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Unbounded dynamics in dissipative flows: Rössler model
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1.
1. B. Eckhardt and C. Jung, “ Regular and irregular potential scattering,” J. Phys. A 19, L829L833 (1986).
http://dx.doi.org/10.1088/0305-4470/19/14/002
2.
2. Y.-C. Lai and T. Tél, “ Complex dynamics on finite-time scales,” Transient Chaos, Applied Mathematical Sciences Vol. 173 (Springer, New York, 2011), pp. xiv+497.
3.
3. J. M. Seoane and M. A. F. Sanjuán, “ New developments in classical chaotic scattering,” Rep. Prog. Phys. 76, 016001 (2013).
http://dx.doi.org/10.1088/0034-4885/76/1/016001
4.
4. T. Tél and M. Gruiz, Chaotic Dynamics: An Introduction Based on Classical Mechanics (Cambridge University Press, 2006).
5.
5. C. Grebogi, E. Ott, and J. A. Yorke, “ Chaotic attractors in crisis,” Phys. Rev. Lett. 48, 1507 (1982).
http://dx.doi.org/10.1103/PhysRevLett.48.1507
6.
6. C. Grebogi, E. Ott, and J. A. Yorke, “ Crises, sudden changes in chaotic attractors, and transient chaos,” Physica D 7, 181200 (1983).
http://dx.doi.org/10.1016/0167-2789(83)90126-4
7.
7. R. Barrio, “ Sensitivity tools vs. Poincaré sections,” Chaos Solitons Fractals 25, 711726 (2005).
http://dx.doi.org/10.1016/j.chaos.2004.11.092
8.
8. R. Barrio, “ Painting chaos: A gallery of sensitivity plots of classical problems,” Int. J. Bifur. Chaos 16, 27772798 (2006).
http://dx.doi.org/10.1142/S021812740601646X
9.
9. O. E. Rössler, “ An equation for continuous chaos,” Phys. Lett. A 57, 397398 (1976).
http://dx.doi.org/10.1016/0375-9601(76)90101-8
10.
10. P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progress in Physics Vol. 1 (Birkhäuser, Boston, Mass., 1980).
11.
11. M. Hénon, “ A two-dimensional mapping with a strange attractor,” Commun. Math. Phys. 50, 6977 (1976).
http://dx.doi.org/10.1007/BF01608556
12.
12. C. Simó, “ On the Hénon-Pomeau attractor,” J. Stat. Phys. 21, 465494 (1979).
http://dx.doi.org/10.1007/BF01009612
13.
13. R. Barrio, F. Blesa, and S. Serrano, “ Qualitative analysis of the Rössler equations: Bifurcations of limit cycles and chaotic attractors,” Phys. D 238, 10871100 (2009).
http://dx.doi.org/10.1016/j.physd.2009.03.010
14.
14. R. Barrio, F. Blesa, S. Serrano, and A. Shilnikov, “ Global organization of spiral structures in biparameter space of dissipative systems with Shilnikov saddle-foci,” Phys. Rev. E 84, 035201 (2011).
http://dx.doi.org/10.1103/PhysRevE.84.035201
15.
15. R. Barrio, F. Blesa, and S. Serrano, “ Topological changes in periodicity hubs of dissipative systems,” Phys. Rev. Lett. 108, 214102 (2012).
http://dx.doi.org/10.1103/PhysRevLett.108.214102
16.
16. R. Genesio, G. Innocenti, and F. Gualdani, “ A global qualitative view of bifurcations and dynamics in the Rössler system,” Phys. Lett. A 372, 17991809 (2008).
http://dx.doi.org/10.1016/j.physleta.2007.10.063
17.
17. L. Gardini, “ Hopf bifurcations and period-doubling transitions in Rössler model,” Nuovo Cimento B 89(11 ), 139160 (1985).
http://dx.doi.org/10.1007/BF02723543
18.
18. E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede, and X. J. Wang, Auto2000: Continuation and Bifurcation Software for Ordinary Differential Equations (2000).
19.
19. L. A. Belyakov, “ Bifurcation set in a system with homoclinic saddle curve,” Math. Notes USSR Acad. Sci. 28(6), 910916 (1980).
http://dx.doi.org/10.1007/BF01709154
20.
20. A. R. Champneys and Y. A. Kuznetsov, “ Numerical detection and continuation of codimension-two homoclinic bifurcations,” Int. J. Bifur. Chaos 4, 785822 (1994).
