4 Second-Order Systems

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1 4 Second-Order Systems Second-order autonomous systems occupy an important place in the study of nonlinear systems because solution trajectories can be represented in the plane. This allows for easy visualization of the qualitative behavior of the system. A second-order autonomous system is represented by two scalar differential equations (, ) (, ) = f x x = f x x x x ( ) ( ) = x = x The locus in the x x x t for all t is a curve that passes through the point x. The x x plane is usually called the state plane or the phase plane. f and f plane of the solution ( ) expresses the tangent vector ( t) to the curve. =, then at x = (,), we draw an arrow pointing from (, ) to (,) + (,) = ( 3, ). Repeating this at every point in a grid covering the plane, we obtain Example 4.: If f ( x) ( x, x) a vector field diagram. Example 4.: Pendulum without friction = x = sin x Δ Δ 7

2 4. Qualitative Behavior of -Order Systems Near Equilibrium Points Consider the linear time-invariant system = Ax where A is a real matrix. The solution of the equation for a given state x is given by ( ) = exp( ) x t M J t M x r where J r is the real Jordan form of A and M is a real nonsingular matrix such that M AM = J r. Depending on the eigenvalues of A, the real Jordan form may take one of three forms, k, and α β β α where k is either or. The first form corresponds to the case when the eigenvalues and are real and distinct, the second form corresponds to the case when the eigenvalues are real and equal, and the third form corresponds to the case of complex eigenvalues = α ± β., j 4.. Real Distinct Eigenvalues In this case and M = v, v, where v and v are the real eigenvectors associated with and. The change of coordinates z = M x transforms the system into two decoupled first-order differential equations, z = z, z = z are different from zero and [ ] whose solution, for a given initial state (, ) t ( ) =, z () t = z e z t z e t z z, is given by Eliminating t between the two equations, we obtain z = cz / / c= z z. where ( ) Three cases: 8

3 Stable node: Both eigenvalues are negative Unstable node: Both eigenvalues are positive Saddle point: Eigenvalues have different sign 9

4 4.. Complex Eigenvalues The change of coordinates z M x = transforms the system into the form z = α z β z, z = β z+ α z The solution of these equations is oscillatory and can be expressed more conveniently in polar coordinates. z r = z + z, θ = tan z where we have two uncoupled first-order differential equation: r = αr and θ = β The solution for a given initial state (, ) r( t) = re αt and ( t) θ = θ + βt r θ is given by When α <, the spiral converges to the origin; when α >, it diverges away from the origin. When α =, the trajectory is a circle of radius r. Three cases Stable focus: α <

5 Unstable focus: α > Circle: α = 4..3 Nonzero Multiple Eigenvalues The change of coordinates z M x = transforms the system into the form z = z+ kz, z = z whose solution, for a given initial state (, ) t ( ) = ( + ), ( ) z t e z kz t z t = e z z z, is given by t Eliminating t, we obtain the trajectory equation

6 z k z z = z + ln z z Two cases: k = ( <, > ) : k = ( <, > ) : 4..4 One or more Eigenvalues are zero When one or both eigenvalues of A are zero, the phase portrait is in some sense degenerate. Here, the matrix A has a nontrivial null space. Any vector in the null space of A is an equilibrium point for the system; that is, the system has an equilibrium subspace, rather than an equilibrium point. The dimension of the null space could be one or two; if it is two, the matrix A will be the zero matrix. When the dimension of the null space is one, the shape of the Jordan form of A will depend on the multiplicity of the zero eigenvalue. When = and, the matrix M is given by M = [ v, v] where v and v are the associated eigenvectors.

7 Two cases: = and (, ) < > : = = : 3

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