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A two-dimensional nonlinear dynamical system is governed by the following set of equations: x = f(x, y) y = g(x, y) where x = x(t), y = y(t) are time-dependent functions. (3) (a) Suppose the dynamical system in Eq. (3) is defined according to the functions f(x, y) = y and g(x, y) = 5x34x - x5. dV (x) + = 0, and hence dx (i) Rewrite this 2D dynamical system as a single second order ODE, identify the potential V(x). Sketch the function V(x) by hand. (ii) Find all fixed points of this system, and using 2D linear stability analysis, classify their nature and stability in the linearised case. (iii) Show that the dynamical system in Eq. (3) is a conservative dynamical system and find a conserved quantity. (iv) Using your answers to parts (ii) and (iii), deduce the behaviour of the fixed points in the nonlinear case. Hence sketch the full phase portrait of the dynamical system. Be sure to draw all fixed points, and any homoclinic/heteroclinic orbits.

A two-dimensional nonlinear dynamical system is governed by the following set of equations: x = f(x, y) y = g(x, y) where x = x(t), y = y(t) are time-dependent functions. (3) (a) Suppose the dynamical system in Eq. (3) is defined according to the functions f(x, y) = y and g(x, y) = 5x34x - x5. dV (x) + = 0, and hence dx (i) Rewrite this 2D dynamical system as a single second order ODE, identify the potential V(x). Sketch the function V(x) by hand. (ii) Find all fixed points of this system, and using 2D linear stability analysis, classify their nature and stability in the linearised case. (iii) Show that the dynamical system in Eq. (3) is a conservative dynamical system and find a conserved quantity. (iv) Using your answers to parts (ii) and (iii), deduce the behaviour of the fixed points in the nonlinear case. Hence sketch the full phase portrait of the dynamical system. Be sure to draw all fixed points, and any homoclinic/heteroclinic orbits.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.Very very grateful!Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer. and and Very very grateful!
A two-dimensional nonlinear dynamical system is governed by the following set of equations: x = f(x, y) y = g(x, y) where x = x(t), y = y(t) are time-dependent functions. (3) (a) Suppose the dynamical system in Eq. (3) is defined according to the functions f(x, y) = y and g(x, y) = 5x34x - x5. dV (x) + = 0, and hence dx (i) Rewrite this 2D dynamical system as a single second order ODE, identify the potential V(x). Sketch the function V(x) by hand. (ii) Find all fixed points of this system, and using 2D linear stability analysis, classify their nature and stability in the linearised case. (iii) Show that the dynamical system in Eq. (3) is a conservative dynamical system and find a conserved quantity. (iv) Using your answers to parts (ii) and (iii), deduce the behaviour of the fixed points in the nonlinear case. Hence sketch the full phase portrait of the dynamical system. Be sure to draw all fixed points, and any homoclinic/heteroclinic orbits.
Transcribed Image Text:A two-dimensional nonlinear dynamical system is governed by the following set of equations: x = f(x, y) y = g(x, y) where x = x(t), y = y(t) are time-dependent functions. (3) (a) Suppose the dynamical system in Eq. (3) is defined according to the functions f(x, y) = y and g(x, y) = 5x34x - x5. dV (x) + = 0, and hence dx (i) Rewrite this 2D dynamical system as a single second order ODE, identify the potential V(x). Sketch the function V(x) by hand. (ii) Find all fixed points of this system, and using 2D linear stability analysis, classify their nature and stability in the linearised case. (iii) Show that the dynamical system in Eq. (3) is a conservative dynamical system and find a conserved quantity. (iv) Using your answers to parts (ii) and (iii), deduce the behaviour of the fixed points in the nonlinear case. Hence sketch the full phase portrait of the dynamical system. Be sure to draw all fixed points, and any homoclinic/heteroclinic orbits.
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