Solved: BifurcationsThe term bifurcation generally refers to something splitting apart

Chapter 6, Problem 37

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BifurcationsThe term bifurcation generally refers to something splitting apart. With regard to differential equations or systems involving a parameter, it refers to abrupt changes in the character of the solutions as the parameter is changed continuously. 33 through 36 illustrate sensitive cases in which small perturbations in the coefcients of a linear or almost linear system can change the type or stability (or both) of a critical point.In the case of a two-dimensional system that is not almost linear, the trajectories near an isolated critical point can exhibit a considerably more complicated structure than those near the nodes, centers, saddle points, and spiral points discussed in this section. For example, consider the systemdx dt D x.x3 !2y3/; dy dt D y.2x3 !y3/(16)having .0;0/ as an isolated critical point. This system is not almost linear because .0;0/ is not an isolated critical point of the trivial associated linear system x0D0, y0D0. Solve the homogeneous rst-order equationdy dx Dy.2x3 !y3/ x.x3 !2y3/to show that the trajectories of the system in (16) are folia of Descartes of the form x3 Cy3 D 3cxy; where c is an arbitrary constant (Fig. 6.2.14).