The Lyapunov dimension formula for the global attractor of the Lorenz system

The exact Lyapunov dimension formula for the Lorenz system has been analytically obtained first due to G.A.Leonov in 2002 under certain restrictions on parameters, permitting classical values. He used the construction technique of special Lyapunov-type functions developed by him in 1991 year. Later it was shown that the consideration of larger class of Lyapunov-type functions permits proving the validity of this formula for all parameters of the system such that all the equilibria of the system are hyperbolically unstable. In the present work it is proved the validity of the formula for Lyapunov dimension for a wider variety of parameters values, which include all parameters satisfying the classical physical limitations. One of the motivation of this work is the possibility of computing a chaotic attractor in the Lorenz system in the case of one unstable and two stable equilibria.


Introduction
The exact Lyapunov dimension formula for the Lorenz system has been analytically obtained first due to G.A.Leonov in 2002 [1] under certain restrictions on parameters, permitting classical values. In his work it was used the technique of special Lyapunov-type functions, which had been created in 1991 year [2] and then was developed in [3,4]. Later in the works [5,6,7] it was shown that the consideration of a wider class of Lyapunov-type func-tions allows to provide the validity of the formula for such parameters of the Lorenz system that all its equilibria are hyperbolically unstable.
In this study it is proved the validity of the formula under classical restrictions on the parameters. The motivation for this investigation is a numerical localization of chaotic attractor in the Lorenz system in the case of one unstable and two stable equilibria [8,9].

The Lorenz system
Consider the classical Lorenz system suggested in the original work of Edward Lorenz [10]: E. Lorenz obtained his system as a truncated model of thermal convection in a fluid layer. The parameters of this system are positive: because of their physical meaning (e.g., b = 4(1 + a 2 ) −1 is positive and bounded).
Active study of the Lorenz system gave rise to the appearance and subsequent consideration of various Lorenz-like systems (see, e.g., [11,12,13,14]). A recent discussion of the equivalence of some Lorenz-like systems and the possibility of universal consideration of their behavior can be found, e.g. in [15,16].
Since the system is dissipative and generates a dynamical system for ∀t ≥ 0 (to verify this, it suffices to consider the Lyapunov function V (x, y, z) = 1 2 (x 2 + y 2 + (z − r − σ) 2 ); see, e.g., [10,4]), it possesses a global attractor (a bounded closed invariant set, which is globally attractive) [17,4]. For the Lorenz system, the following classical scenario of transition to chaos is known [8]. Suppose that σ and b are fixed (we use the classical parameters σ = 10, b = 8/3) and r varies. Then, as r increases, the phase space of the Lorenz system is subject to the following sequence of bifurcations. For 0 < r < 1, there is globally asymptotically stable zero equilibrium S 0 . For r > 1, equilibrium S 0 is a saddle and a pair of symmetric equilibria S 1,2 appears. For 1 < r < r h ≈ 13.9, the separatrices Γ 1,2 of equilibria S 0 are attracted to the equilibria S 1,2 . For r = r h ≈ 13.9, the separatrices Γ 1,2 form two homoclinic trajectories of equilibria S 0 (homoclinic butterfly). For r h < r < r c ≈ 24.06, the separatrices Γ 1 and Γ 2 tend to S 2 and S 1 , respectively. For r c < r < r a ≈ 24.74, the equilibria S 1,2 are stable and the separatrices Γ 1,2 may be attracted to a local chaotic attractor (see, e.g., [8,9]). This attractor is self-excited 1 and can be found using the standard computational procedure, i.e. by constructing a solution using initial data from a small neighborhood of zero equilibrium, observing how it is attracted, and visualizing the attractor (see Fig 1). For r > r a , the equilibria S 1,2 become unstable. The value r = 28 corre-1 An oscillation can generally be easily numerically localized if the initial data from its open neighborhood in the phase space (with the exception of a minor set of points) lead to a long-term behavior that approaches the oscillation. Therefore, from a computational perspective, it is natural to suggest the following classification of attractors [18,19,20,21], which is based on the simplicity of finding their basins of attraction in the phase space: An attractor is called a self-excited attractor if its basin of attraction intersects with any open neighborhood of an equilibrium, otherwise it is called a hidden attractor [18,19,20,21]. Up to now in such Lorenz-like systems as Lorenz, Chen, Lu and Tigan systems only self-excited chaotic attractors were found. In such Lorenz-like systems as Glukhovsky-Dolghansky and Rabinovich systems both self-excited and hidden attractors can be found [22,23,24]. Recent examples of hidden attractors can be found in The European Physical Journal Special Topics: Multistability: Uncovering Hidden Attractors, 2015 (see [25,26,27,28,29,30,31,32,33,34,35,36]). Note that while coexisting self-excited attractors can be found by the standard computational procedure, there is no regular way to predict the existence or coexistence of hidden attractors. sponds to the classical self-excited local attractor (see Fig. 2). : Numerical visualization of the classical self-excited local attractor in the Lorenz system by using the trajectories that start in small neighborhoods of the unstable equilibria S 0,1,2 . Here the separation of the trajectory into transition process (green) and approximation of attractor (blue) is rough.

