The following dynamical systems model was recently proposed by Garcia and Redondo (2022) Dynamical systems approach...

 

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The following dynamical systems model was recently proposed by Garcia and Redondo (2022) "Dynamical systems approach in automobiles technological transition from environmental drivers to represent the technological transition of automobiles from conventional (x) to electric (y): dx dy ax - 8xy, = -2xy + By dt dt The change over time of the number of automobiles of a given type is modelled as the difference between the automobiles of that type entering the market and coming out of the market per year. Phase portraits involving a saddle-node at (xo, yo), xo> 0, yo> 0 were analysed to better understand scenarios where both technologies grow initially but with one type growing much faster than the other (e.g., electric cars dominating over conventional cars) with time. (a) Solve the linear system below using Laplace Transforms dX dt dY -aX+Y, = X-aY dt for some real constant a, and X(0) = 0, Y(0) = 1. (b) Find the range of values of a where (X, Y)=(0,0) is an unstable saddle. (c) If a 0.1, what initial conditions x(0) > xo, y(0)> yo near (xo, yo) will guarantee a faster rate of consumer uptake of electric cars than conventional cars? [Hint: Plot the phase portrait!] The following dynamical systems model was recently proposed by Garcia and Redondo (2022) "Dynamical systems approach in automobiles technological transition from environmental drivers to represent the technological transition of automobiles from conventional (x) to electric (y): dx dy ax - 8xy, = -2xy + By dt dt The change over time of the number of automobiles of a given type is modelled as the difference between the automobiles of that type entering the market and coming out of the market per year. Phase portraits involving a saddle-node at (xo, yo), xo> 0, yo> 0 were analysed to better understand scenarios where both technologies grow initially but with one type growing much faster than the other (e.g., electric cars dominating over conventional cars) with time. (a) Solve the linear system below using Laplace Transforms dX dt dY -aX+Y, = X-aY dt for some real constant a, and X(0) = 0, Y(0) = 1. (b) Find the range of values of a where (X, Y)=(0,0) is an unstable saddle. (c) If a 0.1, what initial conditions x(0) > xo, y(0)> yo near (xo, yo) will guarantee a faster rate of consumer uptake of electric cars than conventional cars? [Hint: Plot the phase portrait!]

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a To solve the linear system using Laplace Transforms we first take the Laplace Transform of both eq... View the full answer