Discusión

.

Certainly αI commutes with B, which allows us to use the binomial theorem when computing Mⁿ = (αI + B)ⁿ. Moreover, since B² = −β²I, we may calculate that B³ = −β²B and B⁴ = β⁴I. Since B⁴ is a constant multiple of the identity matrix, we might expect the same sorts of cyclic oscillations that we saw in the example above.

Finally, we remark that solving inhomogeneous constant-coefficient difference equations is straightforward, but we shall not discuss the techniques here. Indeed, for the inhomogeneous equation

x_n+1 = a₁x_n+ a₂x_n−1+ · · · + a_kx_n−k+1+ g(n),

it is possible to state an analogue of the variation of parameters formula (2.18) for ODEs. Due to the discrete nature of difference equations, the solution contains a summation involving g(n), as opposed to the integral in (2.18).

7.3 First-Order Nonlinear Equations and Stability

It is almost never possible to present a closed formula for the solution of a nonlinear difference equation. As with nonlinear ODEs, we typically settle for a qualitative un-derstanding of how solutions behave. Our development of the qualitative analysis of nonlinear difference equations perfectly parallels the methodology we introduced for nonlinear ODEs. We will begin by analyzing constant solutions, which are analogous to equilibria for ODEs. Afterwards, we will study more exotic dynamical behavior, including periodic solutions, bifurcations, and chaos. We restrict our initial discus-sion to first-order nonlinear difference equations, later generalizing our results to higher-order systems. The material in this section is based heavily on Chapter 10 of Strogatz [11].

Example. The behavior of the iterates of the nonlinear equation x_n+1= x²_ndepends greatly upon our choice of initial condition x₀. For example, if x₀ = 2, then x_n → ∞ as n → ∞. On the other hand, if x0 = 1/2, then the sequence of iterates xnconverges rapidly to 0. Notice also that if x₀ = 1, then x_n = 1 for all n ≥ 0. This constant solution of the difference equation is analogous to an equilibrium for an ODE, and such solutions have a special name.

Definition 7.3.1. A fixed point of the first-order difference equation xn+1 = f (xn) is any number x^∗ such that x^∗ = f (x^∗).

Notice that, by definition, if we start out by using a fixed point as our initial condition, then we will remain stuck at that fixed point for all future iterates.

Example. To find all fixed points x of the difference equation x_n+1 = 2x_n − 2x²_n, we should solve the equation x = 2x − 2x². By algebra, we have 2x(x − 1/2) = 0, a quadratic equation with two roots: x = 0 and x = 1/2. These are the two fixed points of this nonlinear difference equation.

As with equilibria of ODEs, fixed points of difference equations can be stable or unstable. Roughly speaking, a fixed point x^∗ is locally asymptotically stable if whenever we start from an initial condition x₀ that is appropriately “close” to x^∗, the sequence {x_n} of iterates converges to x^∗ as n → ∞. Fixed points can also be repellers—i.e., the gap between x^∗ and x_n may grow as n increases, no matter how close the initial condition x₀ is to x^∗.

Example. Fixed points of the mapping x_n+1 = x²_n satisfy the equation x = x². This quadratic equation has two solutions, x = 0 and x = 1. The fixed point x = 0 is locally asymptotically stable, because if we start from any initial condition x₀ that is “close” to 0, then the sequence of iterates will converge to 0. Specifically, if x₀ ∈ (−1, 1), then x_n→ 0 as n → ∞. In contrast, the fixed point x = 1 is unstable, because if we start from any initial condition other than x₀ = 1, the iterates will be repelled from 1.

We now devise a test to determine whether a fixed point of a difference equation is locally stable or not. As with differential equations, the stability test involves the use of Jacobian matrices. However, the conditions that eigenvalues must satisfy will be different—stability will depend upon more than just the real part of the eigenvalues.

First, consider the first-order difference equation xn+1 = f (xn), and assume that the function f : R → R is continuously differentiable. Suppose that x^∗ is an isolated fixed point of our equation. To determine whether x^∗ is an attractor or repeller, we need to investigate how the iterates of the mapping would behave if we start from an initial condition that is “near” x^∗. Suppose that our initial condition is x₀ = x^∗+ ²₀, where |²0| is a very small number. We will estimate the gap ²1 between the value of x₁ (the first iterate) and the fixed point x^∗ in order to see whether x₁ is closer to the fixed point than x0 was. More exactly, suppose x1 = x^∗+ ²1. We also know that x₁ = f (x₀) and, since x₀ = x^∗+ ²₀, we may use the tangent line approximation at x^∗ to estimate

x₁ = f (x^∗+ ²₀) ≈ f (x^∗) + ²₀f⁰(x^∗).

Recalling that x1 = x^∗+ ²1, we equate our two expressions for x1 to obtain

∗ ∗ 0 ∗

The fact that x^∗ is a fixed point implies that f (x^∗) = x^∗, and therefore x^∗+ ²₁ ≈ x^∗+ ²₀f⁰(x^∗).

Subtracting x^∗ from both sides and taking absolute values, we find that

¯¯

¯¯²₁

²₀

¯¯

¯¯ ≈ |f⁰(x^∗)|.

Interpreting this approximation in words will give rise to our first stability criterion.

