Further Analysis of Lorenz’s Maximum Simplification Equations

a. Dependence on the parameter r

It can be verified that the model (2.1) depends only on the ratio r = k/l = L/K and can be written as

View Expanded

where

View Expanded

View Expanded

and

View Expanded

A plot of the values of a = a(r), b = b(r), and c = c(r) as a function of r > 0 is given in Fig. 1. It can be verified that b = c when r = 41/3 = 0.7598 and a = c when r = 43 = 1.3161. Further, b → 0⁺ (from above) as r → 0 and a → 0⁻ (from below) as r increases.

Let η = (A, F, G)^T ∈ ℝ³ be a column (real) vector of three spectral components, where T denotes the transpose. That is, η₁ = A, η₂ = F, and η₃ = G. Then (2.6) can be succinctly written as

View Expanded

where g: ℝ³ → ℝ³ is such that g(η) = [g₁(η), g₂(η), g₃(η)]^T with g₁(η) = aFG, g₂(η) = bAG, and g₃(η) = cAF.

It should be interesting to note that the Eq. (2.6) is equivalent to the Euler’s equation of motion for a rigid body and is known as the Euler gyrostat (Obukhov 1973). Derivation and analysis of Euler gyrostat can be found in textbooks on classical mechanics (Landau and Lifshitz 1960; Wittenburg 1977; Arnol’d 1989). Recently Gluhovsky and his collaborators (Gluhovsky and Agee 1997; Gluhovsky and Tong 1999; Gluhovsky et al. 2002) have shown that a Volterra gyrostat (of which the Euler gyrostat is a special case) can be used as a building block for generating a wide variety of low-order models of interest in several application domains.

Implicit in the above description are the two sets of coordinate systems. First is the three-dimensional physical space consisting of the two space variables, x and y, and the time variable t, in which the physical quantities such as the streamfunction ψ, vorticity ∇²ψ, and the wind components u and υ are described. The second is the three-dimensional spectral space in which the spectral amplitudes A, F, and G evolve as functions of time according to (2.6) or (2.10). The goal is to characterize the long-term behavior of the physical quantities from those of the spectral components.

b. Special case of r = 1

It can be verified that a = −½, b = ½ and c = 0 when r = 1. In this case, Ġ = 0 and G(t) = G(0) for all t. Hence, if G(0) ≠ 0, (2.6) reduces to a linear harmonic oscillator

View Expanded

whose solution is periodic (Hirsch and Smale 1974). When G(0) = 0 and r = 1, then η(t) ≡ 0. In the following it is tacitly assumed that r ≠ 1.

c. Conservation of kinetic energy and vorticity

Let 𝗗 = diag(d₁, d₂, d₃) be a diagonal matrix where d_i > 0, i = 1, 2, and 3. Consider the functional f: ℝ³ → ℝ defined by

View Expanded

It can be verified that the time derivative of this functional evaluated along the trajectory of (2.6) is given by

View Expanded

Thus, given any parameter vector p = (a, b, c)^T of the model, for any d = (d₁, d₂, d₃)^T that is orthogonal to p, ḟ(η) ≡ 0 and for this choice of the vector d or matrix 𝗗, f (η) is conserved by the model. By direct computation it can be verified that the mean kinetic energy

View Expanded

and the mean square vorticity

View Expanded

are conserved by the model (2.1). That is, E(t) = E(0) and V(t) = V(0) for all t ≥ 0. Refer to Lorenz (1960) and Krishnamurti et al. (1998) for more detail.

d. Solution space and degree of freedom

Let η(0) = [A(0), F(0), G(0)]^T be the given initial condition for (2.10). Then (2.15) implies that the solution η(t) of (2.10) is confined to the surface of the V ellipsoid, which is centered at the origin and is given by

View Expanded

where

View Expanded

Clearly, f_V(η) depends only on η(0) and not on r. Similarly, from (2.14) it follows that the solution η(t) of (2.10) is confined to the surface of the E ellipsoid, which is also centered at the origin, and is given by

View Expanded

where

View Expanded

Substituting (2.19) into (2.18) it can be verified that f_E(η) depends only on η(0) and r.

Combining these, it follows that the solution η(t) of (2.10) must lie in the set

View Expanded

which is the intersection of the two concentric ellipsoids. That is, this set S is the solution space for the model Eq. (2.10). Refer to Fig. 2 for an example of the two intersecting ellipsoids.