Lyapunov dimension of attractors
Consider a dynamical systemẋ where f : R n → R n is a smooth vector-function. Since the initial time is not important for dynamical systems, without loss of generality, we consider a solution x(t, x 0 ) : x(0, x 0 ) = x 0 . The linearized system along the solution x(t, x 0 ) is as followsu where is the (n×n) Jacobian matrix evaluated along the trajectory x(t, x 0 ) of system (2). A fundamental matrix X(t, x 0 ) of linearized system (3) is defined by the variational equationẊ We usually set X(0, x 0 ) = I n , where I n is the identity matrix. Then u(t, u 0 ) = X(t, x 0 )u 0 . In the general case, u(t, u 0 ) = X(t, x 0 )X −1 (0, x 0 )u 0 . Note that if a solution of nonlinear system (2) is known, then we have Consider the transformation of the unit ball B into the ellipsoid X(t, x 0 )B and the exponential growth rates of its principal semiaxes lengths. The principal semiaxes of the ellipsoid X(t, x 0 )B coincides with singular values of the matrix X(t, x 0 ),which are defined as the square roots of the eigenvalues of matrix X(t, x 0 ) * X(t, x 0 ). Let σ 1 (X(t, x 0 )) ≥ · · · ≥ σ n (X(t, x 0 )) > 0 denote the singular values of the fundamental matrix X(t, x 0 ) reordered for each t. Introduce the operator X (·) = 1 t ln | · |, where | · | is the Euclidian norm. Define the decreasing sequence (for all considered t) of Lyapunov exponents LEs are commonly used 2 in the theory of dynamical systems and dimension theory [46,47,48,49,50,4,51]. 2 Two widely used definitions of Lyapunov exponents are the upper bounds of the exponential growth rate of the norms of linearized system solutions (LCEs) [38] and the upper bounds of the exponential growth rate of the singular values of fundamental matrix of linearized system (LEs) [37]. The LCEs [38] and LEs [37] are "often" equal, e.g. for a "typical" system that satisfies the conditions of the Oseledec theorem [37]. However, there are no effective rigorous analytical methods for checking the Oseledec conditions for a given system [39, p.118] (a numerical approach is discussed in [40]). For particular system, LCEs and LEs may be different. For example, for the fundamental matrix X(t) = 1 g(t) − g −1 (t) 0 1 we have the following ordered values:  Consider the largest integer j(t, x 0 ) ∈ {1, .., n} such that Following [53,49], introduce the following definition: the function LD(t, Definition 2. A local Lyapunov dimension at the point x 0 is as follows The Lyapunov dimension of invariant compact set K of dynamical system is defined by the relation Note that, from an applications perspective, an important property of the Lyapunov dimension is the chain of inequalities [49,54,4] where dim T , dim H K, and dim F K are topological, Hausdorff, and fractal dimensions of K, respectively.
necessary imply instability or chaos, because for non-regular linearization there are wellknown Perron effects of Lyapunov exponent sign reversal [41,42,43]. Therefore, in general, for the computation of the Lyapunov dimension of attractor we have to consider a grid of points on the attractor and corresponding local Lyapunov dimensions [44]. More detailed discussion and examples can be found in [45,41].

