Duffing’s equationd2xdt2+kx+x3=Bcos(t)describes the chaotic dynamics of a circuit with a nonlinear inductor or an elastic pen-dulum with a spring stiffness that is more complex than Hooke’s law. In the contextof a spring, suppose the spring is offset from the originx(0) = 1 and is released freelyx′(0) = 0. a) Convert the second order differential equation to a system of first orderequations. Also convert the initial conditions. Write the resulting system of equa-tions in vector notation~y′(t) =~f(t, ~y) and~y(0) =~y0. Clearly define~y(t),~f(t, ~y),and~y0. b) we approximated the solutions to the system in part (a), I get twofunctions. Which function is the solution to Duffing’s equation, and why? Whatdoes the other function represent?
Duffing’s equationd2xdt2+kx+x3=Bcos(t)describes the chaotic dynamics of a circuit with a nonlinear inductor or an elastic pen-dulum with a spring stiffness that is more complex than Hooke’s law. In the contextof a spring, suppose the spring is offset from the originx(0) = 1 and is released freelyx′(0) = 0.
a) Convert the second order differential equation to a system of first orderequations. Also convert the initial conditions. Write the resulting system of equa-tions in vector notation~y′(t) =~f(t, ~y) and~y(0) =~y0. Clearly define~y(t),~f(t, ~y),and~y0.
b) we approximated the solutions to the system in part (a), I get twofunctions. Which function is the solution to Duffing’s equation, and why? Whatdoes the other function represent?
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