On asymptotic behaviour of solution of Lorenz system^{1}

G.A. Leonov ^{2}

leonov@math.spbu.ru

1 Introduction

This paper considers the Lorenz system and gives for it localization estimates of global attractors and criteria of existence of homoclinic orbits. It is demonstrated that in certain cases these localization estimates are asymptotically exact. Since the estimates are uniform with respect to parameters it became possible to prove the existence of homoclinic orbits with the use of formulas of asymptotic integration. The Lorenz system

\begin{matrix}ẋ = −d(x − y), \\ ẏ = rx − y − xz, \\ ż = −bz + xy, \end{matrix}  (1)

which is a three-mode approximation of a two-dimensional thermal convection, is now one of the classical models for the transition of global stability into chaotic oscillations. In the process of such transformation the role of homoclinic orbits, which appear for bifurcational values of parameters is very important. These orbits and the attractors of Lorenz system are situated in certain domains of phase space which can be estimated with the help of localization technique. The obtaining of localization estimates and the investigating of homoclinic orbits is the subject of this paper. We shall suppose further that d, r, b are positive numbers. Let, furthermore, r > 1 and 2d > b. Note that if one of these restrictions is violated then system (1) is globally stable [1, 2], i.e. any its solution tends to a certain equilibrium when t → +∞. Along with system (1) we consider the equivalent system

\begin{matrix}σ̇ = η, \\ η̇ = −μη − ξσ − φ(σ), \\ ξ̇ = −αξ − βση, \end{matrix}  (2)

where φ(σ) = −σ+γσ^{3}, σ = εx/\sqrt{2d}, η = ε^{2}(y−x)/\sqrt{2} , ξ = ε^{2}(z −x^{2}/b), t = t_{1}\sqrt{d} /ε, μ = ε(d + 1)/\sqrt{d}, α = εb/\sqrt{d}, ε = (r − 1)^{−1/2}, β = 2(2d − b)/b, γ = 2d/b.

2 Localization estimates of global trajectories of Lorenz system

In this section we shall obtain localization estimates which are for b ≤ 2 and great r asymptotically exact for global B-attractors with respect to coordinates σ and η. From these estimates if follows that global attractors of systems (2) are situated in domains which are uniformly bounded with respect to parameter r ∈ (1, +∞). This fact will be used for the demonstration of the existence of homoclinic orbits. We remind here certain definitions for the system

1Based on a lecture hold at the DFG-Workshop “Dimensions of attractors”, St. Petersburg, Russia, October 11-12, 1999

2Partly supported by the Russian Research Fund 98-01-01005

\frac{dX}{dt} = f(X),  X ∈ ℝ^{n},  (3)

where f(X) is a smooth vector-function and ℝ^{n} is n-dimensional Euclidean space. Definition 2.1 A set K ⊂ ℝ^{n} is called invariant if for any X_{0} ∈ K the following inclusion is true: X(t, X_{0}) ∈ K, ∀ t ∈ (−∞, +∞).

Here X(t, X_{0}) is the solution of (3) with initial data X(0, X_{0}) = X_{0}. Definition 2.2 A bounded set K ⊂ ℝ^{n} is called globally attracting, if for any solution X(t, X_{0}) of system (3) the following relation is true:

lim_{t→+∞} inf_{u∈K} |X(t, X_{0}) − u| = 0.

Here | · | is the Euclidean norm in ℝ^{n}. Definition 2.3 A bounded set K is called uniformly globally attracting if for any sphere B ⊂ ℝ^{n} and any number ε > 0 there exists a number τ(B, ε) > 0, such that X(t, X_{0}) ∈ K_{ε}, ∀ t ≥ τ(B, ε), X_{0} ∈ B.

Here K_{ε} is an ε-neighbourhood of the set K. Definition 2.4 A globally attracting invariant bounded set is called a global attractor of system (3).

