Phase Portraits (DP IB Applications & Interpretation (AI)): Revision Note

Phase Portraits

What is a phase portrait for a system of coupled differential equations?

• Here we are again considering systems of coupled equations that can be represented in the matrix form , where , , and 

• A phase portrait is a diagram showing how the values of x and y change over time

  • On a phase portrait we will usually sketch several typical solution trajectories

  • The precise trajectory that the solution for a particular system will travel along is determined by the initial conditions for the system

• Let  and be the eigenvalues of the matrix 

  • The overall nature of the phase portrait depends in large part on the values of and 

What does the phase portrait look like when  and  are real numbers?

• Recall that for real distinct eigenvalues the solution to a system of the above form is , where  and  are the eigenvalues of  and  and  are the corresponding eigenvectors

  • AI HL only considers cases where  and  are distinct (i.e., ) and non-zero

• A phase portrait will always include two ‘eigenvector lines’ through the origin, each one parallel to one of the eigenvectors

  • So if  and , for example, then these lines through the origin will have equations and , respectively

  • These lines will define two sets of solution trajectories

  • If the eigenvalue corresponding to a line’s eigenvector is positive, then there will be solution trajectories along the line away from the origin in both directions as t increases

  • If the eigenvalue corresponding to a line’s eigenvector is negative, then there will be solution trajectories along the line towards the origin in both directions as t increases

  • No solution trajectory will ever cross an eigenvector line

• If both eigenvalues are positive then all solution trajectories will be directed away from the origin as t increases

  • In between the ‘eigenvector lines’ the trajectories as they move away from the origin will all curve to become approximately parallel to the line whose eigenvector corresponds to the larger eigenvalue

• If both eigenvalues are negative then all solution trajectories will be directed towards the origin as t increases

  • In between the ‘eigenvector lines’ the trajectories will all curve so that at points further away from the origin they are approximately parallel to the line whose eigenvector corresponds to the more negative eigenvalue

    • They will then converge on the other eigenvalue line as they move in towards the origin

• If one eigenvalue is positive and one eigenvalue is negative then solution trajectories will generally start by heading in towards the origin before curving to head out away again from the origin as t increases

  • In between the ‘eigenvector lines’ the solution trajectories will all move in towards the origin along the direction of the eigenvector line that corresponds to the negative eigenvalue, before curving away and converging on the eigenvector line that corresponds to the positive eigenvalue as they head away from the origin

What does the phase portrait look like when  and  are imaginary numbers?

• Here the solution trajectories will all be either circles or ellipses with their centres at the origin

• You can determine the direction (clockwise or anticlockwise) and the shape (circular or elliptical) of the trajectories by considering the values of  and  for points on the coordinate axes

  • For example, consider the system 

    • The eigenvalues of  are  and , so the trajectories will be elliptical or circular

  • When ­ and , and 

    • This shows that from a point on the positive -axis the solution trajectory will be moving ‘to the right and up’ in the direction of the vector 

  • When ­ and , and 

    • This shows that from a point on the positive -axis the solution trajectory will be moving ‘to the left and down’ in the direction of the vector 

  • The directions of the trajectories at those points tell us that the directions of the trajectories will be anticlockwise

  • They also tell us that the trajectories will be ellipses

    • For circular trajectories, the direction of the trajectories when they cross a coordinate axis will be perpendicular to that coordinate axis

What does the phase portrait look like when  and  are complex numbers?

• In this case  and  will be complex conjugates of the form , where and are non-zero real numbers

  • If , , then we have the imaginary eigenvalues case above

• Here the solution trajectories will all be spirals

  • If the real part of the eigenvalues is positive (i.e., if ), then the trajectories will spiral away from the origin

  • If the real part of the eigenvalues is negative (i.e., if ), then the trajectories will spiral towards the origin

• You can determine the direction (clockwise or anticlockwise) of the trajectories  by considering the values of  and  for points on the coordinate axes

  • For example, consider the system 

    • The eigenvalues of  are  and , so the trajectories will be spirals

    • Because the real part of the eigenvalues is positive, the trajectories will spiral away from the origin

  • When ­ and , and 

    • This shows that from a point on the positive -axis the solution trajectory will be moving ‘to the right and down’ in the direction of the vector 

  • When ­ and , and 

    • This shows that from a point on the positive -axis the solution trajectory will be moving ‘to the right and up’ in the direction of the vector 

  • The directions of the trajectories at those points tell us that the directions of the trajectory spirals will be clockwise