STABILITY OF PERIODIC ORBITS IN THE AVERAGING THEORY: APPLICATIONS TO LORENZ AND THOMAS’ DIFFERENTIAL SYSTEMS

We study the kind of stability of the periodic orbits provided by higher order averaging theory. We apply these results for determining the k−hyperbolicity of some periodic orbits of the Lorenz and Thoma’s differential system.


Introduction and statement of our main result
The averaging theory is a classical method for studying the solutions of the non-linear dynamical systems, and in particular their periodic solutions.For a general introduction to the averaging theory see the book of Sanders, Verhulst and Murdock [9], and the references quoted there.Recently many works extending and improving the averaging method for computing periodic solutions were presented, see for instance [1,5,4,3].Most of these results enhance the number of periodic solutions that can be detected by averaging method.Although few comments are made about the stability of these periodic solutions.To fill this gap the present work provides an strategy to determine the stability of the periodic orbits that bifurcate from of periodic orbits which form a manifold, or from points inside a continuum set that vanish some averaging functions, see Theorem 2. The detection of such bifurcations is possible by applying the Lyapunov-Schimdt reduction method over higher order averaging functions, see Theorem 1.This theorem was already used in [2] and [4] without the stability analysis.
This hypothesis is always true when the unperturbed system has a manifold of Tperiodic solutions.The standard method of averaging for finding periodic solutions consists in write the displacement map (2) as power series of ε in the following way Where for i = 0, 1, 2, 3, 4 we have The functions g 1 , g 2 , g 3 and g 4 will be called here the averaged functions of order 1, 2, 3 and 4 respectively of system (1).
We say that system (1) has a periodic solution bifurcating from the point z 0 if there exists a branch of solutions z(ε) for the displacement function such that d(z(ε), ε) = 0 and z(0) = z 0 .Now we shall present our result about the existence and stability of the periodic solutions of system (1).The methodology used here was introduced for studying differential systems such that the unperturbed part has a sub-manifold of T -periodic solutions, see for instance [1] and [4].The main difference of this work with the previous ones is that the first nonzero averaged function vanishes over a graph.
Let π : R m × R n−m → R m and π ⊥ : R m × R n−m → R n−m denote the projections onto the first m coordinates and onto the last n − m coordinates, respectively.For a point z ∈ U we also consider z = (a, b) ∈ R m × R n−m .Consider the graph The next theorem provides sufficient conditions for the existence of periodic solutions in the differential system (1).This theorem was proved in [3] here we also provide an scheme of its proof.We need this theorem for the statement of our main result in Theorem 2.
Theorem 1.Let r ∈ {0, 1, 2} such that r is the first subindex such that g r ≡ 0. In addition to hypothesis (H) assume that (i) the averaged function g r vanishes on the graph (3).That is g r (z α ) = 0 for all α ∈ V , and (ii) the Jacobian matrix We define the functions Then the following statements hold.
(a) If there exists α * ∈ V such that f 1 (α * ) = 0 and det (Df Theorem 1 shows that the function f 1 and f 2 provides sufficient conditions for the existence of periodic solutions of the differential system (1).
For periodic solutions detected by statement (a) of Theorem 1 the next result reveals how the higher order function f 2 can be used for determining the stability of the periodic solution x(t, z(ε), ε).
The same class of result can be obtained for periodic orbits detected by statement (b) of Theorem 1 using the bifurcation function of order 3.The expressions of such functions are explicitly given in [3].