Definition 2.5 An uniformly globally attracting invariant bounded set K is called global B-attractor of system (3).

It is well known [1–3] that the surfaces ψ_{1} = {(r − z)^{2} + y^{2} = M^{2} + ρ} and ψ_{2} = {z − x^{2}/(2d) = −ρ}, where ρ > 0 and

M = \{ \begin{matrix}r & for b ≤ 2, \\ br/(2\sqrt{b − 1} ) & for b ≥ 2 \end{matrix}

are surfaces without contact for solutions of system (1). That is why for any attractor of system (1) the inequalities

(r − z)^{2} + y^{2} ≤ M^{2},  (4)
z ≥ x^{2}/(2d)  (5)

are true. Hence it follows that on the attractors of system (2) the inequalities

−\frac{M}{\sqrt{2} (r − 1)} − \frac{\sqrt{d} σ}{\sqrt{r − 1}} ≤ η ≤ \frac{M}{\sqrt{2} (r − 1)} − \frac{\sqrt{d} σ}{\sqrt{r − 1}} ,  (6)
ξ > −βσ^{2}/2, ∀ σ ≠ 0  (7)

hold. With the help of estimate (5) we introduce the comparison system [1, 4]

σ̇ = η,  η̇ = −μη + σ − σ^{3},  (8)

which is equivalent to the first-order equation

P \frac{dP}{dσ} + μP − σ + σ^{3} = 0.  (9)

Let us consider positive solutions P_{1}(σ) of (9) on the set [0, σ_{0}) with the initial data P_{1}(σ_{0}) = 0. They define for system (2) in the half-space {σ ≥ 0} the following surfaces without contact:

{η = P_{1}(σ), η > 0, σ ∈ [0, σ_{0}]},  (10)
{η < 0, σ = σ_{0}}.  (11)

Negative solutions P_{2}(σ) of (9) on (−σ_{0}, 0] with the initial data P_{2}(−σ_{0}) = 0 define for system (2) in the half-space {σ ≤ 0} the following surfaces without contact:

{η = P_{2}(σ), η < 0, σ ∈ [−σ_{0}, 0]},  (12)
{η > 0, σ = −σ_{0}}.  (13)

From this and from estimate (6) it follows that if the graph of function P = P_{1}(σ) intersects the straight line

P = \frac{M}{\sqrt{2} (r − 1)} − \begin{matrix}√ & \overline{d} \\ √ & \overline{r − 1} \end{matrix} σ

in a certain point σ_{1} on the interval (0, σ_{0}) then the inequalities

σ < σ_{0},  η < P_{1}(σ) for σ ∈ [σ_{1}, σ_{0}]  (14)

are true on the attractor K of system (2). Similarly, if the graph P = P_{2}(σ) intersects the straight line

P = −\frac{M}{\sqrt{2} (r − 1)} − \frac{\sqrt{d} σ}{\sqrt{r − 1}}

in a certain point σ_{2} on the interval (−σ_{0}, 0) then the inequalities

σ > −σ_{0}, η > P_{2}(σ) for [−σ_{0}, σ_{2}]  (15)

are true on the attractor K of system (2). Note that the surfaces {ξ = C − βσ^{2}/2, C > βσ_{0}^{2}/2} are surfaces without contact for system (2) in the band {|σ| ≤ σ_{0}}. That is why the estimate

ξ ≤ β(σ_{0}^{2} − σ^{2})/2  (16)

holds on attractors of system (2). So we have proved the following result here.

Theorem 2.1 Estimates (4)–(7), (14)–(16) are valid on global attractors of system (1).

Let us give now a simple estimate of σ_{0}. We note for that purpose that the inequalities (14) are true if the graph of P = P_{1}(σ) intersects the straight line P = M/(\sqrt{2} (r − 1)). From the positiveness of μ in the equation (9) it follows that

P_{1}(σ)^{2} > (σ^{2} − σ_{0}^{2}) − \frac{1}{2} (σ^{4} − σ_{0}^{4}).