Estimation of Lyapunov dimension by Lyapunov functions
Along with commonly used numerical methods for estimating and computing the Lyapunov dimension (see, e.g., [44,24]), there is an analytical approach that was proposed by G.A.Leonov [2,3,4,55,5,15]. It is based on the direct Lyapunov method and uses Lyapunov-like functions. The advantage of this method is that it allows to estimate the Lyapunov dimension of invariant set without numerical localization of the set. This is especially important for the systems with hidden attractors when numerical finding of all local attractors may be a challenging task [24].
Since LEs and LD are invariant under the linear changes of variables (see, e.g., [45]), we can apply the linear variable change y = Sx with a nonsingular n × n-matrix S. Then system (2) is transformed into the systeṁ Consider the linearization along the corresponding solution y(t, Here the Jacobian matrix is as follows J(y(t, y 0 )) = S J(x(t, x 0 )) S −1 (9) and the corresponding fundamental matrix satisfies Y (t, y 0 ) = SX(t, x 0 ). For simplicity, let J(x) = J(x(t, x 0 )). Suppose that λ 1 (x, S) · · · λ n (x, S) are eigenvalues of the symmetrized Jacobian matrix (9) . Given an integer j ∈ [1, n] and s ∈ [0, 1], suppose that there are a continuously differentiable scalar function ϑ : R n → R and a nonsingular matrix S such that Then dim L K j + s.
Hereθ is the derivative of ϑ with respect to the vector field f: Theorem 2 ([2, 3, 4, 5]). Assume that there are a continuously differentiable scalar function ϑ and a nonsingular matrix S such that Then any solution of system (2) bounded on [0, +∞) tends to a certain equilibrium as t → +∞.

Main result: Lyapunov dimension of the global Lorenz attractor
By Theorems 1 and 2, for the Lorenz system we can obtain the following result.
Theorem 3. Assume that the following inequalities are satisfied. Let one of the following two conditions be satisfied: b. there are two distinct real roots of equations and where γ (II) is a greater root of equation (16).
In this case then any bounded on [0; +∞) solution of system (1) tends to a certain equilibrium as t → +∞.

2.
If then where K is a bounded invariant set.
For numerical experiments (see, e.g., [56,57]) it is known that the Lyapunov dimension of any invariant compact set of the Lorenz system is bounded from above by the local Lyapunov dimension of the zero equilibrium.
Comparing estimation (20) with the local dimension in the origin, we obtain a dimension formula for a global Lorenz attractor. In this case if Remark 2. It can be easily checked numerically that if all three equilibria are hyperbolic, then the conditions of Theorem 4 (see Fig. 3b) are satisfied. For example, for the standard parameters σ = 10 and b = 8 3 the formula (21) is valid for r > 209 45 .

Numerical analysis
We perform a numerical analysis of parameter domain, which does not satisfy the conditions of Theorem 4.
2. Consider r > 1. In the case of conditions (13)-(19) the following assertion is valid. Thus, for each fixed r > 1 it remains to check numerically the behavior of system with the parameters from the bounded domain [0 < b < 4, 0 < σ < 7]. For the considered increasing sequence r k > 1, numerical simulation shows the lack of chaotic attractor (the trajectories with initial data from the corresponding compact absorbing set [58] have been simulated).
I. In the case of classical Lorenz system the matrix J has the following form Following [4], for the condition we introduce a matrix Then We find the eigenvalues of the matrix 1 2 (SJS −1 + (SJS −1 ) * ). The characteristic polynomial of the matrix takes the form: This implies that the eigenvalues λ i = λ i (x, y, z, S), i = 1, 2, 3 of the matrix 1 2 (SJS −1 + (SJS −1 ) * ) are the numbers: Such a choice of the matrix S provides a simple form of eigenvalues.
II. We shall show that λ 1 ≥ λ 2 ≥ λ 3 , ∀x, y, z: At the left there is a sum of two total squares, i.e. the inequality is always satisfied.