Recall that ²₀ and ²₁ measure the gaps x₀−x^∗and x₁−x^∗, respectively. Thus, the left hand side of the above approximation measures the ratio of these gaps. If the fixed point is an attractor, we would need this ratio to be smaller than 1 in magnitude, implying that the gaps between iterates and the fixed point x^∗ will shrink as we generate more iterates. Conversely, if the ratio exceeds 1 in magnitude, then the gaps between the iterates x_n and the fixed point x^∗ will grow as n increases. Thus, we have provided a heuristic proof of

Theorem 7.3.2. Suppose x^∗ is an isolated fixed point of the first-order difference equation x_n+1 = f (x_n), where f is continuously differentiable. Then x^∗ is lo-cally asymptotilo-cally stable (attracting) if |f⁰(x^∗)| < 1 and is unstable (repelling) if

|f⁰(x^∗)| > 1. If |f⁰(x^∗)| = 1, this test is inconclusive.

Warning: Although unstable fixed points are locally repelling, we must exercise caution when drawing conclusions about long-term behavior of iterates (particularly if f is not as smooth as required by the conditions of Theorem 7.3.2). If f is merely piecewise continuous, it is possible for x_n+1= f (x_n) to have an unstable fixed point which is globally attracting (see exercises).

Example. Consider the difference equation x_n+1 = cos(x_n). We claim that this equation has exactly one fixed point. Fixed points satisfy the transcendental equation x = cos x, which is impossible to solve algebraically. Equivalently, fixed points are roots of the function g(x) = x − cos x. Notice that g(x) is a continuous function and that g(0) = −1 whereas g(π/2) = π/2. Since g is continuous and its values change from negative to positive between x = 0 and x = π/2, the intermediate value theorem from calculus guarantees that g has at least one root in the interval (0, π/2). Next, we must show that g has exactly one real root. To see why, observe that g⁰(x) = 1 + sin x is non-negative because sin x ≥ −1. Thus, the function g(x) is non-decreasing (it is actually a one-to-one function). It follows that the equation g(x) = 0 can have at most one root. Letting x^∗ denote this root, we conclude that x^∗ is the only fixed point of this difference equation.

Again, it is impossible to find the value of x^∗ algebraically. However, the above remarks indicate that 0 < x^∗ < π/2, and this is actually enough information for us to use Theorem 7.3.2 to test the local stability of x^∗. Our difference equation has

the form x_n+1 = f (x_n) where f (x) = cos x. According to the Theorem, we should check the magnitude of f⁰(x^∗). In this case, f⁰(x) = − sin x, from which we calculate

|f⁰(x^∗)| = | − sin(x^∗)|.

Since 0 < x^∗ < π/2, we have | − sin x^∗| < 1. Therefore, Theorem 7.3.2 guarantees that our fixed point x^∗ is locally asymptotically stable. In this example, we never needed to know the exact value of x^∗ in order to test its stability.

Remarkably, the fixed point x^∗ of this difference equation is actually globally asymptotically stable. That is, for any choice of initial condition x0, the sequence of iterates of this mapping will converge to the fixed point! You should test this out by picking any number you like and then using a calculator or computer to repeatedly take the cosine of the number you chose. Make sure your calculator is measuring angles in radians, not degrees. You will find that the value of the fixed point is x^∗ = 0.739085..., which is the only solution of the equation x = cos x.

Example. Recall that the difference equation x_n+1= x²_nhas two fixed points, 0 and 1. In this case, f (x) = x², so f⁰(x) = 2x. Since f⁰(0) = 0 < 1, we see that 0 is a locally asymptotically stable fixed point, and since f⁰(1) = 2 > 1, we conclude that 1 is an unstable fixed point.

Example. By algebra, you can check that the only fixed points of x_n+1 = 3x_n(1−x_n) are 0 and 2/3. Here, f (x) = 3x − 3x², so f⁰(x) = 3 − 6x. Since |f⁰(0)| = 3 > 1, we see that 0 is an unstable fixed point. On the other hand, since |f⁰(2/3)| = 1, we cannot use Theorem 7.3.2 to draw any conclusions regarding the stability of that fixed point.

Definition 7.3.3. A fixed point x^∗ of the equation x_n+1 = f (x_n) is called hyperbolic if |f⁰(x^∗)| 6= 1. Otherwise, the fixed point is called non-hyperbolic.

Our local stability Theorem 7.3.2 can only be used to classify stability of hy-perbolic fixed points. In order to determine whether a non-hyhy-perbolic fixed point is stable, we need a finer approach. After all, the derivation of Theorem 7.3.2 was based upon linear approximation of the function f in the vicinity of a fixed point x^∗. If f has a continuous third derivative, then we can obtain the following theorems regarding stability of non-hyperbolic equilibria:

Theorem 7.3.4. Suppose that x^∗ is an isolated non-hyperbolic equilibrium point of x_n+1 = f (x_n) and, more specifically, that f⁰(x^∗) = 1. Then x^∗ is unstable if f⁰⁰(x^∗) 6= 0. If f⁰⁰(x^∗) = 0 and f⁰⁰⁰(x^∗) > 0 then, again, x^∗ is unstable. Finally, if

In order to state the corresponding theorem for the case f⁰(x^∗) = −1, it is helpful to introduce the notion of the Schwarzian derivative.

Definition 7.3.5. The Schwarzian derivative of a function f is defined as Sf (x) = f⁰⁰⁰(x)

f⁰(x) − 3 2

·f⁰⁰(x) f⁰(x)

¸₂ .

Theorem 7.3.6. Suppose that x^∗ is an isolated non-hyperbolic equilibrium point of x_n+1= f (x_n) 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 x_n+1= 3x_n(1 − x_n). 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.