That the reason why if

(1 − σ_{0}^{2}) − \frac{1}{2} (1 − σ_{0}^{4}) = \frac{M^{2}}{2(r − 1)^{2}}

then the intersection mentoined above takes place. From the latter equation we have

σ_{0} = \sqrt{1 + \frac{M}{r − 1}} .  (17)

Similar argument is true for estimate (15). It follows from relation (17) that any global attractor of (2) is situated in the domain that is uniformly bounded with respect to the parameter r ∈ (1, +∞). For the global B-attractor in the case of b ≤ 2 the estimates (4)–(7), (14)–(16) are asymptotically exact as r → +∞. Indeed, in this case we have on B-attractor

|η| ≤ 1/\sqrt{2}, |σ| ≤ \sqrt{2}.

as r → +∞. Remind that unstable manifolds of zero equilibrium which in null approximation (for small ε) represented by formulas

{ξ = −βσ^{2}/2, η^{2} = σ^{2} − σ^{4}/2}

are a part of B-attractor. So for great r the B-attractor has points close to the planes \{|σ| = \sqrt{2}\}, \{|η| = 1/\sqrt{2}\}.

3 Existence of homoclinic orbits

Denote by σ(t)^{+}, ξ(t)^{+}, η(t)^{+} the separatrix of the saddle σ = ξ = η = 0, which goes into the half-plane {σ > 0}. That is we consider such a solution of (2) that

lim_{t→−∞} σ(t)^{+} = lim_{t→−∞} ξ(t)^{+} = lim_{t→−∞} η(t)^{+} = 0

and σ(t)^{+} > 0 for t ∈ (−∞, T). Here T is a certain number or +∞. It is well known [1, 4, 5] that for fixed values of d and b and the value of r close enough to 1 the value T is equal to +∞. Definition 3.1 If

lim_{t→+∞} σ(t)^{+} = lim_{t→+∞} ξ(t)^{+} = lim_{t→+∞} η(t)^{+} = 0,

the trajectory σ(t)^{+}, ξ(t)^{+}, η(t)^{+} is called the homoclinic orbit.

Let us consider a smooth path b(s), d(s), r(s) (s ∈ [0, 1]) in the space {b, d, r}. The main result of this section is the following assertion.

Theorem 3.1 Suppose that for system (2) with parameters b(0), d(0), r(0) there exist numbers T > τ such that the relations

σ(T)^{+} = η(τ)^{+} = 0,  (18)
σ(t)^{+} > 0, ∀ t < T,  (19)

η(t)^{+} ≠ 0, ∀ t < T, t ≠ τ  (20)

hold. Suppose also that for system (2) with parameters b(1), d(1), r(1) the inequality

σ(t)^{+} > 0, ∀ t ∈ (−∞, +∞)  (21)

is true. Then there exists a number s_{0} ∈ [0, 1] such that system (2) with parameters b(s_{0}), d(s_{0}), r(s_{0}) has a homoclinic orbit σ(t)^{+}, η(t)^{+}, ξ(t)^{+}.

In order to prove this assertion we shall need the following lemmas.

Lemma 3.1 If the hypotheses

η(τ)^{+} = 0, η(t)^{+} > 0, ∀ t ∈ (−∞, τ),

are true for system (2), then η̇(τ)^{+} < 0.

Proof. Suppose the opposite, i.e. η̇(τ)^{+} = 0. In this case the two last equations of theorem (2) imply the equality

η̈(τ)^{+} = αξ(τ)^{+} σ(τ)^{+}.  (22)

From the relations η(t)^{+} > 0, σ(t)^{+} > 0, ∀ t ∈ (−∞, τ) and from the last equality of (2) we have that ξ(t)^{+} < 0 ∀ t ∈ (−∞, τ]. From this and from (22) the inequality η̈(τ)^{+} < 0 follows. The latter contradicts the assumption η̇(τ)^{+} = 0 and the hypotheses of the lemma. This contradiction proves Lemma 1.

