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a) Consider a system of two first-order ordinary differential equations: y, = 2e-2+ ay2, inearise the system of ODE close to the (y1, y2) = (0,0) equilibria. For a = 1 the phase portrait of the linearised system is: y2 = -2 tanh(yı +y2 + y). Ostable node OUnstable node OSaddle OUnstable focus with spiral out OCentre Ostable focus with spiral in

a) Consider a system of two first-order ordinary differential equations: y, = 2e-2+ ay2, inearise the system of ODE close to the (y1, y2) = (0,0) equilibria. For a = 1 the phase portrait of the linearised system is: y2 = -2 tanh(yı +y2 + y). Ostable node OUnstable node OSaddle OUnstable focus with spiral out OCentre Ostable focus with spiral in

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.5: Nonlinear Systems Of Differential Equations
Problem 13E
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a) Consider a system of two first-order ordinary differential equations: y, = 2e -2+ay, linearise the system of ODE close to the (y1, y2) = (0,0) equilibria. For a = I the phase portrait of the linearised system is: y2 = -2 tanh(y + y2 + y). OStable node OUnstable node OSaddle Ounstable focus with spiral out OCentre Ostable focus with spiral in b) For which real value of a the system of ODES = a sin(y + y2) + tanh(y), y2 = - tanh(y, +y)-ay when linearised, displays a centre around the equilibria (y1, y2) = (0,0)? OAny value of a %3D Oa > 0 Oa < 0, a > -1 Oa < -1
Transcribed Image Text:a) Consider a system of two first-order ordinary differential equations: y, = 2e -2+ay, linearise the system of ODE close to the (y1, y2) = (0,0) equilibria. For a = I the phase portrait of the linearised system is: y2 = -2 tanh(y + y2 + y). OStable node OUnstable node OSaddle Ounstable focus with spiral out OCentre Ostable focus with spiral in b) For which real value of a the system of ODES = a sin(y + y2) + tanh(y), y2 = - tanh(y, +y)-ay when linearised, displays a centre around the equilibria (y1, y2) = (0,0)? OAny value of a %3D Oa > 0 Oa < 0, a > -1 Oa < -1
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