Geodesic
Spiral chaos in Lotka-Volterra like models
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 2, pp. 13-24.

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In this work investigations are made of spiral chaos in generalized Lotka-Volterra systems and Rosenzweig-MacArthur systems that describe the interaction of three species. It is shown that in systems under study the spiral chaos appears in agreement with Shilnikov's scenario. When changing a parameter in the system a stable limiting cycle and a saddle-focus equilibrium are born from stable equilibrium. Then the unstable invariant manifold of saddle-focus winds on the stable limit cycle and forms a whirlpool. For some parameter's value the unstable invariant manifold touches one-dimensional stable invariant manifold and forms homoclinic trajectory to saddle-focus. If in this case the limiting cycle loses stability (for example, as result of sequence of period-doubling bifurcations) and saddle value of the saddle-focus is negative then strange attractor appears on base of homoclinic trajectory.
Mots-clés : spiral chaos
Keywords: Lotka-Volterra-like systems, strange attractor. 
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Y. V. Bakhanova; A. O. Kazakov; A. G. Korotkov. Spiral chaos in Lotka-Volterra like models. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 2, pp. 13-24. http://geodesic.mathdoc.fr/item/SVMO_2017_19_2_a0/

[1] L. P. Shilnikov, “A case of the existence of a countable number of periodic motions(Point mapping proof of existence theorem showing neighborhood of trajectory which departs from and returns to saddle-point focus contains denumerable set of periodic motions)”, Soviet Mathematics., 160:3 (1965), 558-561 (In Russ.) | Zbl

[2] A. Arneodo, P. Coullet, C. Tresser, “Occurence of strange attractors in three-dimensional Volterra equations”, Physics Letters A, 79:4 (1980), 259-263 | DOI | MR

[3] A. Arneodo, P. Coullet, C. Tresser, “Possible new strange attractors with spiral structure”, Commun. Math. Phys., 79 (1981), 573-579 | DOI | MR | Zbl

[4] A. Arneodo, P. Coullet, C. Tresser, “Oscillators with chaotic behavior: an illustration of a theorem by Shilńikov”, Journal of Statistical Physics, 27:1 (1982), 171-182 | DOI | MR | Zbl

[5] L. O. Chua, M. Komuro, T. Matsumoto, “The double scroll family”, IEEE Transactions on Circuits and Systems, 33:11 (1986), 1072-1118 | DOI | MR | Zbl

[6] V. S. Anishchenko, “Complex oscillations in simple systems.”, IEEE Transactions on Circuits and Systems, 1990 (In Russ.) | MR

[7] M. T. M. Koper, P. Gaspard, J. H. Sluyters, “Mixed-mode oscillations and incomplete homoclinic scenarios to a saddle focus in the indium / thiocyanate electrochemical oscillator”, The Journal of chemical physics, 97:11 (1992), 8250-8260 | DOI

[8] R. M. May, W. J. Leonard, “Nonlinear aspects of competition between three species”, SIAM journal on applied mathematics, 29:2 (1975), 243–253 | DOI | MR | Zbl

[9] J. A. Vano, J. C. Wildenberg, M. B. Anderson, J. K. Noel, J. C. Sprott, “Chaos in low-dimensional Lotka-Volterra models of competition”, Nonlinearity, 19 (2006), 2391–2404 | DOI | MR | Zbl

[10] L. P. Shilnikov, “The theory of bifurcations and turbulence”, Methods of the qualitative theory of differential equations, 1986, 150-163 (In Russ.) | MR

[11] M. L. Rosenzweig, R. H. Macarthur, “Graphical representation and stability conditions of predator-prey interactions”, American Naturalist, 97:895 (1963), 209-223 | DOI

[12] B. Deng, “Food chain chaos due to junction-fold point”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 11:3 (2001), 514-525 | DOI | Zbl

[13] B. Deng, G. Hines, “Food chain chaos due to Shilnikov’s orbit”, Chaos, 12:3 (2002), 533-538 | DOI | MR | Zbl

[14] B. Deng, “Food chain chaos with canard explosion”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 14:4 (2004), 1083-1092 | DOI | MR | Zbl

[15] Krishna pada Das, “A study of chaotic dynamics and its possible control in a predator-prey model with disease in the predator”, Journal of Dynamical and Control Systems, 21:4 (2002), 605-624 | MR

[16] O. E. Rössler, “An equation for continuous chaos”, Physics Letters A., 57:5 (1976), 397-398 | DOI | MR | Zbl

[17] L. P. Shilnikov, “The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus”, Soviet Math. Dokl., 172:2 (1967), 298-301 (In Russ.) | Zbl

[18] L. P. Shilnikov, “A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type”, Mathematics of the USSR-Sbornik, 81:1 (1970), 92-103 (In Russ.) | Zbl

[19] P. Coullet, C. Tresser, A. Arneodo, “Transition to stochasticity for a class of forced oscillators”, Physics letters A, 72:4-5 (1979), 268-270 | DOI | MR

[20] V. S. Afraimovich, L. P. Shilnikov, “On invariant two-dimensional tori, their disintegration and stochasticity”, Methods of the qualitative theory of differential equations, Gos. Univ. Gorkij, Gor'kov., 1983, 3-26, Gor'kij (In Russ.)