## Dynamical analysis of a new autonomous 3-D chaotic system only with stable equilibria.(English)Zbl 1213.37061

The authors introduce a new chaotic system described by a system of three ordinary differential equations with six terms, one of which is nonlinear (exponential function). They study its properties with the goal to compare the topological structure of a new system with that of a Lorenz system and some other known Lorenz-like chaotic systems. The existence of singularly degenerate heteroclinic cycles, periodic solutions and chaotic attractors are investigated. It is shown that in the case when all equilibria of a new system are stable, the system gives rise to a double-scroll chaotic attractor which does not satisfy the conditions of the Sil’nikov homoclinic theorem. The results of numerical simulations are also provided.

### MSC:

 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 34C28 Complex behavior and chaotic systems of ordinary differential equations
Full Text:

### References:

 [1] Lorenz, E.N., Deterministic non-periodic flow, J. atmospheric sci., 20, 130-141, (1963) · Zbl 1417.37129 [2] Rössler, O.E., An equation for continuous chaos, Phys. lett. A, 57, 397-398, (1976) · Zbl 1371.37062 [3] Sprott, J.C., Some simple chaotic flows, Phys. rev. E, 50, 647-650, (1994) [4] Sprott, J.C., A new class of chaotic circuit, Phys. lett. A, 266, 19-23, (2000) [5] Sprott, J.C., Simplest dissipative chaotic flow, Phys. lett. A, 228, 271-274, (1997) · Zbl 1043.37504 [6] Chen, G.R.; Ueta, T., Yet another chaotic attractor, Internat. J. bifur. chaos, 9, 1465-1466, (1999) · Zbl 0962.37013 [7] Lü, J.H.; Chen, G.R., A new chaotic attractor conined, Internat. J. bifur. chaos, 12, 659-661, (2002) · Zbl 1063.34510 [8] Yang, Q.G.; Chen, G.R.; Huang, K.F., Chaotic attractors of the conjugate Lorenz-type system, Internat. J. bifur. chaos, 17, 3929-3949, (2007) · Zbl 1149.37308 [9] van der Schrier, G.; Maas, L.R.M., The diffusionless Lorenz equations: sil’nikov bifurcations and reduction to an explicit map, Physica D, 141, 19-36, (2000) · Zbl 0956.37038 [10] Shaw, R., Strange attractor, chaotic behaviour and information flow, Z. naturforsch., 36A, 80-112, (1981) · Zbl 0599.58033 [11] Yang, Q.G.; Chen, G.R., A chaotic system with one saddle and two stable node-foci, Internat. J. bifur. chaos, 18, 1393-1414, (2008) · Zbl 1147.34306 [12] Dias, F.S.; Mello, L.F.; Zhang, J.G., Nonlinear analysis in a Lorenz-like system, Nonlinear anal. RWA, (2009) [13] Sparrow, C., The Lorenz equations: bifurcation, chaos, and strange attractor, (1982), Springer-Verlag New York [14] Zhou, T.S.; Chen, G.R.; Tang, Y., Complex dynamical behaviors of the chaotic chen’s system, Internat. J. bifur. chaos, 13, 2561-2574, (2003) · Zbl 1046.37018 [15] Yang, Q.G.; Chen, G.R.; Zhou, T.S., A unified Lorenz-type system and its canonical form, Internat. J. bifur. chaos, 16, 2855-2871, (2006) · Zbl 1185.37088 [16] Kokubu, H.; Roussarie, R., Existence of a singularly degenerate heterclinic cycle in the Lorenz system and its dynamical consequences: part 1*, J. dyn. differ. equ., 16, 513-557, (2004) · Zbl 1061.34036 [17] 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 [18] Mello, L.F.; Coelho, S.F., Degenerate Hopf bifurcations in the Lü system, Phys. lett. A, 373, 1116-1120, (2009) · Zbl 1228.70014 [19] 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 [20] Li, J.; Zhang, J., New treatment on bifurcation of periodic solutions and homoclinic orbits at high $$r$$ in the Lorenz equations, SIAM J. appl. math., 53, 1059-1071, (1993) · Zbl 0781.34031 [21] Yu, Y.; Zhang, S., Hopf bifurcation analysis in the Lü system, Chaos solitons fractals, 21, 1215-1220, (2004) · Zbl 1061.37029 [22] Huang, D., Periodic orbits and bomoclinic orbits of the diffusionless Lorenz equations, Phys. lett. A, 309, 248-253, (2003) · Zbl 1009.37010 [23] Wei, Z.C.; Yang, Q.G., Controlling the diffusionless Lorenz equations with periodic parametric perturbation, Comput. math. appl., 58, 1979-1987, (2009) · Zbl 1189.34118 [24] Sil’nikov, L.P., A case of the existence of a countable number of periodic motions, Sov. math. docklady, 6, 163-166, (1965) · Zbl 0136.08202 [25] Sil’nikov, L.P., A contribution of the problem of the structure of an extended neighborhood of rough equilibrium state of saddle – focus type, Math. USSR-shornik, 10, 91-102, (1970) · Zbl 0216.11201 [26] Silva, C.P., Sil’nikov theorem-a tutorial, IEEE trans. circuits syst. I, 40, 657-682, (1993) [27] Hardy, Y.; Steeb, W.H., The Rikitake two-disk dynamo system and domains with periodic orbits, Int. J. theor. phys., 38, 2413-2417, (1999) · Zbl 0980.86005 [28] Llibre, J.; Messias, M., Global dynamics of Rikitake system, Physica D, 238, 241-252, (2009) · Zbl 1162.37017 [29] Llibre, J.; Zhang, X., Invariant algebraic surfaces of the Rikitake system, J. phys. A: math. gen., 33, 7613-7635, (2000) · Zbl 0967.34002
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.