MATLAB ONLY**

3. 13 points] Unlike the previous problem where a simple nonlinear mapping gave rise to distortions whose properties were directly tied to the nature of the nonlinearity (e.g., a cubic created 3rd harmonic distortion only), generation of distortion products in a differential system is more nuanced. Consider the driven Duffing oscillator: mikr BAsin (ut) where a, ?, ?, and w are constants. a. First, solve the linear case to demonstrate a lack of distortion. Solve the differential equation numerically using a fourth order Runge-Kutta (RK4) method on the interval E [0, 1000]. Note that you'll need to make some choice for the values of the initial conditions. For the parameters use m- 1, k 1, B-0,b-0.3, A 1.5, andw3.0/(2). Make a plot of the associated time waveform (i.e., r versus t) and phase space (i.e., z versus ). Also plot the amplitude of the spectra after the system has come to a steady-state (i.e., initial transients have settled down). b. Second, the nonlinear case to demonstrate even and odd distortions for a harmonic input. Solve as 1.5/(2m). Similarly, produce before, but use m 1, k 1, B-10, b 0.3. A 1.5, and w the same three plots as before. Describe any distortions you see. c. Third, repeat the last but with b-0.1. How has the spectrum changed? d. Lastly, show the Duffing system can behave chaotically. Use m = 1, k =-1, ?-1, b = 0.3, A 0.5, and-1.2/(2m).Hint: You should get a phase plane portrait similar to that shown in Fig.1.] What does the spectrum look like now? Figure 1: Chaotic phase plane portrait for the Duffing oscillator. You should submit your code, as well as an images created