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Lecture8 - 1 Ch8Lecs 8.1 Bifurcations of Equilibria...

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1 Ch8Lecs 8.1 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies. In general, one could study the bifurcation theory of ODEs, PDEs, integro-differential equations, discrete mappings etc. Of course, we are concerned with ODEs. Local bifurcations refer to qualitative changes occurring in a neighborhood of an equilibrium point of a differential equation or a fixed point of an associated Poincaré map. These can be studied by expanding the equations of motion in power series about the point. Consider an ODE depending on a parameter : , , . Assume this system has an equilibrium point that is a sink for where is a critical value of the parameter. Graph the evls as a function of . Real and imaginary axes, region of evls with negative real parts for , arrows showing that the real parts of some evls are increasing with A real evl or a cc pair may traverse the imaginary axis as increases through . The sink becomes a source or a saddle. Or the equilibrium point may simply disappear when it has a zero evl! Example . Fishery model with constant harvesting. Recall the logistic model of population growth with an additional constant term. Interpret as a model of a fishery with the number of fish, time, the fish growth rate, the carrying capacity, and a constant rate of harvesting. The model may be simplified by measuring the quantity of fish and the harvest rate in units of the carrying capacity: and . Further simplify by introducing a dimensionless time variable: . Then obtain . [1] Equilibria correspond to . Analyze graphically
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2 Ch8Lecs sketch -, -, , The direction of the arrows on the -axis now depends on the sign of . When there are two equilibrium points, when there is one, and when there are none. Alternately, we can study the equilibria algebraically. Let . Then equilibrium points of [1] correspond to or . The equilibria are , These exist only for . add , arrows to graph We can write [1] in the form , [2] where the dot now refers to the derivative with respect to dimensionless time. The eigenvalues of the equilibrium points are , which depends on through . Since , its eigenvalue is negative and is a stable equilibrium point. Since , its eigenvalue is positive and is an unstable equilibrium point. As increases toward the equilibria converge and their eigenvalues approach 0. At the equilibria merge as a single non-hyperbolic equilibrium point and for they disappear. Draw a bifurcation diagram for the logistic model with constant harvesting by plotting the equilibria as a function of . sketch -, -, the two branches of the saddle-node A saddle-node bifurcation occurs at a critical value of the parameter, . Simplify the logistic model with constant harvesting by centering the bifurcation at the origin.
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