Marc D. Benayoun
?copyright 2005 Marc Benayoun
Introduction to Phase Plane Analysis:
In the last lab, we discovered how to solve a simple homogeneous second-order
linear system of differential equations. As an example, we examined a model of retinal
feedback between cone cells and horizontal cells of the retina. Today we will be
analyzing the same system but from a more qualitative perspective using phase plane
analysis. Although the explicit solutions determined in lab last week are more
informative, many more complicated systems (such as the non-linear systems that will be
explored in our final lab) can only be qualitatively described. When we describe a
system qualitatively, we look for steady-state values of the solutions (often called fixed
points) and try to classify the dynamics of the solution that lead to these steady state
values. In the previous lab we considered the example
which has the solution
This solution can be described qualitatively. The only finite steady-state solution to this
limx(t)?limy(t)?0system is (x, y) = (0,0). For example, if C=0, then, , 1t??t??
therefore one says that (x, y) = (0,0) is a steady-state or fixed point of the system. For
limx(t)?limy(t)??C?0, then . Regardless of how we choose C and C, 12t??t??1
there are no other steady-state values for this system. Since the initial conditions
determine C and C, then those initial conditions that lead to C=0 will have solutions 121
that steadily tend towards the fixed point (0,0), while all others will steadily tend towards
infinity (ie away from the fixed point). A fixed point with this property (ie some initial
conditions lead to the fixed point and others lead away from it) is called a saddle point.
This simple qualitative description of identifying the steady-state(s) of the solution, the
dynamics of what initial conditions lead to the steady-state(s), and how it is reached
steadily or in an oscillatory fashion can all be determined from a phase plane analysis of
The first step in phase plane analysis is to set up a phase plane. The axes for the
plane are chosen to be the solutions to the system being considered. In the example
above, the phase plane is constructed with y as the ordinate and x as the abscissa. Next,
the x- and y-nullclines are plotted. The x-nullcline is the curve in the x-y plane, where
dx?0. A similar definition applies for the y-nullcline. Intersections of these nullclines dt
dxdy, so x and y are no longer changing with time. In represent points where ??0dtdt
other words, these intersections represent steady-state values or fixed points of the system.
Next, a vector field is constructed by assigning to every point on the x-y plane the vector
Tdxdy??. Notice that this vector field can be determined without knowing the ??dtdt??
dydxdysolution to the system. Since the slope of these vectors is by the chain m?/?dtdtdxrule, the vector field must be tangent to any solution (x, y) of the system. This allows us to use the vector field to calculate the solution of the system for any initial condition
(x,y). Such a solution when plotted on the phase plane is called a trajectory. The phase oo
plane, nullclines, vector field, and several trajectories are shown in the figure below for
the system in Eq.’s 1 and 2.
The nullclines are plotted as dashed lines; the x-nullcline is plotted in red, and the y-nullcline is plotted in orange. Notice that these nullclines intersect at the point (x, y) = (0,0) indicating that this is the steady-state of the system in agreemet with what was
predicted by considering the explicit solutions (Eq.’s 3 and 4). Any linear system of ordinary differential equations described by a matrix with real eigenvalues of opposite
sign (recall that the eigenvalues for this system are -1 and 3) will have a saddle point at
the intersection of its nullclines.
If the matrix describing the linear system has real eigenvalues, which are both
negative, then the fixed point is called a nodal sink. The classic phase portrait of a nodal sink is shown below.
If the matrix describing the linear system has real eigenvalues, which are both positive, then the fixed point is called a nodal source. The classic phase portrait of a
nodal source is shown below.
Notice the difference in the direction of the arrows in the vector field.
If the matrix describing the linear system has imaginary eigenvalues, which have negative real parts, then the fixed point is called a spiral sink. The classic phase portrait of a spiral sink is shown below.
If the matrix describing the linear system has imaginary eigenvalues, which have positive real parts, then the fixed point is called a spiral source. The classic phase portrait of a spiral source is shown below.
These five types of equilibria are collectively known as the generic equilibria. There are
five nongeneric equilibria as well. The most important nongeneric equilibrium is called a
center. It occurs when the eigenvalues of the matrix are purely imaginary. The classic
phase portrait of a center is shown below.
The Biological Model:
We will be studying the same system as last week, the retinal feedback model.
The system from the last lab is represented below:
??0.025sec,??0.08sec,andk?4.Typical values for these parameters are Let us CHassume that the light intensity, L=10 (ie daylight). For our initial conditions we will
~LL~choose that C(0)=H(0)=0. Finally, recall that C?C?,andH?H?. k?1k?1
1. Write a function phase_plane(A, init) that takes a matrix and performs a phase plane
analysis for the linear system, u’=Au. The function should plot a phase plane with axes x
and y, plot the nullclines, create the vector field, and plot the phase plane trajectory that
passes through the initial condition. Finally, the program should output the type of
equilibrium point, saddle point, spiral sink, etc. If the equilibrium point is nongeneric, then the program can just output nongeneric as the class type. This is quite a complicated program, so you may want to write several smaller functions that can be called within phase_plane. For example, a separate function that will simply classify the fixed point, and then another that will create a vector field, etc.
2. Use your phase_plane program to analyze the biological model shown above. What kind of behavior does the fixed point exhibit? Repeat using the parameters for dim light ??0.1sec,??0.5sec,andk?0.5. What is the behavior of the fixed point now? CH
3. Sketch a set of functions x(t) and y(t) versus t that would exhibit a nodal sink behavior
if a phase analysis were performed on the system with these solutions. Repeat for the other four generic equilibria.
4. An interesting area of linear and especially nonlinear dynamics is called bifurcation theory. Bifurcation theory is a study of how parameter values can influence dynamics. Hopefully, you saw above that the parameters during daylight lead to a different type of dynamics for the fixed point than the parameters for dim light. Calculate under what combinations of parameter values each of the different generic equilibria could arise in the retinal feedback model. Are any of the equilibria not permitted by this model?