Taking derivatives of both sides of (2.16) and (2.18) with regard to time, we get

View Expanded

where ∇f_V(η) = (1/V(0))(A, F, 2G)^T is the gradient of f_V(η) and

View Expanded

where ∇f_E(η) = [1/2E(0)][(A/l²), (F/k²), (2G/l² + k²)]^T is the gradient of f_E(η). It can be verified that the gradient vectors ∇f_V and ∇f_E are linearly independent unless r = 1 and G = 0. Thus, the flow of the dynamics in (2.10) is always orthogonal to the plane defined by these two gradient vectors. Since ∇f_V and ∇f_E define the outward normal to the surface of the ellipsoids, the flow always lies along the tangent to the points in S in (2.20).

Lorenz (1960) has shown that (2.6) can be solved analytically where the solution A(t), F(t), and G(t) are expressed as elliptic functions of time. In principle, we can use the tables of elliptic functions (Abramowitz and Stegun 1965) to characterize the long-term behavior of the solution of (2.6). However, following Lorenz (1960), we use the standard numerical methods to solve (2.6). While Lorenz (1960) used the central difference scheme that involves three time steps, we use the fourth-order adaptive Runge–Kutta (RK) method.

e. Boundedness of the trajectories

An immediate consequence of the property described in section 2b is that the trajectories η(t) of (2.10) remains bounded. Let η(0) = [A(0), F(0), G(0)]^T be the initial state. Then from (2.14) and (2.15) we get

View Expanded

For example, when A(0) = 10⁻³ = G(0) and F(0) = 5, K = 5000 and L = 10 000, it can be verified that |A(t)| ≤ 2.5, |F(t)| ≤ 5.0, and |G(t)| ≤ 3.5355, for all t ≥ 0.

This property in turn implies that the model (2.1) is (externally) stable in the sense of Lyapunov (Hirsch and Smale 1974; Sparrow 1982). However, in view of the nonlinear interaction between modes, there is transfer of energy between modes leading to (internal) instability.

An example illustrating this transfer of energy between modes is given in Fig. 3 where the system (2.6) is solved using the fourth-order adaptive Runge–Kutta method. This figure provides a complete picture of the model trajectories, namely the variation of A(t), F(t), and G(t) with time; two dimensional plots of A(t) − F(t), A(t) − G(t) and F(t) − G(t), and the three-dimensional view of the solution.

a. Stability of equilibrium points in ε_A

Let η* ∈ ε_A. Then, the Jacobian (4.1) becomes

View Expanded

Let r < 1. Then, a < 0, b > 0, and c > 0. It can be verified that the eigenvalues of 𝗗_g(η*) are given by

View Expanded

and the corresponding eigenvectors are given by

View Expanded

For latter reference, define the diagonal matrix Λ⁽¹⁾ of eigenvalues and 𝗩⁽¹⁾, the nonsingular matrix of the corresponding eigenvectors as

View Expanded

and

View Expanded

where (it follows from the definition)

View Expanded

Notice that while the eigenvalues λ⁽¹⁾₂ and λ⁽¹⁾₃ in (4.3) depend on A and the ratio r, the eigenvectors v⁽¹⁾₂ and v⁽¹⁾₃ are independent of A. That is, every point in ε_A shares the same set of eigenvectors that depends only on the value of the ratio r. For purposes of comparison, the eigenvalues and vectors of 𝗗_g(η*) when η* belongs to each of the three equilibria are summarized in Table 1.

Let r > 1. Then a < 0, b > 0, and c < 0. It can be verified that the complex eigenvalues of 𝗗_g(η*) are given by

View Expanded

and the corresponding complex eigenvectors are v⁽¹⁾₁, w⁽¹⁾₂ = u₁ + iu₂ and w⁽¹⁾₃ = u₁ − iu₂, respectively, where

View Expanded

We leave it to the reader to verify that this eigen decomposition in the complex domain can be equivalently expressed in real domain using the standard canonical form (Hirsch and Smale 1974) as

View Expanded

where

View Expanded

and

View Expanded

That is, the eigenvectors that are the columns of 𝗩⁽¹⁾ are given in (4.8). In this paper we use this latter form of eigen decomposition when the eigenvalues are complex. Refer to Table 1 for a summary.

