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Example text
21. Phase portraits of a violin string starting from different initial conditions force exceeds the frictional force. Hence the acceleration becomes negative and the velocity starts to decrease. 3. Slipping. The sudden drop of friction caused by the onset of slipping yields the tug of war to the spring, causing the block to slip faster-the acceleration becomes negative, the velocity becomes negative, and the block starts moving to the left. 4. Grabbing. When the leftward motion of the block has decreased the spring force to a value at which it is less than the slipping friction, the motion to the left slows, and the velocity starts going positive.
Linear vs. 12), it follows that there exists S > 0 such that 21x I r(x, to)I - xol V Ix - xol < S. 11) along with the bound on I A (to)x I yields I f (x, to) I IA(to)(x - xo)I - r(x, to) I >c/2Ix-xol V IX - xol0 V x#Xo EB(xo,S). This completes the proof. Remarks: The conditions of the preceding proposition are sufficient, but not necessary, for the existence of isolated equilibria: For example, the system x=x3, X ER, has linearization 0 at the equilibrium 0, which is nonetheless isolated. However, the conditions of the proposition are tight in the sense that there are examples of nonlinear systems which do not have isolated equilibria when they have a singular linearization.
As we discussed in Chapter 1, this means that f, (xo) = f2(x0) = 0 and there exists a 8 > 0 such that there are no other equilibrium points in the ball B(x0, 8) centered at x0 of radius B. Further, let A(xo) _ (x0) E 8x1 82x2 ax2 be the Jacobian matrix off at x0 (linearization of the vector field f (x) about xo). 7) if none of the eigenvalues of A(xo) are on the jw axis. Such equilibrium points are referred to as hyperbolic equilibria. 1 Hartman-Grobman Theorem. 7). In particular, h(xo) = 0. Further, the homeomorphism can be chosen to preserve the parameterization by time.
