November  2017, 16(6): 2253-2267. doi: 10.3934/cpaa.2017111

Generalized Lorenz Equations for Acoustic-Gravity Waves in the Atmosphere. Attractors Dimension, Convergence and Homoclinic Trajectories

Saint-Petersburg State University, Universitetsky pr. 28, Saint-Petersburg, Russia, 198504

* Corresponding author

Received  April 2017 Revised  May 2017 Published  July 2017

Fund Project: G. A. Leonov is supported by the Russian Science Foundation project no. 14-21-00041.

Attractors dimension of Lorenz-Stenflo system is estimated. Convergence criteria are proved. Fishing principle for existence of homoclinic trajectory is applied.

Citation: G. A. Leonov. Generalized Lorenz Equations for Acoustic-Gravity Waves in the Atmosphere. Attractors Dimension, Convergence and Homoclinic Trajectories. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2253-2267. doi: 10.3934/cpaa.2017111
References:
[1]

L. Stenflo, Generalized Lorenz equations for acoustic-gravity waves, Atmosphere Physics Scripts, 53 (1996), 83-84. 

[2]

E. N. Lorenz, Deterministic nonperiodic flow, Atmos. Sci., 20 (1963), 130-141. 

[3]

O. A. Ladyzhenskaya, Determination of minimal global attractors for the Navier-Stokes equations and other particl, Differential Equations. Russian Mathematical Surveys, 42 (1987), 25-60. 

[4]

J. Kaplan and J. Yorke, Chaotic behavior of multidimensional difference equations, Functional Differential Equations and Approximations of Fixed Points, Springer, Berlin (H. Peitgen and H. Walter eds. ), (1979), 204-227.

[5]

G. A. Leonov, Lyapunov dimension formulas for Henon and Lorenz attractors, St. Petersburg Math. J., 13 (2002), 453-464. 

[6]

G. A. LeonovN. V. KuznetsovN. Korzhemanova and D. Kusakin, Lyapunov dimension formula for the global attractor of the Lorenz system, Communications in Nonlinear Science and Numerical Simulation, 41 (2016), 84-103.  doi: 10.1016/j.cnsns.2016.04.032.

[7]

G. A. Leonov, Lyapunov functions in the attractors dimension theory, Appl. Math. and Mech., 76 (2012), 129-141.  doi: 10.1016/j.jappmathmech.2012.05.002.

[8]

G. A. Leonov, Fishing principle for homoclinic and heteroclinic trajectories, Nonlinear Dynamics, 78 (2014), 2751-2758.  doi: 10.1007/s11071-014-1622-8.

[9]

G.A. Leonov, General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu-Morioka, Lu and Chen systems, Phys. Lett. A, 376 (2012), 3045-3050.  doi: 10.1016/j.physleta.2012.07.003.

[10]

G. A. Leonov, Rössler systems: estimates for the dimension of attractors and homoclinic orbits, Dokl. Math., 89 (2014), 369-371. 

[11]

G. A. Leonov, Existence criterion of homoclinic trajectories in the Glukhovsky-Dolzhansky system, Physics Letters A, 379 (2015), 524-528.  doi: 10.1016/j.physleta.2014.12.005.

[12]

G. A. Leonov, Strange Attractors and Classical Stability Theory St. Petersburg University Press, St. Petersburg, 2008.

show all references

References:
[1]

L. Stenflo, Generalized Lorenz equations for acoustic-gravity waves, Atmosphere Physics Scripts, 53 (1996), 83-84. 

[2]

E. N. Lorenz, Deterministic nonperiodic flow, Atmos. Sci., 20 (1963), 130-141. 

[3]

O. A. Ladyzhenskaya, Determination of minimal global attractors for the Navier-Stokes equations and other particl, Differential Equations. Russian Mathematical Surveys, 42 (1987), 25-60. 

[4]

J. Kaplan and J. Yorke, Chaotic behavior of multidimensional difference equations, Functional Differential Equations and Approximations of Fixed Points, Springer, Berlin (H. Peitgen and H. Walter eds. ), (1979), 204-227.

[5]

G. A. Leonov, Lyapunov dimension formulas for Henon and Lorenz attractors, St. Petersburg Math. J., 13 (2002), 453-464. 

[6]

G. A. LeonovN. V. KuznetsovN. Korzhemanova and D. Kusakin, Lyapunov dimension formula for the global attractor of the Lorenz system, Communications in Nonlinear Science and Numerical Simulation, 41 (2016), 84-103.  doi: 10.1016/j.cnsns.2016.04.032.

[7]

G. A. Leonov, Lyapunov functions in the attractors dimension theory, Appl. Math. and Mech., 76 (2012), 129-141.  doi: 10.1016/j.jappmathmech.2012.05.002.

[8]

G. A. Leonov, Fishing principle for homoclinic and heteroclinic trajectories, Nonlinear Dynamics, 78 (2014), 2751-2758.  doi: 10.1007/s11071-014-1622-8.

[9]

G.A. Leonov, General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu-Morioka, Lu and Chen systems, Phys. Lett. A, 376 (2012), 3045-3050.  doi: 10.1016/j.physleta.2012.07.003.

[10]

G. A. Leonov, Rössler systems: estimates for the dimension of attractors and homoclinic orbits, Dokl. Math., 89 (2014), 369-371. 

[11]

G. A. Leonov, Existence criterion of homoclinic trajectories in the Glukhovsky-Dolzhansky system, Physics Letters A, 379 (2015), 524-528.  doi: 10.1016/j.physleta.2014.12.005.

[12]

G. A. Leonov, Strange Attractors and Classical Stability Theory St. Petersburg University Press, St. Petersburg, 2008.

Figure 13.  Separatrix $X(t,s)^+$, $s\in[0,s_0]$.
Figure 14.  Separatrix $X(t,s)^+$, $s=s_0$.
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