The product bc and the ratio c/b in (4.3) and (4.4) are given by

View Expanded

and

View Expanded

Variation of bc and c/b as a function of r are given in Figs. 4a,b, respectively. It follows from (4.3)–(4.4) and (4.8)–(4.9) that the eigenvalues and eigenvectors depend on two factors: the value of the first coordinate A of the equilibrium point η* = (A, 0, 0)^T and the value of the ratio r. For completeness, we consider several subcases.

1) Case 1: Let A > 0 and r < 1

In this case, it can be verified that λ⁽¹⁾₂ > 0 and λ⁽¹⁾₃ < 0, and the equilibrium point, η* = (A, 0, 0) is called a saddle point, with the eigenvector v⁽¹⁾₂ corresponding to the unstable (repelling) direction and the eigenvector v⁽¹⁾₃ to the stable (attracting) direction (Hirsch and Smale 1974). Refer to Table 2 for a summary. In verifying these claims, let η = η* + ξ where ξ is a small perturbation. Then, using the first-order Taylor series expansion, we obtain

View Expanded

or

View Expanded

as the local dynamics of evolution of the perturbation. Now change the variables from the standard (A, F, G) orthogonal coordinate system to oblique coordinate system defined by the eigenvectors v⁽¹⁾₁, v⁽¹⁾₂ and v⁽¹⁾₃ in (4.6). Define a new vector z = (z₁, z₂, z₃)^T where

View Expanded

Substituting (4.13) into (4.12) and using (4.7) we get

View Expanded

That is,

View Expanded

and z₂(t) = z₂(0)e^λ(1)t₂ and z₃(t) = z₃(0)e^λ(1)t₃. Since λ⁽¹⁾₂ > 0, indeed z₂ grows exponentially and λ⁽¹⁾₃ < 0 implies that z₃ decreases exponentially as well. Refer to Fig. 5 for a geometric exposition of these ideas. Thus, any perturbation with a nonzero component along v⁽¹⁾₂ will grow. Hence, in this case η* = (A, 0, 0) is an unstable equilibrium point. Referring to Fig. 5, the angle θ that v⁽¹⁾₂ makes with the positive F axis (same as the angle v⁽¹⁾₃ makes with the negative F axis) is given by

View Expanded

Thus, as r increases from 0 to 1, θ decreases from 90° to 0°.

2) Case 2: Let A < 0 and r < 1

In this case since λ⁽¹⁾₂ < 0 and λ⁽¹⁾₃ > 0, the role of the eigenvalue–eigenvector pairs (λ⁽¹⁾₂, v⁽¹⁾₂) and (λ⁽¹⁾₃, v⁽¹⁾₃) are interchanged and the above analysis and conclusions hold. Thus, the equilibrium point η* = (A, 0, 0) is an unstable saddle point. Refer to Fig. 5.

(i) Effect of changing A

Thus, keeping r(<1) fixed, the force of attraction along v⁽¹⁾₃ and the force of repulsion along v⁽¹⁾₂ as measured by the magnitudes of the eigenvalues |λ⁽¹⁾₃| = Abc and |λ⁽¹⁾₂| = Abc, respectively, diminish to zero as A decreases from 1 to 0. As A increases in value in the negative direction, while the magnitudes of these forces increase, their role is reversed, with v⁽¹⁾₂ being the new direction of attraction and v⁽¹⁾₃ being the new direction of repulsion. This reversal of the role between v⁽¹⁾₂ and v⁽¹⁾₃ occurs when A = 0.

(ii) Effect of changing r

Let A > 0 be fixed. From (4.15) it follows that as r → 0⁺, c/b → ∞, and θ → 90°. Combining this with (4.4) we get a degenerate limit, namely

View Expanded

As r increases form 0⁺, the angle θ decreases from 90°. When r → 1⁻, (c/b) → 0⁺ and θ → 0⁺. Again we get a degenerate limit

View Expanded

When r = 1, c = 0 and the character of the equilibrium η* = (A, 0, 0)^T changes from being a saddle point to a center. Refer to the next subsection for more details.

3) Case 3: Let A > 0 and r > 1 or A < 0 and r > 1

From (4.8) it follows that the nonzero eigenvalues are complex conjugate with zero real part. In this case the equilibrium point is called a center for the corresponding linearized problem (Hirsch and Smale 1974). A geometric view of the two new eigenvectors in (4.9) is shown in Fig. 6. It turns out that while the new eigenvector v⁽¹⁾₃ = (0, 1, 1)^T is independent of r and makes a fixed 45° angle with the positive F axis, the new eigenvector v⁽¹⁾₂ = (0, −|(b/c)|, |(c/b)|)^T makes an angle θ with the negative F axis where

View Expanded

As r increases from one to infinity, |c/b| increases form 0 to ½ and θ from 0 to 26.57°.

