MATH 4B Lecture Notes - Lecture 16: Equilibrium Point, Phase Portrait
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MATH 4B Full Course Notes
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Remember: if the eigenvalues have different signs, we get a saddle. Real part of the eigenvalues is positive --> solution trajectories will spiral out. Positive determinant --> counterclockwise rotation for solution trajectories (?) Negative determinant --> clockwise rotation for solution trajectories (?) First order ode is autonomous because the function depends on the dependent variable but not the independent variable. The function is a vector function of the phase variables but not the independent variable. The unknown solution will be a function of the independent variable. Constant vector such that is called an equilibrium point or critical point. If is an equilibrium point, then is a constant equilibrium solution of the system. Example: find the equilibrium points (critical points) of the vector field. For the homogeneous linear ode, is always an equilibrium solution to the system. Its stability is determined by the sign of the eigenvalues.
