Math Problem Statement
Classify the critical points of the plane autonomous system corresponding to the second-order nonlinear differential equation: \( \ddot{x} + 2(1 - x^3)\dot{x} - 3x = 0 \).
Solution
To classify the critical points of the given second-order nonlinear differential equation, let's break down the problem.
Step 1: Convert the Equation into a System of First-Order Equations
The equation given is:
Define:
Then, we have: and
This gives us the first-order system:
\dot{x}_1 = x_2 \\ \dot{x}_2 = -2(1 - x_1^3)x_2 + 3x_1 \end{cases}$$ ### Step 2: Find the Critical Points To find the critical points, we set $$\dot{x}_1 = 0$$ and $$\dot{x}_2 = 0$$: $$x_2 = 0$$ and $$-2(1 - x_1^3)x_2 + 3x_1 = 0$$ Substitute $$x_2 = 0$$ into the second equation: $$3x_1 = 0$$ This implies $$x_1 = 0$$. Therefore, the only critical point is: $$(x_1, x_2) = (0, 0)$$ ### Step 3: Linearize the System Around the Critical Point To classify the critical point, we need to linearize the system at $$(0, 0)$$. Calculate the Jacobian matrix $$J$$ of the system at $$(x_1, x_2) = (0, 0)$$. The system is: $$\begin{cases} \dot{x}_1 = x_2 \\ \dot{x}_2 = -2(1 - x_1^3)x_2 + 3x_1 \end{cases}$$ The partial derivatives are: $$\frac{\partial \dot{x}_1}{\partial x_1} = 0, \quad \frac{\partial \dot{x}_1}{\partial x_2} = 1$$ $$\frac{\partial \dot{x}_2}{\partial x_1} = 3, \quad \frac{\partial \dot{x}_2}{\partial x_2} = -2(1 - x_1^3)$$ At $$(0, 0)$$, the Jacobian becomes: $$J = \begin{pmatrix} 0 & 1 \\ 3 & -2 \end{pmatrix}$$ ### Step 4: Analyze the Eigenvalues of the Jacobian To classify the critical point, we find the eigenvalues of the Jacobian matrix $$J$$: $$\det\begin{pmatrix} 0 - \lambda & 1 \\ 3 & -2 - \lambda \end{pmatrix} = 0$$ Expanding the determinant: $$\lambda^2 + 2\lambda + 3 = 0$$ Solve for $$\lambda$$: $$\lambda = \frac{-2 \pm \sqrt{4 - 12}}{2} = -1 \pm i\sqrt{2}$$ The eigenvalues are complex with a negative real part, which implies a **stable spiral** at the critical point $$(0, 0)$$. ### Conclusion The critical point $$(0, 0)$$ is a **stable spiral**. Would you like more details or have any questions? ### Related Questions: 1. How do you interpret the stability of a spiral in a dynamical system? 2. What other types of critical points exist in nonlinear systems? 3. How does the behavior change if the real part of the eigenvalues were positive? 4. Why does the Jacobian matrix help in linearizing a nonlinear system? 5. Can we classify critical points for systems of higher-order differential equations in a similar way? ### Tip When classifying critical points, always check the signs and types of eigenvalues, as they directly indicate the nature of the critical point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Nonlinear Differential Equations
Autonomous Systems
Critical Points
Stability Analysis
Formulas
Jacobian matrix of the system
Eigenvalue analysis for stability
Theorems
Stability of critical points
Classification of fixed points
Suitable Grade Level
Undergraduate level