Bifurcations in Linear Systems
MATH 228
3/24/00
Carl Staab
Abstract

	The problem was to classify the linear system of two ordinary autonomous differential equations:
dx/dt = ax + by
dy/dt = -x + -y
	This was accomplished by determining that the trace T = a – 1 and the determinant D = b – a and mapping the trace-determinant plane onto the a-b plane.
	The classification of the system according to the values of a and b is this:

For a <1:
If b > .25a2 + .5a +.25 then the system is a stable spiral
If a < b < .25a2 + .5a +.25 then the system is a stable node
If b < a then the system is a saddle

If a = 1 and b > 1 then the system is a center

For a > 1:
If b > .25a2 + .5a +.25 then the system is a unstable spiral
If a < b < .25a2 + .5a +.25 then the system is a unstable node
If b < a then the system is a saddle

	Almost all of the bifurcations are significant as t approaches positive infinity.  The change from stable spiral to stable node is not significant for large t; both x(t) and y(t) are virtually equal to zero either way.  However, the change between the stable spiral to a center to an unstable spiral is very important for large t, in both the phase plane and the x(t) and y(t) graphs.  The change is between oscillating toward zero (the origin in the phase plane), oscillating without a change of magnitude, and oscillating larger (outward in the phase plane), respectively.  The change between a stable node and a saddle is very important for large t, both in the phase plane and for x(t) and y(t).  The change is between all solutions approaching the origin and most of the solutions approaching infinity.  The change between an unstable spiral and an unstable node is pronounced, for both the phase plane and x(t) and y(t); the solutions go from oscillating wildly to not at all, and two straight-line solutions in the phase plane appear.  Finally, the bifurcation between an unstable node and a saddle, although not very pronounced in the x(t) and y(t) graphs, is important in the phase plane.  All of these are true in reverse as well.

Introduction
When using the system of differential equations:
dx/dt = ax + by
dy/dt = cx + dy
there is an easy way to predict the general type of the solution.  By simply taking the trace (equal to a + d) and the determinant (equal to ad – bc) one can plot the point and determine where it lies on the trace-determinant plane.
The trace-determinant plane is a tool used to help determine the general nature of solutions to linear systems of ordinary differential equations with two dependent variables.  The roots or eigenvalues (signified by ) of the characteristic polynomial
2 –T* + D = 0
determine the nature of the solutions.  If D > .25T2 then these roots are complex (by the quadratic equation) corresponding to a spiral in the phase plane (which is the graph of several solutions in the x-y plane).  If the real term of these roots is positive, then the solution is an unstable spiral in the phase plane; the solutions oscillate, growing larger.  If the real term of these roots is negative, then they correspond to a stable spiral; the solutions oscillate, growing smaller.  If the real term is equal to zero then the solutions simply oscillate forever without increasing or decreasing; this is known as a center.  If D < 0 then (T2-4D)1/2 > T, corresponding to one root being negative and the other positive.  This solution, the saddle, looks much like a set of hyperbolae (on the phase plane).  The negative root is more dominant for small t and the positive one is more dominant for large t.  (Each completely real root corresponds to one straight-line solution on the phase plane.)  Finally, if both roots are negative but real, then all solutions decrease to zero (dominated by the larger root for larger t and the smaller root for smaller t); this is called a stable node.  If both roots are positive but real, all solutions go to infinity; this is called an unstable node (the dominance of roots is like that of the stable node and saddle).  All linear systems of differential equations described here have one equilibrium point that will not be mentioned anymore: this is x = 0, y = 0, the origin.
The problem here is to classify a system of the form
dx/dt = ax + by
dy/dt = -x + -y
and to determine where bifurcations (sharp changes between classes) are significant for large time.

Body

	We learned in class that the classification of a linear system is determined by its position on the trace-determinant plane.  Given a system
dx/dt = ax + by
dy/dt = cx + dy
the trace T = a – d and the determinant D = ad – bc.
In particular, there are three different boundaries that determine the class of a system.  One is the line T = 0, where D >= 0.  This is the boundary between unstable spirals (for positive T) and stable spirals (for negative T).  Any systems that lie on this line are centers.  Another is the parabola D = .25*T2.  This is the boundary between spirals and nodes (for negative T, this is a boundary between stable spirals and stable nodes; for positive T, this is a boundary between unstable spirals and unstable nodes).  Systems that lie on this boundary have one straight-line solution, and all other solutions are parallel to this line for small t, make a U-turn, and are parallel to this line in the other direction for large t.  The last important boundary on the trace-determinant plane is the line D = 0.  This is the boundary between nodes (positive D) and saddles (negative D).
	Since the matrix A (DX = AX; X = (x,y)) for the system is:
(a	b)
(-1	-1)
the trace T = a - 1 and the determinant D = b – a.
Plugging these values into the aforementioned equations for the boundaries yields the following three expressions:
T = 0  a – 1 = 0  a = 1 (for b >= 1)
D = .25*T2  b – a = .25(a2 –2a + 1)  b = .25a2 +.5a +.25
D = 0  b – a = 0  b = a
	This sets up the following possibilities.  For a <1:
If b > .25a2 + .5a +.25 then the system is a stable spiral
If a < b < .25a2 + .5a +.25 then the system is a stable node
If b < a then the system is a saddle

If a = 1 and b > 1 then the system is a center

For a > 1:
If b > .25a2 + .5a +.25 then the system is a unstable spiral
If a < b < .25a2 + .5a +.25 then the system is a unstable node
If b < a then the system is a saddle

	Most of the bifurcations that occur are significant for large time.  The only one that does not seem to be significant is the change between a stable node and a stable spiral, because in both of those the solutions approach zero (the origin, in the phase plane) over large time.
	The rest of the bifurcations are important.  The change from a stable spiral to an unstable spiral quickly changes from solutions approaching zero over large time to solutions oscillating forever without increasing or decreasing to solutions going to infinity at large time.  While the short-term effects may be small, there is a significant difference between zero and infinity.  The phase plane will look much the same, except that solutions will be going outward instead of inward.  However, the x(t) and y(t) graphs will change from oscillations that get ever-smaller to oscillations that get ever-larger.
	Going from an unstable spiral to an unstable node is a significant bifurcation.  The main difference between the two over large time is that in the phase plane, an unstable spiral solution will go around the origin an infinite number of times for infinitely large t, while an unstable node solution will eventually approach a straight line.  In the x(t) and y(t) graphs, the solutions will go from ever-growing oscillation to a simple sum of two exponential curves (which approach either infinity or negative infinity, depending on the constants).
	Going from a stable node to a saddle is a dramatic change.  The limit at infinite (positive) t for a stable node is the origin of the phase plane.  For a saddle, the limit at infinite (positive) t is a straight-line solution in the phase plane, which is an exponential-growth curve for the x(t) and y(t) solutions.
	Going from an unstable node to a saddle is a significant change in the phase plane, although not very important for the individual x(t) and y(t) solutions.  For a saddle, all solutions approach one straight-line solution in the phase plane (in either of its two directions) over large time.  All x(t) and y(t) solutions therefore eventually look like the same exponential curve.  For an unstable node, all solutions end up approximately parallel to the dominant straight-line solution, exactly as with the saddle, but the difference is that they are going away from that straight-line solution, not towards it.  The change between an unstable node going to infinity away from both lines and a saddle going to infinity away from one line and towards another is not significant for small t, but is for large t; however, the difference is much more evident in the phase plane than in the individual x(t) and y(t) solutions, which are affected by the bifurcation only about as much as they would be affected by a small change in parameters without bifurcating.