# Autonomous System

Miran Anwar, mth 235 ss21 1: Hw19-5.4-SDE-2x2NS-PP-NP. Due: 04/26/2021 at 11:00pm EDT. See in LN, § 5.4, See Subsection 5.4.3. 1. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous systemx′ =− 1 2x + xyy′ = 3 2y− 1 2xy(a) The critical points of the system above have the formx0 = [0 0] , x1 =[ x1 y1] .Find these components.x1 =y1 =Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1See in LN, § 5.4, See Subsection 5.4.3. 2. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous systemx′ =− 1 4x + 1 2xyy′ = y− 1 2xy(a) The critical points of the system above have the formx0 = [0 0] , x1 =[ x1 y1] .Find these components.x1 =y1 =Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X01Part 4: The Jacobian Matrix at X1See in LN, § 5.4, See Subsection 5.4.4. 3. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous systemx′ =− 1 4x + 1 2xyy′ = y− 1 2y2 − 1 2xy(a) The critical points of the system above have the formx0 = [0 0] , x1 =[ 0 y1] , x2 =[ x2 y2] .Find these components.y1 =x2 =y2 = Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2See in LN, § 5.4, See Subsection 5.4.4. 4. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous systex′ =−x + xyy′ = 9 8y− y2 − 1 2xy(a) The critical points of the system above have the formx0 = [0 0] , x1 =[ 0 y1] , x2 =[ x2 y2] .Find these components.y1 =x2 =y2 = 2Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2See in LN, § 5.4, See Example 5.4.5. 5. (10 points)

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