## Shil’nikov chaos in the 4D Lorenz-Stenflo system modeling the time evolution of nonlinear acoustic-gravity waves in a rotating atmosphere.(English)Zbl 1284.37022

Summary: The Lorenz-Stenflo system serves as a model of the time evolution of nonlinear acoustic-gravity waves in a rotating atmosphere. In the present paper, we study the Shil’nikov chaos which arises in the 4D Lorenz-Stenflo system. The analytical and numerical results constitute an application of the Shil’nikov theorems to a 4D system (whereas most results present in the literature deal with applying the Shil’nikov theorems to 3D systems), which allows for the study of chaos along homoclinic and heteroclinic orbits arising as solutions to the Lorenz-Stenflo system. We verify the observed chaos via competitive modes analysis – a diagnostic for chaotic systems. We give an analytical test, completely in terms of the model parameters, for the Smale horseshoe chaos near homoclinic orbits of the origin, as well as for the case of specific heteroclinic orbits. Numerical results are shown for other cases in which the general analytical method becomes too complicated to apply. These results can be extended to more complicated higher-dimensional systems governing plasmas, and, in particular, may be used to shed light on period-doubling and Smale horseshoe chaos that arises in such models.

### MSC:

 37C29 Homoclinic and heteroclinic orbits for dynamical systems 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 34C28 Complex behavior and chaotic systems of ordinary differential equations 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text:

