PHASE PLANE DIAGRAMS OF DIFFERENCE EQUATIONS. 1. Introduction

Transcription

1 PHASE PLANE DIAGRAMS OF DIFFERENCE EQUATIONS TANYA DEWLAND, JEROME WESTON, AND RACHEL WEYRENS Abstract. We will be determining qalitatie featres of a discrete dynamical system of homogeneos difference eqations with constant coefficients. By creating phase plane diagrams of or system we can isalize these featres, sch as conergence, eqilibrim points, and stability.. Introdction Continos systems are often approximated as discrete processes, meaning that we look only at the soltions for positie integer inpts. Difference eqations are recrrence relations, and first order difference eqations only depend on the preios ale. Using difference eqations, we can model discrete dynamical systems. The obserations we can determine, by analyzing phase plane diagrams of difference eqations, are if we are modeling decay or growth, conergence, stability, and eqilibrim points. For this paper, we will only consider two dimensional systems. Sppose x(k) and y(k) are two fnctions of the positie integers that satisfy the following system of difference eqations: x(k + ) = ax(k) + by(k) y(k + ) = cx(k) + dy(k). This system is more simply written in matrix from z(k + ) = Az(k), ( ) ( ) x(k) a b where z(k) = is a colmn ector and A =. If z(0) = z y(k) c d 0 is the initial condition then the soltion to the initial ale problem z(k + ) = Az(k), z(0) = z 0,

2 2 TANYA DEWLAND, JEROME WESTON, AND RACHEL WEYRENS is z(k) = A k z 0. Each z(k) is represented as a point (x(k), y(k)) in the Eclidean plane. The set of points {(x(k), y(k)) t N} traces ot a path. The direction of this path is determined by k increasing. The directed path is called a trajectory. By changing initial conditions, we can create different trajectories for the gien system. The phase portrait is a diagram consisting of these mltiple trajectories. The shape of paths, the direction of trajectories, and eqilibrim soltions are some of the qalitatie featres we will explore. The general soltion may seem easy to sole, howeer, A k will be difficlt to compte for large ales of k. We can se a tool called the affine transformation to change or ariables and create a matrix J which will hopeflly be diagonal or close to diagonal. Using the properties of affine transformations we will show any gien matrix A is related to one of only three possible Jordan Canonical forms. 2. Affine Transformation Affine transformations are important becase certain shapes in the (, ) phase plane are presered in the (x, y) phase plane. While the soltions are not continos, the trajectory can be resembled by a continos fnction. More notably, if z = P z is an affine transformation then () a linear trajectory in the (, ) phase plane is transformed to a linear trajectory in the (x, y) phase plane. If the liner soltions in the (, ) phase plane goes throgh the origin, so does the transformed linear soltions. (2) an elliptical trajectory in the (, ) phase plane is transformed to an elliptical trajectory in the (x, y) phase plane. (3) a spiral shaped trajectory in the (, ) phase plane is transformed to a spiral shaped trajectory in the (x, y) phase plane. (4) a power cre shaped trajectory in the (, ) phase plane is transformed to a power cre shaped trajectory in the (x, y) phase plane, for example, parabolas and hyperbolas. (5) a tangent line L to a cre C in the (, ) phase plane is transformed to the tangent like P (L) to the cre P (C) in the (x, y) phase plane.

3 PHASE PLANE DIAGRAMS OF DIFFERENCE EQUATIONS 3 (6) a cred trajectory C that lies in a region R in the (, ) phase plane is transformed to a cred trajectory P (C) that lies in the region P (R) in the (x, y) phase plane. The goal is to find an affine transformation P sch that J = P AP is particlarly simple; specifically, one of the three forms described in Section 3. Two matrices A and J are similar if there exists an inertible matrix P sch that J = P AP. We say the phase portraits of z(k + ) = Az(k) and z(k + ) = Jz(k) are affine eqialent if A and J are similar. To stdy the phase portrait of z(k +) = Az we can consider the affine transformation w(k + ) = Jw, where J is a 2 2 matrix that has a particlarly simple form. Let P be a 2 2 inertible matrix. The change in ariables (2.) z = P w is called the affine transformation. Specifically, if [ ] [ [ p p P = 2 x, z =, and w = p 2 p 22 y] ] then the eqation z = P w becomes x = p + p 2 y = p 2 + p 22. Since P is inertible we also hae w = P z. We are now able to go from one set of ariables to the other. Since P is a constant matrix, we hae (2.2) w(k + ) = P z(k + ) = P Az = P AP w. Setting J = P AP, Eqation 2.2 becomes (2.3) w(k + ) = Jw and the associated initial condition becomes w 0 = w(0) = P z 0. Once w is determined we can find z sing the eqation z = P w. 3. Jordan Canonical Form For eery matrix A we find the similar matrix J throgh the affine transformation. Using the following theorem we know that J will be diagonal or close to diagonal, and therefore mch easier to compte J k for any k.

