Explanation Given: The plane autonomous system is x ′ = − 3 x + y 2 + 2 and y ′ = x 2 − y 2 . Concept used: Stable node: The condition for a critical point to be a stable node is that τ 2 − 4 Δ > 0 , τ < 0 and Δ > 0 where, τ is the trace of the matrix and Δ is the determinant of the matrix. Saddle point: The condition for a critical point to be a saddle point is that τ 2 − 4 Δ > 0 and Δ < 0 where, τ is the trace of the matrix and Δ is the determinant of the matrix. Unstable spiral point: The condition for a critical point to be an unstable spiral point is that τ ≠ 0 , τ 2 − 4 Δ < 0 and α > 0 where, τ is the trace of the matrix, α is the real part of the eigenvalue and Δ is the determinant of the matrix. Calculation: The plane autonomous system is given as follows: x ′ = − 3 x + y 2 + 2 y ′ = x 2 − y 2 Consider that x ′ = P ( x , y ) and y ′ = Q ( x , y ) . The above equations are written as follows: P ( x , y ) = − 3 x + y 2 + 2 Q ( x , y ) = x 2 − y 2 The critical point ( x , y ) is the point for which x ′ = 0 and y ′ = 0 . Solve the equation y ′ = 0 as follows: y ′ = Q ( x , y ) x 2 − y 2 = 0 x 2 = y 2 x = ± y Solve the equation x ′ = 0 as follows: x ′ = P ( x , y ) − 3 x + y 2 + 2 = 0 Substitute x for y in the above equation. x 2 − 3 x + 2 = 0 ( x − 1 ) ( x − 2 ) = 0 The value of x obtained from above equations are x = 1 and x = 2 . Therefore, the critical points for the given plane autonomous system is ( 1 , 1 ) , ( 1 , − 1 ) , ( 2 , 2 ) and ( 2 , − 2 ) . Differentiate P ( x , y ) partially with respect to x as follows: ∂ P ( x , y ) ∂ x = ∂ ( − 3 x + y 2 + 2 ) ∂ x = − 3 Differentiate P ( x , y ) partially with respect to y as follows: ∂ P ( x , y ) ∂ y = ∂ ( − 3 x + y 2 + 2 ) ∂ y = 2 y Differentiate Q ( x , y ) partially with respect to x as follows: ∂ Q ( x , y ) ∂ x = ∂ ( x 2 − y 2 ) ∂ x = 2 x Differentiate Q ( x , y ) partially with respect to y as follows: ∂ Q ( x , y ) ∂ x = ∂ ( x 2 − y 2 ) ∂ x = − 2 y The Jacobian matrix formed for the given system is as follows: g ′ ( x , y ) = ( ∂ P ∂ x ∂ P ∂ y ∂ Q ∂ x ∂ Q ∂ y ) = ( − 3 2 y 2 x − 2 y ) For the critical point ( 2 , − 2 ) , the Jacobian matrix is as follows: g ′ ( 2 , − 2 ) = ( − 3 2 ⋅ ( − 2 ) 2 ⋅ ( 2 ) − 2 ⋅ ( − 2 ) ) = ( − 3 − 4 4 4 ) Consider the trace of the above matrix as τ . Trace of a matrix is the sum of all the elements of the main diagonal. Trace of the above matrix is calculated as follows: τ = − 3 + 4 = 1 The determinant of the matrix is Jacobian matrix is calculated as follows: Δ = | − 3 − 4 4 4 | = − 12 + 16 = 4 The value of τ 2 − 4 Δ is calculated as follows: τ 2 − 4 Δ = ( 1 ) 2 − ( 4 ⋅ 4 ) = 1 − 16 = − 15 The eigenvalues are calculated as follows: λ = τ ± τ 2 − 4 Δ 2 = 1 ± − 15 2 = 1 2 ± i 15 2 The real part ( α ) of the eigenvalue obtain as above is positive

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

See similar textbooks

Concept explainers

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

Videos

Question

Chapter 10.3, Problem 15E

To determine

The category of the critical point for the plane autonomous system x′=−3x+y2+2 and y′=x2−y2.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.3, Problem 14E

Chapter 10.3, Problem 16E

Students have asked these similar questions

Consider an MA(6) model with θ1 = 0.5, θ2 = −25, θ3 = 0.125, θ4 = −0.0625, θ5 = 0.03125, and θ6 = −0.015625. Find a much simpler model that has nearly the same ψ-weights.

Let {Yt} be an AR(2) process of the special form Yt = φ2Yt − 2 + et. Use first principles to find the range of values of φ2 for which the process is stationary.

