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By Constantine H. Houpis, Gary B. Lamont

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Find all the equilibrium points. Show that 1. If x0 < 0, then xk -+ -oo as k -> oo. 2. If x0 > 1, then xk -+ -oo ask -+ oo. 3. For some interval I C ]0, 1 [,if xo E I, then xk -> xti :_ tih lI . Determine the open interval of attraction of Xh. For a game, program the above recursion on your calculator or computer for h = 4 and note that initial conditions starting in ]0, 1[ stay in ]0, 1[. Watch the sequence xk wander all over the interval except for some special initial conditions x0 corresponding to periodic solutions.

In the following model for the spread of disease in a population, xj is the number of infected people, x2 the number not infected, a the death rate, and b the infection rate: zj = -axi + bx1x2, X2 = -bx1X2. The point (0, x2) is a continuum of equilibria for this system and the second column of the corresponding linearization is zero. 3 Some Classical Examples There is a large number of classical and simple examples of nonlinear dynamical systems, that have been studied in detail. 1 The Tunnel Diode Circuit A simple circuit example is useful for illustrating an example of a circuit with multiple equilibria.

In particular, h(xo) = 0. Further, the homeomorphism can be chosen to preserve the parameterization by time. The proof of this theorem is beyond the scope of this chapter, but some remarks are in order. Remarks: 1. Recall from elementary analysis that a homeomorphism is a continuous one to one and onto map with a continuous inverse. 7). What is most surprising about the theorem is that there exists a single map that works for every one of a continuum of trajectories in a neighborhood of x0. In general, it is extremely difficult to compute the homeomorphism h of the theorem.

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