IV.
We perform the analysis of R, choosing the running parameters γ 1 , γ 2 , γ 3 , γ 4 in such a way that (29) is valid. Taking into account (28), we obtain IV. a. Under the conditions A 1 = 0 and B 3 = 0 we transform the relation for R: Then Now we analyze the signs of four addends in (31) (1) (2) Since according to (33), γ 1 > 0, from (35) it follows that 2 is a quadratic polynomial in γ 4 with a negative coefficient of γ 2 4 . Therefore for the inequality 4B 1 B 3 − B 2 2 ≥ 0 to be satisfied, it is necessary that the corresponding quadratic equation has a real root, i.e. a positive discriminant: Since according to (32) the condition B 3 < 0 is satisfied, we obtain D γ 4 ≥ 0 ⇔ B 3 + C 1 + C 2 ≤ 0. We have Thus, for there exists γ 4 such that 4B 1 B 3 − B 2 2 ≥ 0. Note that a part of the right-hand side of the inequality in (37) can be transform in the following way Taking into account (33), (34), (36), (37), (38), conditions (32) become For obtaining (31) it is assumed that A 1 = 0, B 3 = 0. Let us analyze the cases A 1 = 0 and B 3 = 0.
IV. b. Consider B 3 = 0. In this case we have i.e. it is impossible to choose γ 1 , γ 2 , γ 3 , γ 4 in such a way that R ≤ 0 is valid ∀x, y, z.
IV. c. Consider the case γ 1 = A 1 = 0 and B 3 = 0. Then In this case The second and third conditions are similar to the second and fourth conditions in (32). Consequently it remains to consider The latter inequality, obtained under the assumption A 1 = 0, coincides with condition (34) if in (34) we consider γ 1 = 0. Thus, the conditions on the running parameters γ 1 , γ 2 , γ 3 , γ 4 , obtained under the condition A 1 = 0, B 3 = 0, can be joined with those, obtained under the condition A 1 = 0, B 3 = 0, i.e. γ 1 ≥ 0 V. For r − 1 > 0 conditions (41) can be transformed in the following way: For the existence of γ 3 it is necessary and sufficient that We now analyze the obtained inequalities.
V. a. Consider the first inequality from system (43) for different γ 2 . Let be γ 2 = 0. Then the inequality is equivalent to If γ 2 > 0, then we obtain the condition and if γ 2 < 0, then .
In the latter case it is required that since according to (41) we have γ 1 ≥ 0. Thus, since the cases of γ 2 = 0 and γ 2 < 0 impose the same condition on the parameters of the system, they can be joined.
V. b. From the second inequality of system (43) we obtain condition for γ 1 : .
If γ 2 = b − 2σ, then we obtain the following condition for γ 1 .
According to (41), it is valid γ 1 ≥ 0. Therefore it is required the following Thus, inequality (45) holds true also for γ 2 = b − 2σ and for γ 2 = b − 2σ. Condition (45) must be satisfied for any sign of γ 2 . If, in addition, condition (44) is also satisfied, then in function (28) we can take γ 2 ≤ 0 and in this case there exist γ 1 , γ 3 , γ 4 such that the relation (44) is not satisfied, then in function (28) it must be considered γ 2 > 0 and, except for condition (45), it is necessary to find conditions for the existence of γ 1 . In this case we obtain the following family: V. c.
Consider the first condition of family (46): Substituting the value ρ from (23), we obtain V. d. Consider the second condition of family (46): From Item V. c. it is known that the first inequality of system is equivalent to the relation Next we transform the inequality 1 γ 2 (σ + 1) The left-hand side of the inequality is a quadratic polynomial in γ 2 . If coefficient of γ 2 2 is negative, i.e.
If the coefficient is equal to 0, i.e.
then the relation then for the existence of γ 2 > 0, satisfying (48), it is necessary and sufficient that there exist two different real roots of the equation and the largest root must be positive. This condition corresponds to the second condition, stated in the theorem.