18.03, R05NONLINEAR SYSTEMS1 Systems of first order ODEsConsider the autonomous system of ODE’sdx1dt= f1(x1, . . . , xn)dx2dt= f2(x1, . . . , xn)...dxndt= fn(x1, . . . , xn)1. The critical points of the systems are the solutions (x1, . . . , xn) of the systemf1(x1, . . . , xn) = 0f2(x1, . . . , xn) = 0...fn(x1, . . . , xn) = 02. The Jacobian of the system is given by the n × n matrixJ(x1, . . . , xn) =df1dx1df1dx2· · ·df1dxndf2dx1df2dx2· · ·df2dxn.........dfndx1dfndx2· · ·dfndxn.3. The behavior around a critical point (a1, . . . , an) is given by the behavior of linearizationat that critical point. Namely, by the linear systemu0= J(a1, . . . , an)u.118.03, R05How to study the behavior of a nonlinear system1. Find its critical points.2. Find its Jacobian.3. For each critical point (a1, . . . , an):(a) Compute the Jacobian at that point, i.e. the matrix A = J(a1, . . . , an).(b) Find the eigenvalues and eigenvectors of A.(c) Determine the nature of the critical point and the s tability of the system around it.If your system is 2× 2, determine the nature of the critical point (sink, source, saddle,etc...) and draw the phase portrait around it. (Only if asked!)4. Again for a 2 × 2 system, one can determine the long-term behavior of the solutions bytrapping them in a b ox containing all the critical
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