
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

Let us prove that if

We make a replacement

where

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


Instead of z in (4) we substitute expression (2), whence we have an equation

whose solution is the function

where

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

and

Substituting (5) and (6) in (7), we have

Consider the non-autonomous case when


Thus, with

Consider now the case when


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

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


having the following properties:





and




more general case is described in [4] (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

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 [4] of Viktor Iosifovich Yudovich (previously unpublished in scientific journals), which details this issue.
Literature
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.