, 1979), pp. 65–76, Annals of the New York Academy Sciences, vol. 357. New York (1980)
Zgliczyński P.: Fixed point index for iterations of maps, topological horseshoe and chaos. Topol. Methods Nonlinear Anal. 8, 169–177 (1996)
Zgliczyński P., Gidea M.: Covering relations for multidimensional dynamical systems. J. Differ. Equ. 202, 32–58 (2004)
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Dedicated to Professor Kenneth Palmer on his 65th birthday.
The results of this paper have been presented at the conference “Non Autonomous Differential Equations, honoring Ken Palmer on the occasion of his 65th birthday” held in Ancona on June 2011.
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Ruiz-Herrera, A., Zanolin, F. An example of chaotic dynamics in 3D systems via stretching along paths. Annali di Matematica 193, 163–185 (2014). https://doi.org/10.1007/s10231-012-0271-0
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DOI: https://doi.org/10.1007/s10231-012-0271-0
Keywords
- Three-dimensional predator-prey equations
- Seasonal succession
- Poincaré map
- Complex dynamics
- Topological horseshoes