### References:

 [1] Stenflo, L., Acoustic solitary vortices, Phys. Fluids, 30, 3297, (1987) · Zbl 0635.76084 [2] Stenflo, L., Acoustic gravity vortex chains, Phys. Lett. A, 186, 133, (1994) · Zbl 0941.85500 [3] Stenflo, L.; Stepanyants, Yu., Acoustic gravity modons in the atmosphere, Ann. Geophys., 13, 973, (1995) [4] Horton, W.; Kaladze, T.D.; Dam, J.W.; Garner, T.W., A method for the intensification of atomic oxygen Green line emission by internal gravity waves, J. Geophys. Res., 113, (2008) [5] Stenflo, L., Generalized Lorenz equations for acoustic-gravity waves in the atmosphere, Phys. Scr., 53, 83, (1996) · Zbl 1037.76507 [6] Stenflo, L., Nonlinear acoustic-gravity waves, J. Plasma Phys., 75, 841-847, (2009) [7] Banerjee, S.; Saha, P.; Chowdhury, A.R., Chaotic scenario in the stenflo equations, Phys. Scr., 63, 177, (2001) · Zbl 1115.37366 [8] Ekola, T.: A numerical study of the Lorenz and Lorenz-Stenflo systems. Doctoral thesis, Stockholm, ISBN 91-7283-997-x [9] Liu, Z., The first integral of nonlinear acoustic gravity wave equation, Phys. Scr., 61, 526, (2000) [10] Lonngren, K.E.; Bai, E.W., On the synchronization of acoustic gravity waves, Phys. Scr., 64, 489, (2001) · Zbl 1112.76496 [11] Yu, M.Y., Some chaotic aspects of the Lorenz-stenflo equations, Phys. Scr. T, 82, 10, (1999) [12] Yu, M.Y.; Yang, B., Periodic and chaotic solutions of the generalized Lorenz equations, Phys. Scr., 54, 140, (1996) · Zbl 0947.37023 [13] Zhou, C.; Lai, C.H.; Yu, M.Y., Chaos, bifurcation and periodic orbits of the Lorenz-stenflo system, Phys. Scr., 35, 394, (1997) [14] Zhou, C.T.; Lai, C.H.; Yu, M.Y., Bifurcation behavior of the generalized Lorenz equations at large rotation numbers, J. Math. Phys., 38, 5225, (1997) · Zbl 0897.76038 [15] Gorder, R.A.; Choudhury, S.R., Shil’nikov analysis of homoclinic and heteroclinic orbits of the T system, J. Comput. Nonlinear Dyn., 6, (2011) [16] Sun, F.-Y., Shil’nikov heteroclinic orbits in a chaotic system, Int. J. Mod. Phys. B, 21, 4429-4436, (2007) · Zbl 1186.37031 [17] Wang, J.; Zhao, M.; Zhang, Y.; Xiong, X., Si’lnikov-type orbits of Lorenz-family systems, Physica A, 375, 438-446, (2007) [18] Wilczak, D., The existence of shilnikov homoclinic orbits in the michelson system: a computer assisted proof, Found. Comput. Math., 6, 495-535, (2006) · Zbl 1130.37415 [19] Lamb, J.S.W.; Teixeira, M.-A.; Webster, K.N., Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in ℝ\^{}{3}, J. Differ. Equ., 219, 78-115, (2005) · Zbl 1090.34033 [20] Corbera, M.; Llibre, J.; Teixeira, M.-A., Symmetric periodic orbits near a heteroclinic loop in ℝ\^{}{3} formed by two singular points, a semistable periodic orbit and their invariant manifolds, Physica D, 238, 699-705, (2009) · Zbl 1160.37327 [21] Krauskopf, B.; Rieß, T., A lin’s method approach to finding and continuing heteroclinic connections involving periodic orbits, Nonlinearity, 21, 1655-1690, (2008) · Zbl 1163.34028 [22] Wagenknecht, T., Two-heteroclinic orbits emerging in the reversible homoclinic pitchfork bifurcation, Nonlinearity, 18, 527-542, (2005) · Zbl 1067.37023 [23] Jiang, Y.; Sun, J., Si’lnikov homoclinic orbits in a new chaotic system, Chaos Solitons Fractals, 32, 150-159, (2007) · Zbl 1138.37309 [24] Wang, X., Si’lnikov chaos and Hopf bifurcation analysis of rucklidge system, Chaos Solitons Fractals, 42, 2208-2217, (2009) · Zbl 1198.37058 [25] Wang, J.; Zhao, M.; Zhang, Y.; Xiong, X., Si’lnikov-type orbits of Lorenz-family systems, Physica A, 375, 438-446, (2007) [26] Zhou, L.; Chen, Y.; Chen, F., Stability and chaos of a damped satellite partially filled with liquid, Acta Astronaut., 65, 1628-1638, (2009) · Zbl 1184.49030 [27] Zhou, T.; Chen, G.; Celikovský, S., Si’lnikov chaos in the generalized Lorenz canonical form of dynamical systems, Nonlinear Dyn., 39, 319-334, (2005) · Zbl 1142.70012 [28] Wang, J.; Chen, Z.; Yuan, Z., Existence of a new three-dimensional chaotic attractor, Chaos Solitons Fractals, 42, 3053-3057, (2009) · Zbl 1198.65156 [29] Watada, K.; Tetsuro, E.; Seishi, H., Shilnikov orbits in an autonomous third-order chaotic phase-locked loop, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., 45, 979-983, (1998) [30] Wang, Z., Existence of attractor and control of a 3D differential system, Nonlinear Dyn., 60, 369-373, (2010) · Zbl 1189.70103 [31] Cao, Y.Y.; Chung, K.W.; Xu, J., A novel construction of homoclinic and heteroclinic orbits in nonlinear oscillators by a perturbation-incremental method, Nonlinear Dyn., 64, 221-236, (2011) [32] Chen, H.; Xu, Q., Bifurcations and chaos of an inclined cable, Nonlinear Dyn., 57, 37-55, (2009) · Zbl 1176.74089 [33] Chen, H.; Xu, Q., Global bifurcations and