Lemma 3.2 Suppose that relations (18), (19) and inequalities

\begin{matrix}η(t)^{+} > 0,  ∀ t ∈ (−∞, τ), \\ η(t)^{+} ≤ 0,  ∀ t ∈ (τ, T) \end{matrix}  (23)

are true for system (2). Then inequality (20) is true as well.

Proof. Suppose the opposite. Then there exists a certain number ρ ∈ (τ, T), such that the relations

η(ρ)^{+} = η̇(ρ)^{+} = 0,
η̈(ρ)^{+} = ασ(ρ)^{+}ξ(ρ)^{+} < 0,
η(t)^{+} < 0, ∀ t ∈ (ρ, T)

are valid. Note that the trajectories σ(t) = η(t) = 0, ξ(t) = ξ(0) exp(−αt) belong to the stable manifold of the saddle σ = η = ξ = 0. Hence, from the hypotheses (18), (19) and from the relations written out above it follows that the separatrix σ(t)^{+}, η(t)^{+}, ξ(t)^{+} and the stable manifold intersect. Then the separatrix belongs completely to the stable manifold of the saddle, and the relation σ(t)^{+} > 0, ∀ t ≥ ρ is valid. The latter relation contradicts the hypothesis (18). This contradiction proves Lemma 2. It is possible to give the following geometrical interpretation of this proof in the phase space with the coordinates σ, η, ξ. A piece of the stable manifold of the saddle σ = η = ξ = 0 is situated “under” the set {σ > 0, η = 0, ξ ≤ 1−γσ^{2}}. This property does not allow the trajectory with the initial data from the set to reach the plane σ = 0 if it remains in the quadrant {σ ≥ 0, η ≤ 0}. Let us consider the polynomial

p^{3} + ap^{2} + bp + c,  (24)

where a, b, c are positive numbers.

Lemma 3.3 Either all the zeros of (24) have negative real parts or two zeros of (24) have non-zero imaginary parts.

Proof. It is well-known [6] that all the zeros of (24) have negative real parts if ab > c. If ab = c the polynomial (24) has two pure imaginary zeros. Suppose now that for certain a, b, c such that ab < c the polynomial (24) has only real zeros. It follows from the positiveness of coefficients that these zeros are negative. The latter leads to the inequality ab > c which contradicts our assumption. Proof of Theorem 3.1 It is well known [7] that the half-trajectory {σ(t)^{+}, η(t)^{+}, ξ(t)^{+} | t ∈ (−∞, t_{0})} depends continuously on parameter s. Here t_{0} is an arbitrary fixed number. Hence and from Lemma 1 follows that if relations (18)–(20) are fulfilled for system (2) with parameters b(s_{1}), d(s_{1}), r(s_{1}) then these relations are fulfilled also for b(s), d(s), r(s) where s ∈ (s_{1} − δ, s_{1} + δ). Here δ is a certain sufficiently small number and numbers τ and T depend on parameter s. It follows from the above reasoning that the relations (18)–(20) are valid for a certain interval (0, s_{0}). Further we shall assume that (0, s_{0}) is the maximal interval where the relations (18)–(20) are valid. Let us demonstrate that there exists a certain homoclinic orbit which corresponds to the values b(s_{0}), d(s_{0}), r(s_{0}). Note first of all that for a certain value of τ these parameters satisfy the relations

\begin{matrix}η(t)^{+} > 0, ∀ t < τ, η(t)^{+} ≤ 0, ∀ t ≥ τ, \\ σ(t)^{+} > 0, ∀ t ∈ (−∞, +∞). \end{matrix}  (25)

