Theorem 7.3.6. Suppose that x∗ is an isolated non-hyperbolic equilibrium point of xn+1= f (xn) and that f0(x∗) = −1. Then x∗ is unstable if Sf (x∗) > 0 and is locally asymptotically stable if Sf (x∗) < 0.
Example. In our previous example, we found that x∗ = 2/3 is a non-hyperbolic fixed point of the difference equation xn+1= 3xn(1 − xn). Since f (x) = 3x − 3x2, we compute that the first three derivatives of f are
f0(x) = 3 − 6x f00(x) = −6 and f000(x) = 0.
Since f0(x∗) = −1, we may use Theorem 7.3.6. The expression for the Schwarzian derivative reduces to Sf (x∗) = −f000(x∗) − 32f00(x∗)2, and we find that Sf (x∗) =
−54 < 0. Theorem 7.3.6 tells us that the non-hyperbolic fixed point x∗ = 2/3 is locally asymptotically stable.
In the preceding example, we were still able to classify the stability of the fixed point even though Theorem 7.3.2 was inconclusive. Usually, Theorems 7.3.2, 7.3.4 and 7.3.6 are enough to classify stability of fixed points, although there are cases in which all three theorems are inconclusive.
7.4 Systems of Nonlinear Equations and Stability
Fixed points for higher-order difference equations can be analyzed via techniques that are very similar to the ones we developed when considering equilibria of non-linear systems of ODEs. To motivate our definition for fixed points of higher-order equations, let us study a specific second-order equation
xn+1 = [3 + xn− xn−1]xn(1 − xn).
As usual, we write this as a system of two first-order equations by introducing a new variable yn = xn−1. The resulting system is
xn+1 = [3 + xn− yn]xn(1 − xn) and yn+1 = xn.
A fixed point of such a system should correspond to a constant solution of the differ-ence equation. In this example, a fixed point would be any constant vector (x∗, y∗)
such that whenever (xn, yn) = (x∗, y∗), we have (xn+1, yn+1) = (x∗, y∗) as well. More generally,
Definition 7.4.1. Suppose f : Rm → Rm and that xn+1 = f (xn) is a system of m first-order difference equations. A fixed point of the system is any vector x∗ ∈ Rm such that x∗ = f (x∗).
In the above example, we may solve for the fixed points by setting xn+1 = xn = x∗ and yn+1= yn= y∗. The equation yn+1= xn tells us that x∗ = y∗, and substituting this into the equation for xn+1 yields x∗ = 3x∗(1 − x∗). Solving this quadratic equation yields x∗ = 0 and x∗ = 2/3 as the two solutions. Consequently, the system has two fixed points, (x∗, y∗) = (0, 0) and (x∗, y∗) = (2/3, 2/3).
Example. Consider the nonlinear system
xn+1= 2xn− xnyn and yn+1 = xnyn. Fixed points (x, y) must simultaneously satisfy
x = 2x − xy and y = xy.
By algebra, these two equations are x(1 − y) = 0 and y(x − 1) = 0. If x = 0 in the first of these, then the second would force y = 0 as well. Similarly, if y = 1 in the first equation, then this would force x = 1 in the second equation. We have obtained precisely two equilibria, (x, y) = (0, 0) and (x, y) = (1, 1).
Stability. If x∗ is a fixed point of a system xn+1= f (xn) of m first-order difference equations, we need a way of testing the stability of x∗. We mimic exactly what we did when linearizing systems of ODEs at an equilibrium point. Recall that the linear approximation of a function f : Rm → Rm at a point x∗ ∈ Rn is given by
f (x) ≈ f (x∗) + Jf (x∗)(x − x∗).
Using this approximation in the above difference equation,
xn+1≈ f (x∗) + Jf (x∗)(xn− x∗) = x∗+ Jf (x∗)(xn− x∗),
where we have used the fact that f (x∗) = x∗ since x∗ is a fixed point. To measure the gap between iterates of the difference equation and the fixed point, define yn = xn− x∗. Then in the vicinity of x∗, the vectors yn approximately satisfy
yn+1 = Jf (x∗)yn,
a linear, constant-coefficient system. The exact solution of this linearized system is given by
yn = [Jf (x∗)]ny0,
where the vector y0 is a measure of our initial gap x0 − x∗. In order to ensure that x∗ is locally asymptotically stable, we need a criterion which guarantees that the gap yn = xn− x∗ will approach 0 as we let n → ∞. By inspecting the solution of the linearized system, we must insist that the entries of the powers of the Jacobian matrix Jf (x∗) converge to 0 as n → ∞. If this Jacobian matrix is diagonalizable, we could write Jf (x∗) = P DP−1where D is a diagonal matrix containing the eigenvalues, and P is a constant, invertible matrix. The fact that powers of D contain powers of the eigenvalues suggests a stability criterion: each eigenvalue should have “size” smaller than 1 so that their powers will converge to 0 as n → ∞. For real numbers, the absolute value function gives us a notion of “size”. We now generalize the concept of absolute value to include complex numbers.
