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 & 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 $X(t,s)^+$, $s=s_0$.
[1]

Sergey Gonchenko, Ivan Ovsyannikov. Homoclinic tangencies to resonant saddles and discrete Lorenz attractors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 273-288. doi: 10.3934/dcdss.2017013

[2]

Cristina Lizana, Leonardo Mora. Lower bounds for the Hausdorff dimension of the geometric Lorenz attractor: The homoclinic case. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 699-709. doi: 10.3934/dcds.2008.22.699

[3]

Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 481-500. doi: 10.3934/dcds.2007.17.481

[4]

Rodrigo Samprogna, Cláudia B. Gentile Moussa, Tomás Caraballo, Karina Schiabel. Trajectory and global attractors for generalized processes. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-26. doi: 10.3934/dcdsb.2019047

[5]

Kuanysh A. Bekmaganbetov, Gregory A. Chechkin, Vladimir V. Chepyzhov, Andrey Yu. Goritsky. Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2375-2393. doi: 10.3934/dcds.2017103

[6]

Dung Le. Exponential attractors for a chemotaxis growth system on domains of arbitrary dimension. Conference Publications, 2003, 2003 (Special) : 536-543. doi: 10.3934/proc.2003.2003.536

[7]

V. V. Chepyzhov, A. Miranville. Trajectory and global attractors of dissipative hyperbolic equations with memory. Communications on Pure & Applied Analysis, 2005, 4 (1) : 115-142. doi: 10.3934/cpaa.2005.4.115

[8]

Xingbo Liu, Deming Zhu. On the stability of homoclinic loops with higher dimension. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 915-932. doi: 10.3934/dcdsb.2012.17.915

[9]

Xinfu Chen. Lorenz equations part II: "randomly" rotated homoclinic orbits and chaotic trajectories. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 121-140. doi: 10.3934/dcds.1996.2.121

[10]

Vladimir Chepyzhov, Alexei Ilyin, Sergey Zelik. Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1835-1855. doi: 10.3934/dcdsb.2017109

[11]

Gregory A. Chechkin, Vladimir V. Chepyzhov, Leonid S. Pankratov. Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1133-1154. doi: 10.3934/dcdsb.2018145

[12]

Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2019101

[13]

Youngna Choi. Attractors from one dimensional Lorenz-like maps. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 715-730. doi: 10.3934/dcds.2004.11.715

[14]

Narcisse Batangouna, Morgan Pierre. Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system. Communications on Pure & Applied Analysis, 2018, 17 (1) : 1-19. doi: 10.3934/cpaa.2018001

[15]

Martin Wechselberger, Warren Weckesser. Homoclinic clusters and chaos associated with a folded node in a stellate cell model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 829-850. doi: 10.3934/dcdss.2009.2.829

[16]

Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281

[17]

Xianwei Chen, Zhujun Jing, Xiangling Fu. Chaos control in a pendulum system with excitations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 373-383. doi: 10.3934/dcdsb.2015.20.373

[18]

Leonardo Mora. Homoclinic bifurcations, fat attractors and invariant curves. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1133-1148. doi: 10.3934/dcds.2003.9.1133

[19]

Gabriel Deugoue. Approximation of the trajectory attractor of the 3D MHD System. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2119-2144. doi: 10.3934/cpaa.2013.12.2119

[20]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (6)
  • HTML views (1)
  • Cited by (2)

Other articles
by authors

[Back to Top]