On the existence of periodic solutions in the Lorentz system
This is my third topic on Habré ( part 1 and part 2 ), devoted to the dynamic Lorentz system. I continue to study the question of the existence of periodic solutions (cycles) in this system. It was possible to obtain an interesting result with a certain ratio of its parameters.
Consider the system of Lorentz differential equations
where , r and b are some positive numbers, system parameters.
Let us prove that if , then in system (1) there are no periodic solutions (except, of course, the equilibrium position).
We make a replacement
where is some function of time.
We differentiate (2), we obtain
On the left side of expression (3), we substitute the right side of the third equation of system (1), and on the right side (3) - the right side of the first equation of system (1), given that . We obtain
Instead of z in (4) we substitute expression (2), whence we have an equation
whose solution is the function
where is an arbitrary constant.
Now, in the second equation of system (1), we substitute expression (2) instead of z . Moreover, we express y from the first equation of system (1). We get
Substituting (5) and (6) in (7), we have
Consider the non-autonomous case when in equation (8). Suppose that in this case equation (8) has a periodic solution with period T. Since the derivative of a periodic function with period T is a periodic function with period T , then the left side of equation (8) is a periodic function with period T . However, the right-hand side of equation (8) is non-periodic, since it is not a periodic function. Got a contradiction.
Thus, with equation (8) has no periodic solutions.
Consider now the case when . We have an autonomous second-order equation
for which, according to the Bendixson criterion [1, p. 142-143] there are no periodic solutions, which proves their absence in the Lorentz system at .
Note that in this case the parameter rcan take any value. Then, at sufficiently large values, periodic solutions will also be absent in the Lorentz system, which seems rather unobvious, since the parameter r is proportional to the temperature difference between the lower and upper liquid layers during free convection. With an increase in the temperature gradient in the layer, convective shafts should appear in the fluid, and here the fluid eventually becomes stationary (laminar mode). This is also confirmed in a numerical experiment (stable foci were observed at different values of r - figure (projection of the trajectory arc onto the xOy plane) at the beginning of the topic). Most likely, this is explained by the fact that the Lorentz system rather roughly describes this process, although with other relations between and b ( r takes on a rather large value), a stable limit cycle is observed in system (1) [2, p. 291-294].
Despite all the simplicity, in my opinion, this result from the point of view of the theory of differential equations is interesting in that the third-order nonlinear system allowed a decrease in order, which is rare, and the theory of differential equations on the plane is well developed.
Consider another case when . In the literature known to me, it is studied in a linear approximation. We apply the second Lyapunov method. We compose the Lyapunov function
having the following properties:
to effect the right sides of equations (1). Then, by the Barbashin-Krasovsky theorem [3, p. 248-250] any solution of system (1)
more general case is described in  (Lemma 1.2), where the absence of cycles in system (1) is proved (limit regimes are equilibrium positions) for
We also note that the system ( 1) there are always particular solutions of the form
where is an arbitrary constant.
In this topic, the question of the absence of periodic solutions for system (1) was considered. However, analytical studies of the dynamics of system (1) for the presence of cycles are described in the literature, but there are not many sources, since many study the Lorentz system numerically. The following is a list of references where a rigorous proof of the existence of a limit cycle in system (1) was found for large values of the parameter r [2, 4-8]. It includes the manuscript  of Viktor Iosifovich Yudovich (previously unpublished in scientific journals), which details this issue.
1. Nemytsky VV, Stepanov VV Qualitative theory of differential equations. - M .: URSS editorial, 2004.
2. Neymark Yu.I., Landa P.S. Stochastic and chaotic oscillations. - M: LIBROCOM, 2009.
3. Demidovich B.P. Lectures on the mathematical theory of stability. - M .: Nauka, 1967.
4. Yudovich V.I. Asymptotics of limit cycles of the Lorentz system for large Rayleigh numbers // Manuscript of Dep. at VINITI, No. 2611-78. 1978.
5. Robbins KA Periodic Solutions and Bifurcation Structure at High R in the Lorenz Model // SIAM Journal on Applied Mathematics, 36 (3): 457-472, 1979.
6. Shimizu T. Analytic Form of the Simplest Limit Cycle in the Lorenz Model // Physica A: Statistical Mechanics and its Applications, 97 (2): 383-398, 1979. DOI:10.1016 / 0378-4371 (79) 90113-4 .
7. Pokrovsky L.A. Solution of the system of Lorentz equations in the asymptotic limit of a large Rayleigh number. I. Lorentz system in the simplest quantum laser model and application of the averaging method to it // Theoretical and Mathematical Physics, 62 (2): 272-290, 1985.
8. Jibin Li, Jianming Zhang. New Treatment on Bifurcations of Periodic Solutions and Homoclinic Orbits at High in the Lorenz Equations // SIAM Journal on Applied Mathematics, 53 (4): 1059-1071, 1993.