4 4 TANYA DEWLAND, JEROME WESTON, AND RACHEL WEYRENS Theorem 3.. Let A be a two by two real matrix. Then there is a nonsinglar real matrix P so that where: A = P JP, () If A has real eigenales λ, λ 2, not necessarily distinct, with linearly independent eigenectors, then [ ] λ 0 J =. 0 λ 2 (2) If A has a single eigenale λ with a single independent eigenector, then [ ] λ 0 J 2 =. λ (3) If A has complex eigenales α ± iβ, then [ ] α β J 3 =. β α These are the only three simple forms of J. Now that we hae a simple matrix J, we can begin creating or phase plane diagrams for analysis. 4. Analysis The simplest soltion, or triial soltion is when x = 0 and y = 0 simltaneosly. This triial soltion is represented by at point at the origin in the phase plane. The triial soltion is called a critical, or eqilibrim, point. A non triial soltion to the system traces a smooth cre which we called a trajectory. We take a sampling of trajectories to create or phase plane diagram. The geometrical properties of the phase plane diagram are related to the algebraic characteristics of the matrix A, which are presered throgh the affine transformation. We can see that the eigenales of A play a decisie role in determining many of important characteristics of the phase portrait. We will now analyze the possible combinations of eigenales with initial ales. Case : J = [ λ 0 0 λ 2 ].

5 PHASE PLANE DIAGRAMS OF DIFFERENCE EQUATIONS 5 General soltion: w(k) = ( c λ k c 2 λ k 2 ) When 0 < λ < λ 2 <, If c = 0, then as k approaches infinity, w(k) approaches zero, along the -axis. If c 2 = 0, then as k approaches infinity, w(k) approaches zero, along the -axis. When c is non-zero and c 2 is non-zero, the points lie along the line ln( c ) (k) = c 2 λ lnλ 2. In this instance, all trajectories approach zero. The diagram will resemble a in Figre. This is called a sink, or stable node. When λ > λ 2 >, If c = 0, then as k approaches infinity, w(k) dierges to infinity, along the -axis. If c 2 = 0, then as k approaches infinity, w(k) dierges to infinity, along the -axis. When c is non-zero and c 2 is non-zero, the points lie along the line ln( c ) (k) = c 2 λ lnλ 2. In this instance, all trajectories dierge to infinity. The diagram looks like b in Figre. This is called a sorce, or nstable node. (a) Sink (b) Sorce Figre. J Diagrams When λ = and λ 2 < λ, then w(k) = c (k) + c 2 λ 2 (k). The points will trace a ertical line at = c, tending towards = 0. The eqilibrim points will fill ot the whole -axis. This is called a degenerate node.

6 6 TANYA DEWLAND, JEROME WESTON, AND RACHEL WEYRENS When λ < and λ 2 >, then if c = 0, as k approaches infinity, w(k) dierges to infinity, along the -axis. If c 2 = 0, then as k approaches infinity, w(k) dierges to infinity, along the -axis. For c 0 and c 2 0 when k is een, the points lie along the cre in qadrant I and IV. When k is odd, the points lie along the cre in qadrant II and III. For all cres, the points dierge to infinity. This is called a sorce with reflection. When 0 < λ = λ 2 <, if c = 0, then as k approaches infinity, w(k) approaches zero, along the -axis. If c 2 = 0, then as k approaches infinity, w(k) approaches zero, along the -axis. When c 0 and c 2 0 the points lie alone the line (k) = c 2 c (k), and the slope depends on the initial conditions c and c 2. The diagram looks like Figre 2a. This is called a stable star node. When λ = λ 2 >, if c = 0, then as k approaches infinity, w(k) dierges to infinity, along the -axis. If c 2 = 0, then as k approaches infinity, w(k) dierges to infinity, along the -axis. When c 0 and c 2 0 the points lie alone the line (k) = c 2 c (k), and the slope depends on the initial conditions c and c 2. The diagram looks like Figre 2b. This is called an nstable star node. (a) Stable Star (b) Unstable Star Figre 2. J Diagrams When 0 < λ < and λ 2 >, if c = 0 and c 2 = 0 we get the point (0, 0). If c = 0, then as k approaches infinity, w(k) dierges to infinity, along the -axis. If c 2 = 0, then as k approaches infinity, w(k) tends towards zero, along the -axis. When c 0 and c 2 0 the points lie