Applications
Lorenz differential system.Consider the differential system ẋ =a(x − y), ż =x y − cz, with a, b, c being real coefficients.In recent publication [2] the authors have found a periodic orbit bifurcating from the origin of system (9), see Figure 2. The next theorem completes this work giving the stability characterization of that periodic solution.
Theorem 3. Let a = −1 + a 2 ε 2 and c = c 1 ε.Assume that b > 1, a 2 < 0, c 1 = 0 and |ε| = 0 sufficiently small.Then the Lorenz differential system (9) has a periodic orbit bifurcating from the origin.Furthermore for c 1 > 0 this periodic orbit is an attractor, otherwise for c 1 < 0 the periodic orbit has a stable manifold formed by two topological cylinders and an unstable manifold formed by two topological cylinders.Theorem 3 is proved in section 3 using Theorems 1 and 2.
Thomas' systems.A circulant system is a differential system defined by a function f (x, y, z) having the variables cyclically symmetric according to ẋ = f (x, y, z), where the function f (u, v, w) is fixed and the variables are rotated.In 1999 René Thomas propose two circulant systems having cyclic symmetry ẋ = sin y − βx, 10) is defeined by the function f (u, v, w) = −au + sin v and system (11) is defined by f (u, v, w) = −au + bv − v 3 .The chaotic behaviour generated by these systems was presented in [12], system (10) was also studied by Sprott and Chlouverakis in [11].System (10) is sometimes called Thomas' system, see for instance [10,Chapter 3].The next results give sufficient conditions for the existence of periodic solutions on these differential systems.
One can check that the origin is an equilibrium point of system (10), and that it has the eigenvalues 1 origin has a pair of complex eigenvalues on the imaginary axis and the bifurcation of a periodic orbit occurs.
For ε > 0 sufficiently small and β 1 > 0 the differential system (10) has an isolated periodic solution bifurcating from the origin.
Theorem 4 is proved in section 3 using Theorems 1 and 2 taking r = 0. System (11) has 27 stead states but we will be interested into the pair symmetric equilibrium Taking a = 5 √ 3ω/6 and b = √ 3ω/3 with ω > 0, these equilibrium points have the eigenvalues − √ 3ω and ±ωi.The next theorems show that periodic orbits are born at P − and P + .
Theorem 5. Let a = 5 √ 3ω/6+εa 1 , b = √ 3ω/3+εb 1 with ω > 0 and (5b 1 −2a 1 ) < 0. Then for ε > 0 sufficiently small the differential system (11) has two periodic solutions such that φ + (t, ε) bifurcates from P + and φ − (t, ε) bifurcates from P − .Here The periodic orbit analytically found in Theorem 5 was detected numerically by Thomas in [12], he also shows for specific values of a and b that these periodic solutions give born to a strange attractors after a cascade of doubling.The following figures illustrate this phenomena.Here a 1 = 6, b 1 = 1 and ω = 1 the time interval is from 0 to 1000. Figure 2 shows the solution starting at (−0.8, −0.8, −0.45) being attracted by the periodic orbit φ − (t, ε), see equation ( 12).As we increase ε the periodic orbit grows in size and complexity, see Figures 3, 4. The approximation to the periodic orbit provided by ( 12) can be seen as a dashed curve.Figures 5, 6 and 7 shows the appearance of the strange attractor as ε increase.
A fundamental notion in qualitative theory of differential equations is hyperbolicity.A constant matrix will be called hyperbolic if its eigenvalues lie out of the imaginary axis, in which case its index is the number of eigenvalues in the right half-plane.Consider a matrix function A k depending on a parameter ε.If A 0 is hyperbolic of index i, then one can see that for ε > 0 sufficiently small A(ε) will be hyperbolic with the same index i.
If A 0 is not hyperbolic the placement of the eigenvalues of A(ε) may be hard to determine.To deal with this problem we use a method introduced by Murdock and Robinson, see [7] and [8].The matrix A(ε) is called k-hyperbolic of index i if for every smooth matrix function B(ε) there exists an ε 0 > 0 such that A(ε) is hyperbolic of index i for all ε in the interval 0 < ε < ε 0 .The next result will be needed for proving Theorem 2.
Assume that there exists a matrix function S(ε) that block diagonalizes A(ε) into its left, right and center blocks L(ε), C(ε), R(ε) which for ε = 0 have their eigenvalues respectively in the left half-plane, on the imaginary axis, and in the right half-plane.Thus Theorem 6 ([6, Theorem 5.7]).Let C(ε) be the center block of A(ε) and let its size be m × m.Then A(ε) is k-hyperbolic provided that: (a) A 0 has no multiple eigenvalues on the imaginary axis.
Using (21) we can write the Jacobian matrix of the displacement function at z(ε) as a power series of ε around ε = 0 as a classical result about ordinary differential equations says that when (22) is a hyperbolic matrix, the periodic solution x(t, z(ε), ε) will be hyperbolic with the same kind of stability.This is also referred as linear stability.Thus the proof of the theorem follows from applying Theorem 6 in the 1-jet (5) observing that hypothesis (s 1 ) and (s 2 ) are equivalent with the hypothesis (a ) and (b ) respectively.Thus the matrix is 2−hyperbolic and the theorem is proved.
Proof of Theorem 3. The existence of such periodic orbit is proved in Theorem 4 of [2].Following the ideas of this proof we see that, after some changes of variables, system (9) can be put into the normal form for applying Theorem 1, given by equation ( 22) of [2], with z = (ρ, z) and the derivative with respect to θ.Thus calculating the higher order averaging functions of this system for i = 0, 1, 2, 3 we have g i (z) = (g i1 (z), g i2 (z)) where g 0 (z) ≡ 0 and Thus we can calculate the functions f i (α) for i = 1, 2 with respect to the averaging functions above and the graph .
By the hypothesis of Theorem 3 one can check that α * = 2ω √ −2a 2 is a simple zero of function f 1 (α).Then we can apply Theorem 2 with r = 1.By (21) we can write the initial point of the periodic solution as z(ε) = z α * + εz 1 with and the matrix (5) becomes The matrix A(ε) has the two distinct eigenvalues As a 2 is negative by hypothesis, we have that for ε > 0 sufficiently small if c 1 > 0, Re(λ 1 ) < Re(λ 2 ) < 0 consequently the periodic orbit is an attractor.Otherwise, if c 1 < 0, Re(λ 2 ) < 0 < Re(λ 1 ) then the periodic orbit has a stable manifold formed by two topological cylinders, and an unstable manifold formed by two topological cylinders.
System (24) is 2π-periodic and it is into the normal form for applying Theorem 1.