GENERALIZED LORENZ EQUATIONS FOR ACOUSTIC-GRAVITY WAVES IN THE ATMOSPHERE. ATTRACTORS DIMENSION, CONVERGENCE AND HOMOCLINIC TRAJECTORIES

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

Here Ω is an angular frequency of earth's rotation, κ is a thermal diffusion coefficient, k 1 and k 2 are mode parameters [1].
The passage to chaos in system 1 can be another than the Lorenz scenario [2]. Consider Lorenz parameters σ = 10, b = 8/3, and α = 1. In this case a homoclinic trajectory for r > 1.01 is lacking. For the Lorenz system such a trajectory exists for r = 13.926 . . . [2].

2.
Preliminaries. O. A. Ladyzhenckaya has given the following definition of global B-attractor [3] for the differential equation Definition 2.1. We say that the invariant set K is uniformly globally attractive if for any δ > 0 and bounded set B ⊂ R n there exists t(δ, B) > 0 such that Here K(δ) is δ-neighborhood of set K, X(0, B) = B.
Definition 2.2. We say that K is a global B-attractor if it is invariant (X(t, K) = K, ∀t), bounded, closed, uniformly globally attractive set.
Further we consider global B-attractors K. J. Kaplan and J. Yorke [4] have given the following definitions of Lyapunov dimension.    Let F be continuously differentiable mapping of open set U ⊂ R n , K ⊂ U . Denote by T X F the Jacobian matrix of F at the point X. Let K ⊂ U be a bounded invariant set: Let α j (A) denote singular values of n × n matrix A arranged so that α 1 (A) ≥ · · · ≥ α n (A).
where j is a largest integer number from [0, n] such that and a number s ∈ [0, 1) is such that   .
Definition 2.4. The local Lyapunov dimension of one-parameter group of mapping F t at the point X ∈ K is a number Definition 2.5. The Lyapunov dimension of mappings F t on the set K is a number Consider system 2 with a continuously differentiable function f (X). Assume that for any initial data X 0 , system 2 has a solution X(t, X 0 ) defined for t ∈ [0, +∞). Here X(0, X 0 ) = X 0 .
Denote by F t (X 0 ) = X(t, X 0 ) a shift operator with respect to X 0 , and assume that F t K = K, ∀t ∈ R 1 .  Let J(X) be the Jacobian matrix of f (X): Here * is a sign of transposing.
Hereθ(X) = (grad ϑ(X)) * f (X).   Consider a certain set D ⊂ R n , which is diffeomorphic to a closed ball, whose boundary ∂D is transversal to the vectors f (x), x ∈ ∂D. The set D is positive invariant for solutions x(t) of system 3.
Theorem 2.7 ([5]- [7]). Suppose that there exist a continuously differentiable function ϑ(X) and a nonsingular matrix S such that Then any solution of system 2 with the initial data X(0) ∈ D tends to equilibrium as t → +∞.
Consider a differential equation where f (X, q) is a smooth vector-function, R n = {X} is a phase space of system 5, R m = {q} is a parameter space of system 5.
Let γ(s), s ∈ [0, 1] be a smooth path in a space of parameters {q}. Consider the Tricomi problem [8]- [12]: Is there a point q 0 ∈ γ(s) for which system 5 with q 0 has a homoclinic trajectory?
Recall that a trajectory X(t) of system 5 is said to be homoclinic if the following relation Consider system 5 with q = γ(s), and introduce the following notation: point of the first crossing of separatrix X(t, s) + with the closed set Ω: If such a crossing is lacking, then it is assumed that X(s) + = ∅. Here ∅ is empty set.
The fishing principle can be interpreted as follows. Figure 13 shows a fisherman at the point X 0 with the fishing rod X(t, s) + . The manifold Ω is a lake surface and ∂Ω is a shore line.
If s = 0, a fish has been caught with the fishing rod. Then X(t, s) + , s ∈ [0, s 0 ) is the path of the fishing rod with the fish to the shore.
By assumption 5), the fish cannot be taken to the shore ∂Ω/X 0 since ∂Ω/X 0 is a forbidden zone.
Therefore only the situation shown in Fig. 14 is possible (i.e., for s = s 0 , the fisherman has caught a fish). This corresponds to a homoclinic trajectory.
Let us now describe the numerical procedure to define on the path γ(s) the point Γ, which corresponds to a homoclinic trajectory. Here it is assumed that conditions 1), 2), 5) of Fishing Principle are satisfied. Such a sequence can be obtained if the paths γ and γ j are divided sequentially by two parts of the same length and if from two paths it is chosen the path, for the ends of which the opposite conditions 3) and 4) are satisfied.
The Jacobian matrix of the right-hand side of system 1 has the form The latter inequality is satisfied if By Lemma 3.1 in Ω we have Consequently inequality 8 is satisfied if Thus here it is proved the following result.
These inequalities are satisfied if This inequality is satisfied if By Lemma 3.1 in Ω we have Consequently inequality 10 is satisfied in Ω if This implies the following relations In this case by Theorem 2.7 it is valid the following result.

Consider now relations
.
Then by Theorem 2.6 we have the following result.
Then dim L K < 2 + s.
is satisfied.
This result it follows from the last equations of system 1.  is a solution of system 1.