7 PHASE PLANE DIAGRAMS OF DIFFERENCE EQUATIONS 7 (k) ln( )lnλ c alone the line (k) = c 2 λ 2. The diagram looks like Figre 3a. This is called a saddle. When < λ < 0 < < λ 2, if c = 0 and c 2 = 0 we get the point (0, 0). If c = 0, then as k approaches infinity, w(k) dierges to infinity, along the -axis. If c 2 = 0, then as k approaches infinity, w(k) tends towards zero, along the -axis. For c 0 and c 2 0 when k is een, the points lie along the cre in qadrant I and IV, which resemble a reflection oer the -axis. When k is odd, the points lie along the cre in qadrant II and III, which resemble a reflection oer the -axis. For all non-zero initial conditions, the points dierge to infinity. In general, (k) ln( )lnλ c the points lie alone the line (k) = c 2 λ2. The diagram looks like Figre 3b This is called a saddle with reflection. k= k=3 k=4 k=2 k=0 (a) Saddle (b) Saddle with Reflection Figre 3. J Diagrams Case 2: J 2 = General soltion: w(k) = λ k ( For 0 < λ < we can obsere that: [ ] λ 0. λ ) c λ c k + c 2 λ When c = 0, then w(k) tends towards zero along the -axis as k approaches infinity. Specifically, when c = 0, c 2 < 0, then w(k) moes along the negatie -axis, and when c = 0, c 2 > 0, then w(k) moes along the positie -axis.

8 8 TANYA DEWLAND, JEROME WESTON, AND RACHEL WEYRENS If c 0, then as k approaches infinity, w(k) tends towards zero. The shape of trajectories can be represented by the fnction = λ log λ + c 2. c c It is important to note that when < λ < 0, the points oscillate across the -axis. Specifically, the een k ales trace the trajectories on the positie half of the -axis, and the odd k ales trace the trajectories on the negatie half of the -axis. The diagram will look like Figre 4a. For λ = we can obsere that: If c = 0, w(k) dierges to infinity along the -axis as k approaches infinity. Specifically, if c = 0, c 2 < 0, w(k) moes along the negatie -axis, and when c = 0, c 2 > 0, w(k) moes along the positie - axis. If c 0, w(k) dierges to infinity as k approaches infinity. The trajectories can be represented by the fnction = c. It is important to note that when λ = the points oscillate across the -axis. Specifically, the een k ales trace the trajectories on the positie half of the -axis, and the odd k ales trace the trajectories on the negatie half of the -axis. The diagram will look like Figre 4b. Finally, when λ > we can obsere that: If c = 0, w(k) dierges to infinity along the -axis as k approaches infinity. If c 0, then w(k) dierges to infinity as k approaches infinity. The trajectories can be represented by the fnction = λ log λ + c 2. c c It is important to note that when λ <, the points oscillate across the -axis. Specifically, the een k ales trace the trajectories on the positie half of the -axis, and the odd k ales trace the trajectories on the negatie half of the -axis. The diagram will look like Figre 4c.

9 PHASE PLANE DIAGRAMS OF DIFFERENCE EQUATIONS 9 (a) 0 < λ < (b) λ = Figre 4. J 2 Diagrams (c) λ > Case 3: [ ] α β J 3 =. β α ( ) ( ) cos kθ sin kθ General soltion: w(k) = λ k c sin kθ cos kθ c 2 where λ = α 2 + β 2 and θ = tan ( β ) α 2 If α 2 + β 2 = the points lie on a circle, rotating conterclockwise. Under this condition, J is called the rotation matrix. The diagram looks like Figre 5a. When α 2 + β 2 > the phase plane diagram will be an nstable spiral, meaning it will dierge from the origin towards infinity. The diagram looks like Figre 5b. When α 2 + β 2 < the phase plane diagram will be a stable spiral, that is, tending towards the origin. The diagram looks like Figre 5c. 5. Acknowledgements We wold like to thank Dr. Daidson and or gradate mentor, Simon Pfeil, for their knowledge, motiation, patience and gidance with this project. Additionally, we wold like to thank gradate stdent Ben Dribs for all his help drawing the phase plane diagrams sed in or paper.

10 0 TANYA DEWLAND, JEROME WESTON, AND RACHEL WEYRENS (a) Circle (b) Unstable Spiral Figre 5. J 3 Diagrams (c) Stable Spiral References [] Kelley, Walter G., Peterson, Allan C., Difference Eqations: An Introdction with Applications, Academic Press, San Diego, CA, 200, pg [2] Daidson, Mark, Linear Systems of Differential Eqations. [3] Elaydi, Saber, An Introdction to Difference Eqations, Springer, New York, NY, 2005.