Question

1. The Duffing equation is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by -ax+3x3 = cos(wt) at medt dr. where function r = r(t) is the displacement at timet, is the velocity, and is the acceleration. The parameter 8 controls the amount of damping, a controls the linear stiffness, B controls the amount of non-linearity in the restoring force, and 7 and w are the amplitude and angular frequency of the periodic driving force respectively. Consider a system without the driving force, that is, y = 0, and a > 0, B >0, =5. (a) Show that the above equation can be written as a non-linear system of first order differential equations on=-Br+ ar – 5y.

Answer 1

ca). So, yo, Do, 2 > 5 The Jet, do + 5 ax +3*20. Need let y ou 30 we get, and do + 57 -xx +®73 -0. and +57 - +p2 = -23 +02-57. 6 For F Coaticae point de to and does . > y=0 and -82² + ax-5y =o. => @-A2?)x=0 L3=1 k=0 on x=1 Hence the Cabicae points are Co,), (e) u o dx - y = film, ny] de = -323+03 - 5y = $2(x,7). Nows Now Df (2, 4) = DS (2,4) – ( on Southern o so about Caption -5) 8560,0) ) Hence di negozi Zed system ad (0,n) po:ort is $(5) - Of(0,0) (3) - (85) (3)

Again, os c os = 10 s) 30 the linearized system at CV,0) is al) of (TO) () -> den . y. de --sy. similarey the linescized system at de o ry, the = -2-57 © now Of(0,0) = (-s) The seoot of bf 10,0) is a ca+5) - CO. => A^ 6 - 3 = 0. z 2 - 5V25740 2 . since the two roots has reas poort, so by Hartman Czelman theorem the emere Zed System (*) and it's no own nom. limeve system have same behaviowe bimi love fore otheve trvo linemusized System.

when Xa, Pol We Det. 0$10,0) = (25) So the eigen valry are, -st (25+8 2 = -55 33 a the roots are of opposite sign, so the aince point excitiere (0,o) is saddre. (Phase - Pore bait) let, DF (0,0) = A so Ax= -55x33 x,. = (A - - 57159)x=0. from here find xo . Similarly let X₂ is an eigen vector at -5--233.C fond x1, x2 yowisselt) ... The general solletion is (O) = 0,06 5+X3), 8 (-5- kim: eway Solve for other two critical points