×

zbMATH — the first resource for mathematics

Nonlinear analysis in a Lorenz-like system. (English) Zbl 1208.34066
The authors investigate dynamical behaviors of a Lorenz-like system. They study the stability and bifurcations which occur in a new three parameter quadratic chaotic system. Also, the existence of singularly degenerate heteroclinic cycles is obtained via choosing some suitable bifurcation parameters. Finally, the disappearance of these cycles triggers the existence of chaotic attractors.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
34C28 Complex behavior and chaotic systems of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lorenz, E.N., Deterministic nonperiodic flow, J. atmospheric sci., 20, 130-141, (1963) · Zbl 1417.37129
[2] Sparrow, C., The Lorenz equations: bifurcations, chaos and strange attractors, (1982), Springer-Verlag New York · Zbl 0504.58001
[3] Chen, G.; Ueta, T., Yet another chaotic attractor, Internat. J. bifur. chaos, 9, 1465-1466, (1999) · Zbl 0962.37013
[4] Liu, C.; Liu, T.; Liu, L.; Liu, K., A new chaotic attractor, Chaos solitons fractals, 22, 1031-1038, (2004) · Zbl 1060.37027
[5] Rössler, E., An equation for continuous chaos, Phys. lett. A, 57, 397-398, (1976) · Zbl 1371.37062
[6] Rikitake, T., Oscillations of a system of disk dynamos, Proc. R. Cambridge philos. soc., 54, 89-105, (1958) · Zbl 0087.23703
[7] Lü, J.; Chen, G., A new chaotic attractor coined, Internat. J. bifur. chaos, 12, 659-661, (2002) · Zbl 1063.34510
[8] Genesio, R.; Tesi, A., Harmonic balance methods for analysis of chaotic dynamics in nonlinear systems, Automatica, 28, 531-548, (1992) · Zbl 0765.93030
[9] Chua, L.O., Chua’s circuit. an overview ten years later, J. circuits syst. comput., 4, 117-159, (1994)
[10] Gao, S.; Teng, Z.; Nieto, J.J.; Torres, A., Analysis of an SIR epidemic model with pulse vaccination and distributed time delay, J. biomed. biotechnol., 2007, (2007), Article ID 64870
[11] Zhang, H.; Chen, L.; Nieto, J.J., A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear anal. RWA, 9, 1714-1726, (2008) · Zbl 1154.34394
[12] Mello, L.F.; Coelho, S.F., Degenerate Hopf bifurcations in the Lü system, Phys. lett. A, 373, 1116-1120, (2009) · Zbl 1228.70014
[13] Mello, L.F.; Messias, M.; Braga, D.C., Bifurcation analysis of a new Lorenz-like chaotic system, Chaos solitons fractals, 37, 1244-1255, (2008) · Zbl 1153.37356
[14] Écalle, J., Introduction aux functions analysables et preuve constructive de la conjecture de Dulac, (1992), Hermann Paris, (in French) · Zbl 1241.34003
[15] Ilyashenko, Y., Finiteness theorems for limit cycles, (1993), American Mathematical Society Providence, RI
[16] A. Ferragut, J. Llibre, C. Pantazi, Polynomial vector fields in \(\mathbb{R}^3\) with infinitely many limit cycles, preprint · Zbl 1270.34041
[17] Li, X.-F.; Chu, Y.-D.; Zhang, J.-G.; Chang, Y.-X., Nonlinear dynamics and circuit implementation for a new Lorenz-like attractor, Chaos solitons fractals, 41, 2360-2370, (2009) · Zbl 1198.37048
[18] Kokubu, H.; Roussarie, R., Existence of a singularly degenerate heteroclinic cycle in the Lorenz system and its dynamical consequences: part I, J. dynam. differential equations, 16, 513-557, (2004) · Zbl 1061.34036
[19] Messias, M., Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system, J. phys. A: math. theor., 42, 115101, (2009) · Zbl 1181.37019
[20] Pontryagin, L.S., Ordinary differential equations, (1962), Addison-Wesley Publishing Company Inc. Reading · Zbl 0112.05502
[21] Kuznetsov, Y.A., Elements of applied bifurcation theory, (1998), Springer-Verlag New York · Zbl 0914.58025
[22] Sotomayor, J.; Mello, L.F.; Braga, D.C., Bifurcation analysis of the watt governor system, Comput. appl. math., 26, 19-44, (2007) · Zbl 1182.70038
[23] Sotomayor, J.; Mello, L.F.; Braga, D.C., Hopf bifurcations in a watt governor with a spring, J. nonlinear math. phys., 15, 278-289, (2008)
[24] Chu, Y.-D.; Zhang, J.-G.; Li, X.-F.; Chang, Y.-X.; Luo, G.-W., Chaos and chaos synchronization for a non-autonomous rotational machine systems, Nonlinear anal. RWA, 9, 1378-1393, (2008) · Zbl 1154.34333
[25] Wolf, A.; Swift, J.; Swinney, H.; Vastano, J., Determining Lyapunov exponents from a time series, Physica D, 16, 285-317, (1985) · Zbl 0585.58037
[26] Yang, Q.; Chen, G., A chaotic system with one saddle and two stable node-foci, Internat. J. bifur. chaos, 18, 1393-1414, (2008) · Zbl 1147.34306
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.