Indeed, if there exist numbers T_{2} > T_{1} > τ, for which the relations

σ(t)^{+} > 0, ∀ t ∈ (−∞, T_{2}),
σ(T_{2})^{+} = 0, η(T_{1})^{+} > 0,
η(t)^{+} > 0, ∀ t < τ,
η(τ)^{+} = 0, η̇(τ)^{+} < 0

are true then for s sufficiently close to s_{0} and such that s < s_{0} the inequalityη(T_{1})^{+} > 0 remains true. This fact contradicts the definition of the number s_{0}. If there exist numbers T_{1} > τ, for which η(T_{1})^{+} > 0, η(t)^{+} > 0, ∀ t < τ, η(τ)^{+} = 0, η̇(τ)^{+} < 0, and σ(t)^{+} > 0, ∀ t ∈ (−∞, +∞), then again for s sufficiently close to s_{0} and such that s < s_{0} the inequality η(T_{1})^{+} > 0 remains true and again we have the contradiction with the definition of the number s_{0}. If there exist numbers T > τ, for which σ(t)^{+} > 0, ∀ t < T, σ(T)^{+} = 0, η(t)^{+} > 0, ∀ t < τ, η(t)^{+} ≤ 0, ∀ t ∈ [τ, T],then the inequality (20) is true according Lemma 2. Consequently for s = s_{0} relations (18)–(20) are fulfilled and then (0, s_{0}) is not the maximal interval, for which these relations are true. By these contradictions inequalities (25) are proven. It follows from (25) that for s = s_{0} only an equilibrium can be the ω-limit set of the trajectories σ(t)^{+}, η(t)^{+}, ξ(t)^{+}. Let us demonstrate that the equilibrium σ = 1/\sqrt{γ}, η = ξ = 0 can not be an ω-limit point of the trajectory. The linearization in the neighbourhood of this equilibrium gives the following characteristic polynomial

p^{3} + (α + μ)p^{2} + (αμ + 2/γ)p + 2α.

Suppose that for s = s_{0} the ω-limit set of the separatrix σ(t)^{+}, η(t)^{+}, ξ(t)^{+} includes the point σ = 1/\sqrt{γ}, η = ξ = 0. By Lemma 3 and by the fact that half-trajectories {σ(t)^{+}, η(t)^{+}, ξ(t)^{+} | t ∈ (−∞, t_{0})} depend continuously on the parameter s we obtain the following assertion. For the values s which are sufficiently close to s_{0} the separatrices σ(t)^{+}, η(t)^{+}, ξ(t)^{+} either tend to an equilibrium σ = 1/\sqrt{γ}, η = ξ = 0 as t → +∞ or oscillate in a certain time-interval changing the sign of the coordinate η. Each of these possibilities contradicts with the properties (18)–(20). So, for system (2) with parameters b(s_{0}), d(s_{0}), r(s_{0}) the trajectory σ(t)^{+}, η(t)^{+}, ξ(t)^{+} tends to the trivial equilibrium as t → +∞. Note that when proving Theorem 2 we have received the following more strong result. If for s ∈ [0, s_{0}) relations (18)–(20) are satisfied and for s = s_{0} these relations are not satisfied then system (2) with parameters b(s_{0}), d(s_{0}), r(s_{0}) has a homoclinic orbit. Let us apply Theorem 2 to various concrete situations. Let us fix numbers b and d. It is well known [1, 4, 5] that for r sufficiently close to 1 inequality (21) is true. Let us demonstrate that if the inequality

3d − 2b > 1  (26)

is satisfied and r is large enough then the relations (18)–(20) are satisfied as well. To this end we consider the system

\begin{matrix}Q\frac{dQ}{dσ} = −μQ − P σ − φ(σ), \\ Q\frac{dP}{dσ} = −αP − βQσ, \end{matrix}  (27)

which is equivalent to system (2) in the sets {σ ≥ 0, η > 0} and {σ ≥ 0, η < 0}. Here P and Q are solutions of (27) which are functions of σ. It follows from Theorem 1 that values σ(t)^{+}, η(t)^{+}, ξ(t)^{+} are uniformly bounded with respect to parameter r. So it is possible to realize the asymptotic integration of solutions of (27) with a small parameter ε which correspond the separatrix under consideration. The first approximation of these solutions has the form

