Wang, Xiong; Chen, Juan; Lu, Jun-An; Chen, Guanrong A simple yet complex one-parameter family of generalized Lorenz-like systems. (English) Zbl 1258.34112 Int. J. Bifurcation Chaos Appl. Sci. Eng. 22, No. 5, Paper No. 1250116, 16 p. (2012). Summary: This paper reports the finding of a simple one-parameter family of three-dimensional quadratic autonomous chaotic systems. By tuning the only parameter, this system can continuously generate a variety of cascading Lorenz-like attractors, which appears to be richer than the unified chaotic system that contains the Lorenz and the Chen systems as its two extremes. Although this new family of chaotic systems has very rich and complex dynamics, it has a very simple algebraic structure with only two quadratic terms (same as the Lorenz and the Chen systems) and all nonzero coefficients in the linear part being \(-1\) except one \(-0.1\) (thus, simpler than the Lorenz and Chen systems). Surprisingly, although this new system belongs to the Lorenz-type of systems in the classification of the generalized Lorenz canonical form, it can generate not only Lorenz-like attractors but also Chen-like attractors. This suggests that there may exist some other unknown yet more essential algebraic characteristics for describing general three-dimensional quadratic autonomous chaotic systems. Cited in 12 Documents MSC: 34C60 Qualitative investigation and simulation of ordinary differential equation models 34C28 Complex behavior and chaotic systems of ordinary differential equations 34D45 Attractors of solutions to ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations Keywords:chaotic attractor; Lorenz system; Chen system; generalized Lorenz canonical form PDFBibTeX XMLCite \textit{X. Wang} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 22, No. 5, Paper No. 1250116, 16 p. (2012; Zbl 1258.34112) Full Text: DOI arXiv References: [1] DOI: 10.1142/S0218127402005467 · Zbl 1043.37023 · doi:10.1142/S0218127402005467 [2] DOI: 10.1016/j.chaos.2005.02.040 · Zbl 1100.37016 · doi:10.1016/j.chaos.2005.02.040 [3] Čelikovský S., Kybernetika 30 pp 403– [4] DOI: 10.1142/S0218127499001024 · Zbl 0962.37013 · doi:10.1142/S0218127499001024 [5] 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 [6] DOI: 10.1142/S0218127402004620 · Zbl 1063.34510 · doi:10.1142/S0218127402004620 [7] DOI: 10.1142/S021812740200631X · Zbl 1043.37026 · doi:10.1142/S021812740200631X [8] DOI: 10.1016/j.physa.2004.12.040 · doi:10.1016/j.physa.2004.12.040 [9] DOI: 10.1007/978-1-4612-5767-7 · doi:10.1007/978-1-4612-5767-7 [10] DOI: 10.1016/0097-8493(93)90082-K · doi:10.1016/0097-8493(93)90082-K [11] DOI: 10.1103/PhysRevE.50.R647 · doi:10.1103/PhysRevE.50.R647 [12] DOI: 10.1016/S0375-9601(97)00088-1 · Zbl 1043.37504 · doi:10.1016/S0375-9601(97)00088-1 [13] Sprott J. C., J. Chaos Th. Appl. 5 pp 3– [14] Ueta T., Int. J. Bifurcation and Chaos 10 pp 1917– [15] Vanečěk A., Control Systems: From Linear Analysis to Synthesis of Chaos (1996) [16] DOI: 10.1142/S0218127408021063 · Zbl 1147.34306 · doi:10.1142/S0218127408021063 [17] DOI: 10.1142/S0218127407019792 · Zbl 1149.37308 · doi:10.1142/S0218127407019792 [18] DOI: 10.1142/S0218127406016501 · Zbl 1185.37088 · doi:10.1142/S0218127406016501 [19] DOI: 10.1016/S0960-0779(03)00251-0 · Zbl 1053.37015 · doi:10.1016/S0960-0779(03)00251-0 [20] DOI: 10.1142/S0218127403008089 · Zbl 1046.37018 · doi:10.1142/S0218127403008089 [21] DOI: 10.1142/S0218127404011296 · Zbl 1129.37326 · doi:10.1142/S0218127404011296 [22] DOI: 10.1142/S0218127406016203 · Zbl 1185.37092 · doi:10.1142/S0218127406016203 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.