(i) Change of character of the equilibrium η* = (A, 0, 0)^T

As r changes from 1⁻ to 1⁺, it follows from (4.17) that the old eigenvectors v⁽¹⁾₂ and v⁽¹⁾₃ in (4.4) become degenerate and cease to exist. Simultaneously, a new set of eigenvectors v⁽¹⁾₂ and v⁽¹⁾₃ in (4.9) are created. Of these, v⁽¹⁾₃ = (0, 1, 1)^T is fixed. When r → 1⁺, |c/b| → 0 and hence new v⁽¹⁾₂ → (0, −1, 0)^T. Comparing this with (4.17), it looks as though the old v⁽¹⁾₃ is reincarnated as the new v⁽¹⁾₂ as r changes from 1⁻ to 1⁺. Thus, as r changes from 1⁻ to 1⁺, η* = (A, 0, 0)^T changes its character from being a saddle to a center.

b. Stability of equilibrium points in ε_F

Let η* = (0, F, 0)^T ∈ ε_F. Then, the Jacobian in (4.1) becomes

View Expanded

whose eigenvalues and eigenvectors are given by

View Expanded

with λ⁽²⁾₁ = 0, λ⁽²⁾₂ = Fac, λ⁽²⁾₃ = −Fac and

View Expanded

The product ac and the ratio c/a are given by

View Expanded

and

View Expanded

Variation of ac and c/a as a function of r are given in Figs. 7a,b, respectively. From (4.20) and (4.22) it follows that the eigenvalues λ⁽²⁾₂ and λ⁽²⁾₃ depend on two factors: the value of the second coordinate F of the equilibrium point η* = (0, F, 0)^T and the value of the ratio r. A summary of the properties of these eigenvalues for various combinations of values of F and r are given in Table 3.

Again it can be verified that η* = (0, F, 0)^T is an unstable saddle point when r > 1 and F ≠ 0. However, when r < 1 and F ≠ 0, perturbations around this equilibrium will have a local periodic behavior. It is interesting to note that the behavior of the equilibria in ε_A for r < 1 are quite similar to those in ε_F for r > 1 and vice versa.

c. Stability of equilibrium points in ε_G

Let η* = (0, 0, G)^T ∈ ε_G. Then the Jacobian in (4.1) becomes

View Expanded

whose complex eigenvalues are given by λ⁽³⁾₁ = 0, λ⁽³⁾₂ = iG|ab|, λ⁽³⁾₃ = −iG|ab|. Using the standard canonical form, the eigen decomposition becomes

View Expanded

where

View Expanded

and

View Expanded

The product ab and ratio b/a are given by

View Expanded

View Expanded

From (4.25) it follows that the eigenvalues λ⁽³⁾₂ and λ⁽³⁾₃ are always complex conjugates (with zero real parts) for all values of r. Variation of ab and b/a as a function of r are given in Figs. 8a,b, respectively. Refer to Table 4 for a summary of the properties of these eigenvalues.

a. Small perturbation of equilibrium points in ε_A

Let η* = (A, 0, 0)^T ∈ ε_A. Depending on the values of A and r, consistent with section 4, we again consider four subcases.

3) Case 3: A > 0 and r > 1 or A < 0 and r > 1

In this case b > 0 but c < 0 and recall from Table 1 that the eigenvalues λ⁽¹⁾₂ = iA|bc| and λ⁽¹⁾₃ = −iA|bc|. This in turn implies that η* = (A, 0, 0)^T is a center and the components F(t) and G(t) oscillate where the frequency of oscillation depends on both A and r. In verifying this, assume that A is nearly constant. In this case (2.6) becomes

View Expanded

Referring to Hirsch and Smale (1974), we get the solution of (5.2) as

View Expanded

View Expanded

The period T of the harmonic components in (5.3)–(5.4) is given by

View Expanded

Thus, T is a decreasing function of A and r.

