Math 410(510) Notes (4) Qualitative behavior of Linear systems a complete list. Junping Shi
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1 A. Two different positive eigenvalues Math 4(5 Notes (4 Qualitative behavior of Linear systems a complete list Junping Shi Eigenvalues: λ = and λ = 4 Eigenvectors: V = (,, V = (, ( ( General solution: Y = c e t + c ( x + y x + 3y e 4t. (, is the only equilibrium point, and any non-zero solution satisfies lim Y(t = and lim Y(t = (,.. There are two linear independent straight line solutions: ( ( Y = c e t, Y = c which are on the lines y =.5x and y = x respectively. e 4t 3. The non-straight-line solution satisfies (i when t, the solution is tangent to the straight line solution on y =.5x, (ii when t, the solution is almost parallel to the straight line solution on y = x. 5. This type of equilibrium point is a (-dimensional source. B. Two different negative eigenvalues Eigenvalues: λ = and λ = 4 Eigenvectors: V = (,, V = (, ( ( General solution: Y = c e t + c ( x y x 3y e 4t. (, is the only equilibrium point, and any non-zero solution satisfies lim Y(t = and lim Y(t = (,.. There are two linear independent straight line solutions: ( ( Y = c e t, Y = c e 4t
2 which are on the lines y =.5x and y = x respectively. 3. The non-straight-line solution satisfies (i when t, the solution is tangent to the straight line solution on y =.5x, (ii when t, the solution is almost parallel to the straight line solution on y = x. 5. This type of equilibrium point is a (-dimensional sink. C. One negative eigenvalue and one positive eigenvalue ( x + 3y x y Eigenvalues: λ = and λ = Eigenvectors: V = (3,, V = (, ( ( 3 General solution: Y = c e t + c e t. (, is the only equilibrium point.. There are two linear independent straight line solutions: ( ( 3 Y = c e t, Y = c which are on the lines y =.5x and y = x respectively. e t 3. Y is the only solution which satisfies lim Y(t = and lim Y(t = (,. (stable solution 4. Y is the only solution which satisfies lim Y(t = and lim Y(t = (,. (unstable solution 5. The non-straight-line solution satisfies (i lim Y(t = t ± (ii when t, the solution tends to the unstable solution, (ii when t, the solution tends to the stable solution. 6. This type of equilibrium point is a (-dimensional saddle. (there is no saddle in -dimension D. complex eigenvalues a ± bi, a < ( x 3y 3x y Eigenvalues: λ = + 3i and λ = 3i Eigenvectors: V = (i,, V = ( i, General solution: Y = c ( sin(3t cos(3t e t + c ( cos(3t sin(3t e t. (, is the only equilibrium points, and any non-zero solution satisfies lim Y(t = and lim Y(t = (,.
3 . There is no straight line solutions. 3. Any non-zero solution spiral toward the origin, around the origin infinitely many times. 4. The solution curves (t, x(t, (t, y(t are decaying periodic functions. 5. This type of equilibrium point is a (-dimensional spiral sink. 6. Orientation: clockwise or counterclockwise? Qualitative behavior of solutions: E. complex eigenvalues a ± bi, a > Eigenvalues: λ = + 3i and λ = 3i Eigenvectors: V = ( i,, V = (i, General solution: Y = c ( sin(3t cos(3t ( x + 3y 3x + y e t + c ( cos(3t sin(3t. (, is the only equilibrium points, and any non-zero solution satisfies lim Y(t = and lim Y(t = (,.. There is no straight line solutions. e t 3. Any non-zero solution spiral away from the origin, around the origin infinitely many times. 4. The solution curves (t, x(t, (t, y(t are exponential growing periodic functions. 5. This type of equilibrium point is a (-dimensional spiral source. 6. Orientation: clockwise or counterclockwise? Qualitative behavior of solutions: F. complex eigenvalues a ± bi, a = Eigenvalues: λ = i and λ = i Eigenvectors: V = (, i, V = (, i General solution: Y = c ( sin( t cos( t ( y x + c ( cos( t sin( t. (, is the only equilibrium points, and any non-zero solution is a periodic solution.. There is no straight line solutions. 3. Any non-zero solution spiral around the origin infinitely many times, but stay on a periodic orbit. 4. The solution curves (t, x(t, (t, y(t are periodic functions.
4 lim 5. This type of equilibrium point is a (-dimensional center. 6. Orientation: clockwise or counterclockwise? G. Two real eigenvalues: λ > λ = Eigenvalues: λ = and λ = 8 Eigenvectors: V = (3,, V = (, ( ( 3 General solution: Y = c + c e 8t ( x + 3y 4x + 6y. Any point on the line x+3y = is an equilibrium point, and any non-equilibrium solution satisfies Y(t = and lim Y(t is an equilibrium.. Any non-equilibrium solution is a straight line solution. 3. This type of equilibrium point is a (-dimensional degenerate source. H. Two real eigenvalues: = λ > λ degenerate sink I. Repeated eigenvalues: λ = λ > Eigenvalues: λ = λ = Eigenvectors: V = (,, V = (, or any vector ( ( General solution: Y = c e t + c e t ( x y. (, is the only equilibrium point, and any non-zero solution satisfies lim Y(t = and lim Y(t = (,.. Each solution is a straight line solution, and any ray from (, is a solution orbit. 3. This type of equilibrium point is a (-dimensional star source. J. Repeated eigenvalues: λ = λ < star sink K. Repeated eigenvalues: λ = λ > ( x + y y
5 Eigenvalues: λ = λ = Eigenvectors: V = (,, General solution: Y = c ( ( ( e t + c t ( + e t. (, is the only equilibrium point, and any non-zero solution satisfies lim Y(t = and ( lim Y(t = (,. There is one straight line solution Y = c e t, which is on the line y =.. Any non-straight-line solution satisfies (i t, the solution is tangent to the straight line solution, (ii t, the solution is almost parallel to the straight line solution (but in the opposite direction. 3. This type of equilibrium point is a (-dimensional trying to spiral source. L. Repeated eigenvalues: λ = λ < trying to spiral sink M. Repeated eigenvalues: λ = λ = Eigenvalues: λ = λ = Eigenvectors: V = (,, General solution: Y = c ( ( ( + c t ( y ( +. Any point on y = is an equilibrium point, any non-equilibrium solution is a straight line solution which is parallel to the line of equilibrium points.. This type of equilibrium point is a (-dimensional parallel lines. N. Repeated eigenvalues: λ = λ < : a = b = c = d = dumb system Classification: Two real eigenvalues:. λ > λ > : source. λ > λ = : degenerate source 3. λ > > λ : saddle 4. λ = > λ : degenerate sink 5. > λ > λ : sink Two complex eigenvalues: λ ± = a ± bi. a > : spiral source. a = : center 3. a < : spiral sink
6 One real eigenvalue: λ = λ = λ. λ > : star source or trying to spiral source. λ = : parallel lines or dumb system 3. λ < : star sink or trying to spiral sink Stable equilibrium: sink, spiral sink, star sink or trying to spiral sink (negative eigenvalues, or complex eigenvalues with negative real part Others: unstable (neutrally stable: degenerate sink, degenerate source, center
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