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joshmccraney said:The ODE system looks something like this ##y' = A y##. Let's pick ##y_2 = 0, y_1 = 1##, which implies ##y' = \langle-3,2\rangle##. This implies at the point ##(1,0)## in the phase plane there will be a vector pointing in the direction ##\langle -3,2\rangle##, and hence moving in the direction along the spiral away from the center.
Plot several points and you'll see the behavior, such as the point ##(y_1 = a > 0, y_2 = 0)##. Alternatively, look at the sign of the real component of the eigenvalue to determine whether or not solutions converge to the origin or not.e101101 said:Im not quite sure if that means all solns would be approaching the center? Clockwise or Counterclockwise
A phase plane diagram with complex eigenvalues is a graphical representation of the behavior of a system with two variables, where the eigenvalues of the system's matrix have imaginary components. It is used to analyze the stability and behavior of dynamic systems.
A phase plane diagram with complex eigenvalues differs from one with real eigenvalues in that it includes spiral trajectories instead of just straight lines. This is because complex eigenvalues indicate oscillatory behavior in the system, while real eigenvalues indicate exponential growth or decay.
A phase plane diagram with complex eigenvalues can provide information about the stability of a system, the types of equilibrium points, and the behavior of the system over time. It can also help in predicting the response of the system to different initial conditions.
In a phase plane diagram with complex eigenvalues, the eigenvalues determine the shape and orientation of the trajectories. The real part of the eigenvalues determines the direction of the trajectories, while the imaginary part determines the curvature.
Phase plane diagrams with complex eigenvalues are commonly used in fields such as control systems, electrical engineering, and physics to analyze the behavior of dynamic systems. They can also be used in biology and ecology to model population dynamics and in economics to study economic systems.