Definition 7.4.2. If z = x+iy is a complex number, then the modulus of z is defined as |z| =p
x2+ y2.
Notice that if z is real (i.e., y = 0), then |z| =√
x2 = |x| and the modulus of z is given by the usual absolute value function.
Now that we have a notion of “size” for complex numbers, all of our above remarks constitute a heuristic proof of the following stability theorem.
Theorem 7.4.3. Let f : Rm → Rm and suppose that all entries in the Jacobian matrix Jf (x) are continuous. An isolated fixed point x∗ of a system xn+1 = f (xn) of m first-order difference equations is
• locally asymptotically stable if ALL eigenvalues of Jf (x∗) have modulus less than 1;
• unstable if ANY eigenvalue of Jf (x∗) has modulus greater than 1.
Remark: This stability criterion is different (although similar in spirit) from the one we developed when analyzing equilibria of ODEs. An equilibrium x∗ of a system of ODEs is stable if all eigenvalues of Jf (x∗) have negative real part. By contrast, a fixed point x∗ for a system of difference equations is stable if all eigenvalues of Jf (x∗) have modulus less than 1.
Example. Find all fixed points of the system xn+1 = 3xn− xnyn yn+1 = −2yn+ xnyn
and determine whether they are locally stable or unstable.
Solution: Fixed points (x, y) of this system satisfy
x = 3x − xy and y = −2y + xy.
Equivalently, x(2 − y) = 0 and y(x − 3) = 0. If x = 0 in the first of these equations, then we are forced to set y = 0 in order to satisfy the second equation. Likewise, if y = 2 in the first equation, then we must have x = 3 in the second equation. Thus, there are two fixed points, (0, 0) and (3, 2). For this system, the function f : R2 → R2 is defined by
f (x, y) =
· 3x − xy
−2y + xy
¸ , and its Jacobian matrix is
Jf (x, y) =
· 3 − y −x y x − 2
¸ .
At the fixed point (0, 0), the Jacobian matrix Jf (0, 0) is a diagonal matrix, namely diag{3, −2}, and the eigenvalues are the diagonal entries. The eigenvalues are real, so their moduli are simply their absolute values. Since there is an eigenvalue with modulus larger than 1 (in fact both have modulus larger than 1), Theorem 7.4.3 tells us that (0, 0) is an unstable equilibrium.
At the other fixed point (3, 2), the Jacobian matrix is Jf (x, y) =
· 1 −3
2 1
¸ ,
and the associated characteristic equation is λ2 − 2λ + 7 = 0. From the quadratic formula, the eigenvalues are λ± = 1 ±√
6i. These two eigenvalues have modulus
|λ±| = q
12+ (√
6)2 = √
7 > 1.
Again, we conclude that the fixed point is unstable because there are eigenvalues with modulus exceeding 1.
For 2 × 2 matrices, there is actually a quick way to determine whether all eigen-values have modulus less than 1 without actually computing the eigeneigen-values. The following Lemma can be used to test the stability of fixed points for systems of two first-order difference equations.
Lemma 7.4.4. Suppose that A is a 2 × 2 matrix. Then the eigenvalues of A have
modulus less than 1 if and only if
trA + det A > −1, trA − det A < 1, and det A < 1.
The proof of Lemma 7.4.4 is relatively straightforward after you recall how to express the eigenvalues of A in terms of the trace and determinant. This Lemma is a special case of the Jury Stability Test, which provides an iterative procedure by which one may determine whether all eigenvalues of an n × n matrix have modulus less than 1.
Example. The matrix
A =
· 1 −1
1
4 −1
¸
has trace 0 and determinant −3/4. All three criteria of Lemma 7.4.4 are satisfied, which means that the eigenvalues of A have modulus less than 1.
Lemma 7.4.4 tends to be especially useful for systems of difference equations which contain unspecified parameters, because it can provide inequalities that the parameters must satisfy so that equilibria will be stable. Such systems are the subject of the following section.