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Notes-PhasePlane - The Phase Plane Phase portraits type and...

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The Phase Plane Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations Phase Portraits of Linear Systems Consider a systems of linear differential equations x ′ = Ax . Its phase portrait is a representative set of its solutions, plotted as parametric curves (with t as the parameter) on the Cartesian plane tracing the path of each particular solution ( x , y ) = ( x 1 ( t ), x 2 ( t )), -∞ < t < . Similar to a direction field, a phase portrait is a graphical tool to visualize how the solutions of a given system of differential equations would behave in the long run. In this context, the Cartesian plane where the phase portrait resides is called the phase plane . The parametric curves traced by the solutions are sometimes also called their trajectories . Remark : It is quite labor-intensive, but it is possible to sketch the phase portrait by hand without first having to solve the system of equations that it represents. Just like a direction field, a phase portrait can be a tool to predict the behaviors of a system’s solutions. To do so, we draw a grid on the phase plane. Then, at each grid point x = ( α , β ), we can calculate the solution trajectory’s instantaneous direction of motion at that point by using the given system of equations to compute the tangent / velocity vector, x ′. Namely plug in x = ( α , β ) to compute x ′ = Ax . In the first section we will examine the phase portrait of linear system of differential equations. We will classify the type and stability the equilibrium solution of a given linear system by the shape and behavior of its phase portrait.
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Equilibrium Solution (a.k.a. Critical Point, or Stationary Point) An equilibrium solution of the system x ′ = Ax is a point ( x 1 , x 2 ) where x ′ = 0 , that is, where x 1 ′ = 0 = x 2 ′. An equilibrium solution is a constant solution of the system, and is usually called a critical point . For a linear system x ′ = Ax , an equilibrium solution occurs at each solution of the system (of homogeneous algebraic equations) Ax = 0 . As we have seen, such a system has exactly one solution, located at the origin, if det( A ) ≠ 0. If det( A ) = 0, then there are infinitely many solutions. For our purpose, and unless otherwise noted, we will only consider systems of linear differential equations whose coefficient matrix A has nonzero determinant. That is, we will only consider systems where the origin is the only critical point. Note : A matrix could only have zero as one of its eigenvalues if and only if its determinant is also zero. Therefore, since we limit ourselves to consider only those systems where det( A ) ≠ 0, we will not encounter in this section any matrix with zero as an eigenvalue.
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Classification of Critical Points Similar to the earlier discussion on the equilibrium solutions of a single first order differential equation using the direction field, we will presently classify the critical points of various systems of first order linear differential equations by their stability . In addition, due to the truly two-dimensional
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