Mathematical Models, Analysis and Simulation, Part I, Fall 2005 Homework 5, Phase Portraits. (Score: 7.0) Three problems are to be solved in this homework assignment. The first consists in scaling of a homogeneous and a nonhonogeneous differential equation. The second is a study of a dynamical system with a simple bifurcation, and the third problem deals with predator-prey models. Hints for making phaseplanes with Matlab are annexed. Problem 1 (1.0). The differential equation for damped free oscillations is a second order homogeneous linear equation. It can be written in the following form: m d 2 u dt 2 + c du dt + ku =0 . Here the state variable u is the deviation from a stationary position and t is the time. The equation contains three parameters, i.e. the mass m , the damping constant c , and the spring constant k . We first introduce a dimensionless time τ by scaling of the original time variable t . After this is done, we derive a differential equation for u ( τ ) from the equation above for u ( t ) . One of the results is that the equation for nondimensionalized variables will often contain fewer parameters than the original equation. This fact will simplify the formal mathematical treatment of the problem, and it will also lead to improved insight about both the problem and its solution. We reparametrize by introducing two parameters as follows: The undamped (angular) fre-quency ϖ0 is defined by ϖ0 = p k / m and the critical damping c0 is defined by c0 = 2 √ km . Note that these new parameters are not dimensionless. We use them to define the dimensionless damping constant α = c / c0 and the dimensionless time τ = ϖ0 t . Your first task is to derive the differential equation for u ( τ ) and to show that it equals u 00 + 2 α u0 + u =0 . Observe that the scaling of the time has reduced the number of parameters from three to one. With forced oscillations we study the differential equation m d 2 u dt 2 + c du dt + ku = F cos ϖ t . Here there are two additional parameters, the amplitude F and the (angular) frequency ϖ of the external force. It is easy to nondimensionalize the driving frequency ϖ since we already have introduced the reference frequency ϖ0 . Thus we introduce a dimensionless driving frequency β . Furthermore 1
we introduce a dimensionless state variable y by proper choice of scale factor (this should contain F , of course). Your second task is to derive a differential equation for y ( t ) In this case the scaling of both the dependent and the independent variables has reduced the number of parameters from five to two: α and β . Problem 2 (3.0). Consider the following linear system of differential equations: u0 =-± 1 1 P 1 ² u , where P is a real–valued parameter. (0.75) a)
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