×

Classification of chaos in 3-D autonomous quadratic systems. I. Basic framework and methods. (English) Zbl 1185.37092

Summary: This paper is Part I of a series of contributions on the classification problem of chaos in three-dimensional autonomous quadratic systems. We try to classify chaos, based on the Ši’lnikov criteria, in such a large class of systems into the following four types: (1) chaos of the Ši’lnikov homoclinic orbit type; (2) chaos of the Ši’lnikov heteroclinic orbit type; (3) chaos of the hybrid type; i.e. those with both Ši’lnikov homoclinic and homoclinic orbits; and (4) chaos of other types. We are especially interested in finding out all the simplest possible forms of chaotic systems for each type of chaos. Our main contributions are to develop some effective classification methods and to provide a basic classification framework under which each of the four types of chaos can be justified by some examples that are useful for describing the feasibility and procedure of the classification. In particular, we show several novel chaotic attractors, e.g., one hybrid-type chaotic attractor with three equilibria, one heteroclinic orbit and one homoclinic orbit, and one 4-scroll chaotic attractor with five equilibria and two heteroclinic orbits.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C28 Complex behavior and chaotic systems of ordinary differential equations
37C29 Homoclinic and heteroclinic orbits for dynamical systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Čelikovský S., Kybernetika 30 pp 403–
[2] DOI: 10.1142/S0218127402005467 · Zbl 1043.37023 · doi:10.1142/S0218127402005467
[3] DOI: 10.1142/S0218127499001024 · Zbl 0962.37013 · doi:10.1142/S0218127499001024
[4] DOI: 10.1007/b79665 · Zbl 1024.00022 · doi:10.1007/b79665
[5] DOI: 10.1016/0005-1098(92)90177-H · Zbl 0765.93030 · doi:10.1016/0005-1098(92)90177-H
[6] DOI: 10.2307/2695795 · Zbl 0991.37015 · doi:10.2307/2695795
[7] DOI: 10.2307/2318254 · Zbl 0351.92021 · doi:10.2307/2318254
[8] DOI: 10.1142/S0218127403006509 · Zbl 1078.37504 · doi:10.1142/S0218127403006509
[9] DOI: 10.1142/S0218127404009880 · Zbl 1086.37516 · doi:10.1142/S0218127404009880
[10] DOI: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[11] DOI: 10.1016/0375-9601(76)90101-8 · Zbl 1371.37062 · doi:10.1016/0375-9601(76)90101-8
[12] DOI: 10.1017/S0022112092003392 · Zbl 0747.76089 · doi:10.1017/S0022112092003392
[13] DOI: 10.1016/0375-9601(80)90466-1 · doi:10.1016/0375-9601(80)90466-1
[14] Ši’lnikov L. P., Sov. Math. Docklady 6 pp 163–
[15] DOI: 10.1070/SM1970v010n01ABEH001588 · Zbl 0216.11201 · doi:10.1070/SM1970v010n01ABEH001588
[16] DOI: 10.1109/81.246142 · Zbl 0850.93352 · doi:10.1109/81.246142
[17] DOI: 10.1103/PhysRevE.50.R647 · doi:10.1103/PhysRevE.50.R647
[18] DOI: 10.1119/1.19538 · doi:10.1119/1.19538
[19] DOI: 10.1016/S0375-9601(00)00026-8 · doi:10.1016/S0375-9601(00)00026-8
[20] DOI: 10.1016/S0764-4442(99)80439-X · Zbl 0935.34050 · doi:10.1016/S0764-4442(99)80439-X
[21] Ueta T., Int. J. Bifurcation and Chaos 10 pp 1917–
[22] Wang X. F., Int. J. Bifurcation and Chaos 10 pp 549–
[23] DOI: 10.1007/978-1-4612-1042-9 · doi:10.1007/978-1-4612-1042-9
[24] R. Williams, Turbulence Seminar Berkeley 1996/97, eds. P. Bermard and T. Ratiu (Springer-Verlag, Berlin, 1997) pp. 94–112.
[25] DOI: 10.1142/S0218127403008089 · Zbl 1046.37018 · doi:10.1142/S0218127403008089
[26] DOI: 10.1142/S0218127404010175 · Zbl 1129.37325 · doi:10.1142/S0218127404010175
[27] DOI: 10.1016/S0960-0779(03)00251-0 · Zbl 1053.37015 · doi:10.1016/S0960-0779(03)00251-0
[28] DOI: 10.1142/S0218127404011296 · Zbl 1129.37326 · doi:10.1142/S0218127404011296
[29] DOI: 10.1016/S0960-0779(03)00243-1 · Zbl 1053.37016 · doi:10.1016/S0960-0779(03)00243-1
[30] DOI: 10.1016/j.chaos.2003.10.030 · Zbl 1048.37032 · doi:10.1016/j.chaos.2003.10.030
[31] DOI: 10.1007/s11071-005-4195-8 · Zbl 1142.70012 · doi:10.1007/s11071-005-4195-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.