multi-pulse orbits of a parametric excited system with autoparametric resonance, Nonlinear Dyn., 65, 187-216, (2011) · Zbl 1260.34083 [34] Wang, R.; Xiao, D., Bifurcations and chaotic dynamics in a 4-dimensional competitive Lotka-Volterra system, Nonlinear Dyn., 59, 411-422, (2010) · Zbl 1183.92086 [35] Shilnikov, A., Complete dynamical analysis of a neuron model, Nonlinear Dyn., 68, 305-328, (2012) · Zbl 1254.37058 [36] Yagasaki, K., Detection of homoclinic bifurcations in resonance zones of forced oscillators, Nonlinear Dyn., 28, 285-307, (2002) · Zbl 1015.70018 [37] Xu, Y.; Zhu, D., Bifurcations of heterodimensional cycles with one orbit flip and one inclination flip, Nonlinear Dyn., 60, 1-13, (2010) · Zbl 1189.70102 [38] Zhou, T.; Tang, Y.; Chen, G., Chen’s attractor exists, Int. J. Bifurc. Chaos, 9, 3167-3177, (2004) · Zbl 1129.37326 [39] Chen, Z.; Yang, Y.; Yuan, Z., A single three-wing or four-wing chaotic attractor generated from a three-dimensional smooth quadratic autonomous system, Chaos Solitons Fractals, 38, 1187-1196, (2008) · Zbl 1152.37312 [40] Silva, C.P., Shil’nikov theorem—a tutorial, IEEE Trans. Circuits Syst., 40, 675-682, (1993) · Zbl 0850.93352 [41] Shil’nikov, L.P., A case of the existence of a countable number of periodic motions, Sov. Math. Dokl., 6, 163-166, (1965) · Zbl 0136.08202 [42] Shil’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 Sb., 10, 91-102, (1970) · Zbl 0216.11201 [43] Yu, P.; Sun, J.-Q. (ed.); Luo, A.C.J. (ed.), Bifurcation, limit cycles and chaos of nonlinear dynamical systems, 92-120, (2006), Amsterdam [44] Yu, W.; Yu, P.; Essex, C., Estimation of chaotic parameter regimes via generalized competitive mode approach, Commun. Nonlinear Sci. Numer. Simul., 7, 197-205, (2002) · Zbl 1025.34040 [45] Yu, P.; Yao, W.; Chen, G., Analysis on topological properties of the Lorenz and the Chen attractors using GCM, Int. J. Bifurc. Chaos, 17, 2791-2796, (2007) · Zbl 1141.37309 [46] Chen, Z.; Wu, Z.Q.; Yu, P., The critical phenomena in a hysteretic model due to the interaction between hysteretic damping and external force, J. Sound Vib., 284, 783-803, (2005) · Zbl 1237.34055 [47] Gorder, R.A.; Choudhury, S.R., Classification of chaotic regimes in the T system by use of competitive modes, Int. J. Bifurc. Chaos, 20, 3785-3793, (2010) · Zbl 1279.34056 [48] Gorder, R.A., Emergence of chaotic regimes in the generalized Lorenz canonical form: a competitive modes analysis, Nonlinear Dyn., 66, 153-160, (2011) · Zbl 1294.34049 [49] Gorder, R.A., Traveling wave solutions of the $$n$$-dimensional coupled Yukawa equations, Appl. Math. Lett., 25, 1106-1110, (2012) · Zbl 1246.35170 [50] Choudhury, S.R.; Gorder, R.A., Competitive modes as reliable predictors of chaos versus hyperchaos and as geometric mappings accurately delimiting attractors, Nonlinear Dyn., 69, 2255-2267, (2012) [51] Reeves, B.; Gorder, R.A.; Choudhury, S.R., Chaotic regimes, post-bifurcation dynamics, and competitive modes for a generalized double Hopf normal form, Int. J. Bifurc. Chaos, 22, (2012) · Zbl 1258.34082 [52] Ahn, C.K., An answer to the open problem of synchronization for time-delayed chaotic systems, Eur. Phys. J. Plus, 127, 1-9, (2012) [53] Ahn, C.K., A T-S fuzzy model based adaptive exponential synchronization method for uncertain delayed chaotic systems: an LMI approach, J. Inequal. Appl., 2010, (2010) · Zbl 1216.93070 [54] Ahn, C.K., Neural network $$H$$\^{}{∞} chaos synchronization, Nonlinear Dyn., 60, 295-302, (2010) · Zbl 1189.92004 [55] Ahn, C.K.; Jung, S.T.; Kang, S.K.; Joo, S.C., Adaptive $$H$$\^{}{∞} synchronization for uncertain chaotic systems with external disturbance, Commun. Nonlinear Sci. Numer. Simul., 15, 2168-2177, (2010) · Zbl 1222.93079 [56] Ahn, C.K., T-S fuzzy $$H$$\^{}{∞} synchronization for chaotic systems via delayed output feedback control, Nonlinear Dyn., 59, 535-543, (2010) · Zbl 1189.93053 [57] Ahn, C.K., $$L$$_{2}−$$L$$_{∞} chaos synchronization, Prog. Theor. Phys., 123, 421-430, (2010) · Zbl 1203.34097 [58] Ahn, C.K., Fuzzy delayed output feedback synchronization for time-delayed chaotic systems, Nonlinear Anal. Hybrid Syst., 4, 16-24, (2010) · Zbl 1179.93116 [59] Ahn, C.K., Output feedback $$H$$\^{}{∞} synchronization for delayed chaotic neural networks, Nonlinear Dyn., 59, 319-327, (2010) · Zbl 1183.68456 [60] Ahn, C.K., Adaptive $$H$$\^{}{∞} anti-synchronization for time-delayed chaotic neural networks, Prog. Theor. Phys., 122, 1391-1403, (2009) · Zbl 1187.82099 [61] Ahn, C.K., An $$H$$\^{}{∞} approach to anti-synchronization for chaotic systems, Phys. Lett. A, 373, 1729-1733, (2009) · Zbl 1229.34078
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.