The question now is: when will A be nearly a constant? Recall that when r is large, a < 0 but close to zero (a → 0⁻). Hence Ȧ is small in magnitude but its sign depends on those of F and G. Refer to Fig. 16. Thus for large r, A is very slowly varying and the orbits of (2.6) look quite similar to a merry-go-round where the cars go round and round in closed circuits but they reach a maximum when they cross the G axis and a minimum when they cross the F axis. Refer to Fig. 17 for an example of the trajectory in this case.

b. Small perturbation of equilibrium points in ε_F

Comparing entries in Table 1, it turns out that there is a one to one correspondence between the stability properties of equilibria in ε_A and ε_F. In particular, the properties of the eigenvalues and eigenvectors of the Jacobian of equilibria in ε_F for r > 1 (r < 1) are very similar to those of ε_A for r < 1 (r > 1). This in turn implies that the qualitative properties of the trajectories starting close to ε_F when r > 1 (r < 1) are quite similar to those starting close to ε_A when r < 1 (r > 1). In view of this similarity, to save space, we observe that all the remarks and conclusion of section 5a carry over to equilibria in ε_F.

c. Small perturbations of equilibrium points in ε_G

From (2.7) and (2.8) we have a < 0 and b > 0 for all r > 0. In view of this, from Table 1 it turns out that the eigenvalues of the Jacobian at the equilibrium point in ε_G are purely imaginary for all r > 0. This in turn implies that the qualitative properties of trajectories starting close to ε_G are quite similar to those starting close to ε_A when r > 1 or ε_F for r < 1. Hence all the relevant conclusions from section 5a carry over to this case as well.

a. Case 1

In view of (3.2), from (2.14) and (2.15) consider the ratio

View Expanded

This ratio depends on the initial point η = (A, F, G)^T and the value of r. Following Lorenz (1960) we define three types of initial conditions: R₁(l) = 1 is called critical and R₁(l) < (>)1 is called sub (sup) critical cases. From (6.1) it can be verified that

View Expanded

Let

View Expanded

Since G²/F² is nonnegative, we consider two subcases.

1) 0 < r < 1

In this case β is positive. Hence, 0 < R₁(l) < 1 for all −β < δ < 0 and R₁(l) > 1 when δ > 0 with R₁(l) = 1 when δ = 0. As an example, if r = ½, then β = 15. Thus, R₁(l) is subcritical for all −15 < δ < 0, and supercritical for all δ > 0.

The cross sections of the V and E ellipsoids by the condition (6.3) are given by

View Expanded

and

View Expanded

Combining these with (6.1) we get

View Expanded

Thus, under the condition (6.3), R₁(l) = 1 when δ = 0. To check if an equilibrium point in ε_A is reachable, let G = 0 in (6.3). Then F = 0 and from (6.4) and (6.5) we get

View Expanded

which from (6.6) is consistent only when δ = 0. This argument leads to the following inescapable conclusion: orbits starting from initial points lying outside of the pair of straight lines given by

View Expanded

cannot approach points in ε_A.

A number of observations are in order.

1. Critical points and eigenvectors of 𝗗_g(η*), η* ∈ ε_A. Combining (4.25) with (6.8), the latter can be written as

   

   

   View Expanded

   

   Thus, the pair of straight lines given by G = ±(c/b)F in the F–G plane (refer to Fig. 9) indeed corresponds to the eigenvectors v⁽¹⁾₂ = [0, 1, (c/b)]^T and v⁽¹⁾₃ = [0, 1, −(c/b)]^T. Thus, points along these two eigenvectors in the F–G plane in Fig. 9 correspond to the critical case. Similarly, the region R₁ and R₃ correspond to the subcritical and region R₂ and R₄ to the supercritical cases.

2. Critical points and heteroclinic orbits. Since the orbits starting from points along v⁽¹⁾₂ and v⁽¹⁾₃ approach points in ε_A, these orbits are necessarily heteroclinic orbits corresponding to the unstable case in section 5a. However, orbits starting from the sub and super critical regions can never approach an equilibrium point in ε_A.

2) r > 1

Since

View Expanded

it follows from (6.1) that

View Expanded

Also, since β < 0, from (6.3) it follows that δ ≥ |β| > 0 for 2G²/F² to be nonnegative. Hence this corresponds to the supercritical case and in this case orbits starting from any η = (A, F, G)^T ∈ ℝ³ cannot reach an equilibrium point in ε_A. This is to be anticipated since when r > 1, any equilibrium point in ε_A is a center.

b. Case 2

In view of (3.3) using (2.14) and (2.15), consider the ratio

View Expanded

Analogous to (6.2) we readily see that

View Expanded

Let

View Expanded

Since the lhs of (6.13) is nonnegative, we consider two subcases.