Q_{1}(σ)^{2} = σ^{2} − \frac{σ^{4}}{2} − 2μ ∫_{0}^{σ} σ\sqrt{1 − σ^{2}/2} dσ−

−2αβ ∫_{0}^{σ} σ (1 − \sqrt{1 − σ^{2}/2} ) dσ, Q_{1}(σ) ≥ 0,

P_{1}(σ) = − (\frac{β}{2} ) σ^{2} + αβ (1 − \sqrt{1 − σ^{2}/2} )

Q_{2}(σ)^{2} = σ^{2} − \frac{σ^{4}}{2} − 2μ ∫_{σ}^{\sqrt{2}}σ\sqrt{1 − σ^{2}/2} dσ − \frac{4}{3} μ+

\begin{matrix}+2αβ ∫_{σ}^{\sqrt{2}}σ (1 + \sqrt{1 − σ^{2}/2} ) dσ − \frac{2}{3} αβ, Q_{2}(σ) ≤ 0, \\ P_{2}(σ) = − (\frac{β}{2} ) σ^{2} + αβ (1 + \sqrt{1 − σ^{2}/2}) . \end{matrix}

It follows from these formulas that if inequality (26) is fulfilled then for certain T > τ relations (18)–(20) are true, and

\begin{matrix}ξ(T)^{+} = P_{2}(0) = 2αβ, \\ η(T)^{+} = Q_{2}(0) = −\sqrt{8(αβ − μ)/3} = −\sqrt{8ε(3d − 2b − 1)/(3\sqrt{d})} . \end{matrix}

So, for a special path b(s) ≡ b, d(s) ≡ d, r(s) : r(0) = r_{1}, r(1) = r_{2}, where r_{1} is sufficiently large and r_{2} is sufficiently close to 1, all the hypotheses of Theorem 2 are true. That is why we get the following result.

Corollary 3.1 If positive numbers b and d satisfy inequality (26) then there exists a number r ∈ (1, +∞), such that system (2) with these parameters b, d, r has a homoclinic orbit σ(t)^{+}, η(t)^{+}, ξ(t)^{+}.

For the first time this result was obtained in paper [8]. Then it was discussed in papers [9–12]. Let us now fix d = 10 and r = 28 and consider parameter b ∈ (0, +∞). It is well known [12] that for

b > \frac{3d − 1}{2}

condition (21) is fulfilled. In order to consider the case of small b we reduce the system (2) to the form

\begin{matrix}σ̇ = η, \\ η̇ = −μη − uσ + σ − σ^{3}, \\ u̇ = −αu + \frac{ε(2d − b)}{\sqrt{d}} σ^{2}. \end{matrix}  (28)

Here u = ξ +βσ^{2}/2. Since the half-trajectory {σ(t)^{+}, η(t)^{+}, ξ(t)^{+} | t ∈ (−∞, t_{0}} depends continuously on parameter b it follows that for small b it is possible to replace system (28) by the system

\begin{matrix}σ̇ = η, \\ η̇ = −\frac{ε(d + 1)}{\sqrt{d}} η − uσ + σ − σ^{3}, \\ u̇ = 2ε\sqrt{d} σ^{2}. \end{matrix}  (29)

As a result of numerical integration of solution σ(t)^{+}, η(t)^{+}, ξ(t)^{+} of system (29) for d = 10, r = 28 we receive that conditions (18)–(20) are valid. So by Theorem 2 we get the following result from our reasoning.

Corollary 3.2 Suppose d = 10, r = 28. There exists a positive number b such that system (2) with parameters b, d = 10, r = 28 has a homoclinic orbit σ(t)^{+}, η(t)^{+}, ξ(t)^{+}.

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