Diagrams of Difference Equations

Transcription

1 Introdction Dewland, SMILE Weston, Weyrens REU Smmer Phase Plane2011 Diagrams of Difference Eqations of Difference Eqations Tanya Dewland 1 Jerome Weston 2 Rachel Weyrens 3 1 Department of Mathematics Uniersity of Mississippi Oxford, MS 2 Department of Mathematics Loisiana State Uniersity Baton Roge, LA 3 Department of Mathematics Uniersity of Arkansas Fayetteille, AR

2 Introdction y x of Difference Eqations

3 Introdction Otline Introdction Terminology Affine Transformation Jordan Canonical Forms of Difference Eqations

4 Goals Introdction Terminology Affine Transformation Model discrete dynamical systems to determine otcome. Determine qalitatie featres of a system of homogeneos difference eqations with constant coefficients. of Difference Eqations

5 Introdction System of Difference Eqations Terminology Affine Transformation x(k + 1) = ax(k) + by(k) y(k + 1) = cx(k) + dy(k) General soltion: z(k) = A k z(0) ( ) [ ] x(k) a b Where z = and A = y(k) c d of Difference Eqations

6 Introdction Terminology Affine Transformation Each soltion is in the set {(x(k), y(k)) : k N} Trajectory Phase Plane Diagram of Difference Eqations

7 Introdction Affine Transformation Terminology Affine Transformation A tool for changing ariables Preseres collinearity of Difference Eqations

8 Change in Variables Introdction Terminology Affine Transformation w = z(k) = Pw(k) [ ] [ ] (k) p11 p, P = 12 (k) p 21 p 22 Create J = P 1 AP gies w(k + 1) = Jw(k) General soltion: w(k) = J k w(0) x = p 11 + p 12 y = p 21 + p 22. of Difference Eqations

9 Introdction Terminology Affine Transformation Jordan Canonical Form Theorem Let A be a two by two real matrix. Then there is a nonsinglar real matrix P so that A = PJP 1, where: If A has real eigenales λ 1, λ 2, not necessarily distinct, with linearly independent eigenectors, then [ ] λ1 0 J 1 =. 0 λ 2 of Difference Eqations

10 Introdction Terminology Affine Transformation Jordan Canonical Form Theorem cont. If A has a single eigenale λ with a single independent eigenector, then [ ] λ 0 J 2 =. 1 λ If A has complex eigenales α ± iβ, then [ ] α β J 3 =. β α of Difference Eqations

11 Introdction Spectral Radis Theorem Terminology Affine Transformation r(a) = max{ λ : λ is an eigenale} If r(a) < 1, then any soltion to z(k) = A k z(0) has the property lim k Ak z(0) = 0. If r(a) 1, some soltions z(k) does not tend toward the origin as k. of Difference Eqations

12 Case 1 Introdction Jordan Canonical Forms Where λ 1, λ 2 R [ ] λ1 0 J 1 = 0 λ 2 ( c1 λ General soltion: w(k) = k ) 1 c 2 λ k 2 of Difference Eqations

13 Sorce and Sink Introdction Jordan Canonical Forms (c) λ 1 > λ 2 > 1 Figre: Trajectories (d) 0 < λ 1 < λ 2 < 1 of Difference Eqations

14 Sorce and Sink Introdction Jordan Canonical Forms (a) λ 1 > λ 2 > 1 Figre: = c p (b) 0 < λ 1 < λ 2 < 1 of Difference Eqations

15 Introdction Unstable and Stable Star Jordan Canonical Forms (a) λ 1 = λ 2 > 1 Figre: = c 2 c 1 (b) 0 < λ 1 = λ 2 < 1 of Difference Eqations

16 Introdction Jordan Canonical Forms Saddle and Saddle with Reflection k=1 k=3 k=4 k=2 k=0 (a) 0 < λ 1 < 1, λ 2 > 1 (b) 1 < λ 1 < 0 < 1 < λ 2 Figre: = c p of Difference Eqations

17 Degenerate Node Introdction Jordan Canonical Forms w(k) = ( c1 λ k 2 c 2 ) Figre: 0 < λ 2 < λ 1 = 1 of Difference Eqations

18 Case 2 Introdction Jordan Canonical Forms Where λ R J 2 = [ ] λ 0 1 λ General soltion: w(k) = λ k 1 ( c1 λ c 1 k + c 2 λ ) of Difference Eqations

19 Introdction Jordan Canonical Forms (a) λ = 1 (b) λ > 1 Figre: = λ log λ c 1 + c 2 c 1 (c) λ < 1 of Difference Eqations

20 Case 3 Introdction Jordan Canonical Forms Where α, β R [ ] α β J 3 = β α ( cos kθ General soltion: w(k) = λ k sin kθ λ = α 2 + β 2 and θ = tan 1 ( β α ) ) ( ) sin kθ c1 where cos kθ c 2 of Difference Eqations

21 Introdction Jordan Canonical Forms (a) α 2 + β 2 = 1 (b) α 2 + β 2 > 1 Figre: J 3 Diagrams (c) α 2 + β 2 < 1 of Difference Eqations

22 Introdction Here A = [ 1 ] x(k + 1) = x(k) + y(k) y(k + 1) = 0.25x(k) + y(k) The eigenales of A are λ 1 = 1 2 and λ 2 = 3 2. [ ] [ ] P = and J = of Difference Eqations

23 Introdction y x (a) Phase Plane of A (b) Phase Plane of J of Difference Eqations

24 Introdction of Difference Eqations

25 Introdction of Difference Eqations

26 Introdction Thank Yo! of Difference Eqations