Local Stability of Period Two Cycles of Second Order Rational Difference Equation

Conjecture 1.1. Show that the period-two solution of (1.2) is locally asymptotically stable.

Definition 2.1. Let $$ be an equilibrium solution of (2.1).

• (i)

  $$ is called locally stable if for every ϵ > 0, there exists δ > 0 such that for all x₀, x₋₁ ∈ I, with $$, we have

  $$ (2.3)

• (ii)

  $$ is called locally asymptotically stable if it is locally stable, and if there exists γ > 0, such that for all x₀, x₋₁ ∈ I, with $$, we have

  $$ (2.4)

• (iii)

  $$ is called a global attractor if for every x₀, x₋₁ ∈ I, we have

  $$ (2.5)

• (iv)

  $$ is called globally asymptotically stable if it is locally stable and a global attractor.

• (v)

  $$ is called unstable if it is not stable.

• (vi)

  $$ is called a source, or a repeller, if there exists r > 0 such that for all x₀, x₋₁ ∈ I, with $$, there exists N ≥ 1 such that

  $$ (2.6)
  Clearly a source is an unstable equilibrium.

Definition 2.2. Let

denote the partial derivatives of f(u, v) evaluated at the equilibrium $$ of (2.1). Then the equation is called the linearized equation associated with (2.1) about the equilibrium solution $$.

Definition 2.3. A solution {x_n} of (2.1) is said to be periodic with period P if

A solution {x_n} of (2.1) is said to be periodic with prime period P, or a P-cycle if it is periodic with period P and P is the least positive integer for which (2.9) holds.

Theorem 2.4 (Linearized stability). (a) If both roots of the quadratic equation

lie in the open unit disk |λ | < 1, then the equilibrium $$ of (2.1) is locally asymptotically stable.

• (b)

  If at least one of the roots of (2.10) has absolute value greater than one, then the equilibrium $$ of (2.1) is unstable.

• (c)

  A necessary and sufficient condition for both roots of (2.10) to lie in the open unit disk |λ | < 1 is

  $$ (2.11)

•  

  In this case, the locally asymptotically stable equilibrium $$ is also called a sink.

• (d)

  A necessary and sufficient condition for both roots of (2.10) to have absolute value greater than one is

  $$ (2.12)

•  

  In this case, $$ is a repeller.

• (e)

  A necessary and sufficient condition for one root of (2.10) to have absolute value greater than one and for the other to have absolute value less than one is

  $$ (2.13)

•  

  In this case, the unstable equilibrium $$ is called a saddle point.

• (f)

  A necessary and sufficient condition for a root of (2.10) to have absolute value equal to one is

  $$ (2.14)

•  

  or

  $$ (2.15)

•  

  In this case, the equilibrium $$ is called a nonhyperbolic point.

Lemma 3.1. (a) When

Equation (1.4) has no nonnegative prime period-two solution.

• (b)

  When

  $$ (3.5)

•  

  Equation (1.4) has prime period-two solution,

  $$ (3.6)

•  

  if and only if

  $$ (3.7)

•  

  where Φ and Ψ are the positive and distinct solutions of the quadratic equation

  $$ (3.8)

Proof. (a) Suppose p + z ≥ 1 and assume, for the sake of contradiction, there exist distinct positive real numbers Φ and Ψ such that

•  

  is a prime period-two solution of (1.4). Then, Φ and Ψ satisfy

  $$ (3.10)

•  

  Furthermore,

  $$ (3.11)
  $$ (3.12)

•  

  Subtracting (3.11) from (3.12), we have

  $$ (3.13)

•  

  But, p + z ≥ 1 implies that Φ + Ψ ≤ 0 which contradicts the hypothesis that Φ and Ψ are distinct positive real numbers.

• (b)

  Now, suppose p + z < 1, and let Φ and Ψ be two positive real numbers such that

  $$ (3.14)

•  

  is a prime period-two solution of (1.4). Thus, Φ and Ψ satisfy

  $$ (3.15)
  $$ (3.16)

•  

  Moreover,

  $$ (3.17)
  $$ (3.18)

•  

  Subtracting (3.17) from (3.18), we have

  $$ (3.19)

•  

  Furthermore, adding (3.17) to (3.18), we have

  $$ (3.20)

•  

  Hence, Φ and Ψ > 0 satisfy the quadratic equation

  $$ (3.21)

•  

  In other words, Φ and Ψ are given by

  $$ (3.22)

Theorem 3.2. Suppose (1.4) has a prime period-two solution. Then, the period-two solution is locally asymptotically stable.

Proof. To start off, we first vectorize (1.4) by introducing the following change of variables:

and write (1.4) in the following equivalent form: where

Now Φ and Ψ generate a period-two solution of (1.4) if and only if

is a fixed point of T², the second iterate of T. Furthermore, where

The prime period-two solution of (1.4) is asymptotically stable if the eigenvalues of the Jacobian matrix $$, evaluated at $$ lie inside the unit disk. But,

and, hence, its characteristic equation is where By Theorem 2.4(c), the eigenvalues of lie inside the unit disk if and only if

To this end, without loss of generality, assume that 0 < Φ < Ψ. Then, by (3.15),

Hence, Similarly, we observe that Furthermore, since p + z < 1, (3.19) implies the sum of Φ and Ψ is less than 1 and, a fortiori, each is less than 1. Indeed, we have

With that in mind, it is clear that

In addition, with the understanding that we have The second inequality follows immediately since p + z < 1 and However, the proof of the first inequality is a bit tricky.

By (3.19),

Therefore, if and only if which is true if and only if The latter inequality holds true because Φ < Ψ and This completes the proof.