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Consider the non-linear system z' (t) = 2² – y², y' (t) = ax² + 2y – (a + 2) where a is a parameter. (a) Let a = 1. Find all equilibrium points of the system and determine their types (saddle, spiral sink/source, nodal sink/source) if possible. (b) For every a, (1, 1) is an equilibrium point of the system. Can (1, 1) be a saddle point for a positive a? If yes, find the range of a such that this is a saddle point; if no, explain the reason. 2 -2 -6 -4 -2 0 2 The picture shows the computer-generated phase portrait of the non-linear system for a = 1; you may use this picture to check your answer in (a).

Consider the non-linear system z' (t) = 2² – y², y' (t) = ax² + 2y – (a + 2) where a is a parameter. (a) Let a = 1. Find all equilibrium points of the system and determine their types (saddle, spiral sink/source, nodal sink/source) if possible. (b) For every a, (1, 1) is an equilibrium point of the system. Can (1, 1) be a saddle point for a positive a? If yes, find the range of a such that this is a saddle point; if no, explain the reason. 2 -2 -6 -4 -2 0 2 The picture shows the computer-generated phase portrait of the non-linear system for a = 1; you may use this picture to check your answer in (a).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the non-linear system z' (t) = 2² – y², y' (t) = ax² + 2y – (a + 2) where a is a parameter. (a) Let a = 1. Find all equilibrium points of the system and determine their types (saddle, spiral sink/source, nodal sink/source) if possible. (b) For every a, (1, 1) is an equilibrium point of the system. Can (1, 1) be a saddle point for a positive a? If yes, find the range of a such that this is a saddle point; if no, explain the reason. 2 -2 -6 -4 -2 0 2 The picture shows the computer-generated phase portrait of the non-linear system for a = 1; you may use this picture to check your answer in (a).
Transcribed Image Text:Consider the non-linear system z' (t) = 2² – y², y' (t) = ax² + 2y – (a + 2) where a is a parameter. (a) Let a = 1. Find all equilibrium points of the system and determine their types (saddle, spiral sink/source, nodal sink/source) if possible. (b) For every a, (1, 1) is an equilibrium point of the system. Can (1, 1) be a saddle point for a positive a? If yes, find the range of a such that this is a saddle point; if no, explain the reason. 2 -2 -6 -4 -2 0 2 The picture shows the computer-generated phase portrait of the non-linear system for a = 1; you may use this picture to check your answer in (a).
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