1) r > 1

In this case, since α is positive, we get 0 < R₂(k) < 1 (subcritical) when −α < δ < 0 and R₂(k) > 1 (supercritical) when δ > 0. In the critical case (δ = 0), using (4.23), we obtain

View Expanded

Thus, the pair of straight lines corresponding to (6.14) are given by G = ±(c/a)A. These straight lines correspond to the eigenvectors v⁽²⁾₂ = [1, 0, (c/a)]^T and v⁽³⁾₂ = [1, 0, −(c/a)]^T. Thus, the orbits starting from points along v⁽²⁾₂ and v⁽²⁾₃ approach the equilibrium points η* = (0, F*, 0)^T where F* = ±2υ = ±2kE. As in case 1, orbits starting from sub (super) critical regions cannot approach an equilibrium points in ε_F.

2) r < 1

Since

View Expanded

it follows that R₂(k) > 1 for all 0 < r < 1. Hence this corresponds to the supercritical case and orbits starting from any initial point η ∈ ℝ³ cannot approach an equilibrium point in ε_F. Again this is to be anticipated since η* ∈ ε_F is a center when r < 1.

c. Case 3

From (3.3), (2.14), and (2.15), it follows that

View Expanded

Since

View Expanded

for all r, R₃(l, k) < 1. Again it can be verified by following the above arguments that equilibrium η* ∈ ε_G can never be reached by orbits starting from any point η ∈ ℝ³.

a. Index cycles associated with heteroclinic orbits

In this work, we define the start of an index cycle to be the time when A first drops below 99% of A_max and the end of the cycle to be when A rises above 99% of A_max. The period of the model index cycle is defined as the length of time between the start and end of the cycle. Given this definition, we can examine the sensitivity of the index cycle period to changing initial conditions and values of r. Since we use the variable A to define the index cycle, we focus our analysis around the A equilibrium. The results of this sensitivity analysis are summarized in Tables 6 –8

An example of an index cycle portrayed by this model is provided in Fig. 10, given by a small F perturbation imposed upon ε_A. As noted in Fig. 9 this is a perturbation in the R₁ region of the F–G domain, and results in a heteroclinic orbit with properties summarized in Table 5. For several units of time, A remains near its initial value (A = +1) then at approximately t = 15, the F and G variables quickly increase while the value of A rapidly switches from +1 to −1. As A approaches −1, the values of F and G decrease to zero. The solution remains nearly the same for several units of time (around t = 50). The solution suddenly shifts again at approximately t = 85, when F increases and G decreases rapidly, and the value of A switches from −1 to +1. The period of the index cycle in this case is 69.8 units of time.

A similar example is provided in Fig. 11, given by a small G perturbation imposed upon ε_A. This is a perturbation in the R₂ region of the F–G domain, which results in a heteroclinic orbit with properties described in Table 5. As with the R₁ example, the evolution of A is consistent with the index cycle concept: A remains near its initial value (A = +1) for several units of time, switches quickly from +1 to −1, remains near the minimum value for some time, then quickly changes again from −1 to +1. However, the behavior of the F evolution in this example is similar in character to the G evolution in the previous (R₁) example, F changes sign as A switches from +1 to −1 and back again, while G remains positive throughout the evolution. In this case, the period of the index cycle is slightly longer (77.1 units of time) than in the previous example.

b. Index cycles in the stable case

Lorenz (1960) stated that cycling of A was possible in the stable (r > 1) situation if the perturbations were large and V/E was supercritical. As shown in section 6, the critical point for r > 1 is determined by G²/A² = (r⁴ − 1)/2. However, in both the subcritical and supercritical cases, the evolution of A is not consistent with the definition of an index cycle. An example of a subcritical case is shown in Fig. 3. While A varies considerably in this example, it remains positive and does not remain near its maximum or minimum value for any considerable period of time. On the other hand, the F evolution is consistent with the index cycle definition. In the supercritical situation, the behavior of A and G are switched, A varies from A_max to −A_max and G remains positive for initial G > 0 (negative for initial G < 0). While this is consistent with the index cycle definition, A remains at its maximum or minimum value for only an instant, therefore the A evolution is inconsistent with the index cycle definition in the supercritical case (for r > 1) as well.