Describe the important characteristics of the autocorrelation function for the following models: (a) MA(1), (b) MA(2), (c) AR(1), (d) AR(2), and (e) ARMA(1,1).

Chapter 10 Solutions

Differential Equations with Boundary-Value Problems

Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - Prob. 2E Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - Prob. 4E Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - Prob. 15E Ch. 10.1 - In Problems 716 find all critical points of the...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - Prob. 10E Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - Prob. 16E Ch. 10.2 - Prob. 17E Ch. 10.2 - Determine a condition on the real constant so...Ch. 10.2 - Prob. 19E Ch. 10.2 - Prob. 20E Ch. 10.2 - Prob. 21E Ch. 10.2 - Prob. 22E Ch. 10.2 - Prob. 23E Ch. 10.2 - In Problems 23-26 a nonhomogeneous linear system...Ch. 10.2 - Prob. 25E Ch. 10.2 - Prob. 26E Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - Prob. 3E Ch. 10.3 - Prob. 4E Ch. 10.3 - Prob. 5E Ch. 10.3 - In Problems 310, without solving explicitly,...Ch. 10.3 - In Problems 310, without solving explicitly,...Ch. 10.3 - Prob. 8E Ch. 10.3 - Prob. 9E Ch. 10.3 - In Problems 310, without solving explicitly,...Ch. 10.3 - Prob. 11E Ch. 10.3 - In Problems 1120 classify (if possible) each...Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - Prob. 15E Ch. 10.3 - Prob. 16E Ch. 10.3 - Prob. 17E Ch. 10.3 - Prob. 18E Ch. 10.3 - Prob. 19E Ch. 10.3 - Prob. 20E Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Prob. 23E Ch. 10.3 - Prob. 24E Ch. 10.3 - Prob. 25E Ch. 10.3 - Prob. 26E Ch. 10.3 - Prob. 27E Ch. 10.3 - Show that the dynamical system x = x + xy y = 1 y...Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Prob. 31E Ch. 10.3 - Prob. 32E Ch. 10.3 - Prob. 33E Ch. 10.3 - Prob. 34E Ch. 10.3 - Prob. 35E Ch. 10.3 - Prob. 36E Ch. 10.3 - When a nonlinear capacitor is present in an...Ch. 10.3 - Prob. 38E Ch. 10.3 - Prob. 39E Ch. 10.4 - Prob. 1E Ch. 10.4 - Prob. 2E Ch. 10.4 - Prob. 3E Ch. 10.4 - Prob. 4E Ch. 10.4 - Prob. 5E Ch. 10.4 - Prob. 6E Ch. 10.4 - Prob. 7E Ch. 10.4 - Prob. 8E Ch. 10.4 - Prob. 9E Ch. 10.4 - Competition Models A competitive interaction is...Ch. 10.4 - Prob. 12E Ch. 10.4 - Prob. 13E Ch. 10.4 - Prob. 14E Ch. 10.4 - Prob. 15E Ch. 10.4 - Additional Mathematical Models Damped Pendulum If...Ch. 10.4 - Prob. 17E Ch. 10.4 - Prob. 18E Ch. 10.4 - Prob. 19E Ch. 10.4 - Prob. 20E Ch. 10.4 - Prob. 21E Ch. 10.4 - Prob. 22E Ch. 10 - Prob. 1RE Ch. 10 - Prob. 2RE Ch. 10 - Prob. 3RE Ch. 10 - Prob. 4RE Ch. 10 - Prob. 5RE Ch. 10 - Prob. 6RE Ch. 10 - Prob. 7RE Ch. 10 - Prob. 8RE Ch. 10 - Prob. 9RE Ch. 10 - Prob. 10RE Ch. 10 - Prob. 11RE Ch. 10 - Discuss the geometric nature of the solutions to...Ch. 10 - Classify the critical point (0, 0) of the given...Ch. 10 - Prob. 14RE Ch. 10 - Prob. 15RE Ch. 10 - Prob. 16RE Ch. 10 - Prob. 17RE Ch. 10 - Prob. 18RE Ch. 10 - Prob. 19RE Ch. 10 - Prob. 20RE

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The category of the critical point for the plane autonomous system x ′ = − 3 x + y 2 + 2 and y ′ = x 2 − y 2 .

Solution Summary: The author describes the critical points for the plane autonomous system: stable node, saddle point, and unstable spiral point.

Concept explainers

Videos

The category of the critical point for the plane autonomous system x′=−3x+y2+2 and y′=x2−y2.

Want to see the full answer?

Chapter 10 Solutions