http://dx.doi.org/10.1142/S0218127494000587
21.
21. V. Castro, M. Monti, W. B. Pardo, J. A. Walkenstein, and E. Rosa, Jr., “ Characterization of the Rössler system in parameter space,” Int. J. Bifurcation Chaos 17, 965973 (2007).
http://dx.doi.org/10.1142/S0218127407017689
22.
22. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences Vol. 112, 3rd ed. (Springer-Verlag, New York, 2004).
23.
23. J.-M. Ginoux, B. Rossetto, and L. O. Chua, “ Slow invariant manifolds as curvature of the flow of dynamical systems,” Int. J. Bifurcation Chaos 18, 34093430 (2008).
http://dx.doi.org/10.1142/S0218127408022457
24.
24. H. Stewart, J. Thompson, Y. Ueda, and A. Lansbury, “ Optimal escape from potential wells-patterns of regular and chaotic bifurcation,” Physica D 85, 259295 (1995).
http://dx.doi.org/10.1016/0167-2789(95)00172-Z
25.
25. H. B. Stewart, Y. Ueda, C. Grebogi, and J. A. Yorke, “ Double crises in two-parameter dynamical systems,” Phys. Rev. Lett. 75, 24782481 (1995).
http://dx.doi.org/10.1103/PhysRevLett.75.2478
26.
26. N. Fenichel, “ Geometric singular perturbation theory for ordinary differential equations,” J. Differential Equations 31, 5398 (1979).
http://dx.doi.org/10.1016/0022-0396(79)90152-9
27.
27. C. K. R. T. Jones, “ Geometric singular perturbation theory,” Lecture Notes Math. 1609, 44118 (1995).
http://dx.doi.org/10.1007/BFb0095239
28.
28. F. Verhulst, Methods and Applications of Singular Perturbations, Texts in Applied Mathematics Vol. 50 (Springer, New York, 2005).
29.
29. J. Guckenheimer, K. Hoffman, and W. Weckesser, “ The forced van der Pol equation. I. The slow flow and its bifurcations,” SIAM J. Appl. Dyn. Syst. 2, 135 (2003).
http://dx.doi.org/10.1137/S1111111102404738
30.
30. H. G. Kaper and T. J. Kaper, “ Asymptotic analysis of two reduction methods for systems of chemical reactions,” Phys. D 165, 6693 (2002).
http://dx.doi.org/10.1016/S0167-2789(02)00386-X
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/content/aip/journal/chaos/24/2/10.1063/1.4871712
2014-04-25
2014-07-29

Abstract

Transient chaos and unbounded dynamics are two outstanding phenomena that dominate in chaotic systems with large regions of positive and negative divergences. Here, we investigate the mechanism that leads the unbounded dynamics to be the dominant behavior in a dissipative flow. We describe in detail the particular case of boundary crisis related to the generation of unbounded dynamics. The mechanism of the creation of this crisis in flows is related to the existence of an unstable focus-node (or a saddle-focus) equilibrium point and the crossing of a chaotic invariant set of the system with the weak-(un)stable manifold of the equilibrium point. This behavior is illustrated in the well-known Rössler model. The numerical analysis of the system combines different techniques as chaos indicators, the numerical computation of the bounded regions, and bifurcation analysis. For large values of the parameters, the system is studied by means of Fenichel's theory, providing formulas for computing the slow manifold which influences the evolution of the first stages of the orbit.

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Scitation: Unbounded dynamics in dissipative flows: Rössler model
http://aip.metastore.ingenta.com/content/aip/journal/chaos/24/2/10.1063/1.4871712
10.1063/1.4871712
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