Math Problem Statement
calculate the points of equiliber?
Solution
To determine the equilibrium points of the given dynamic system, we must solve the system of differential equations by setting the derivatives , , and to zero. The system is defined as follows:
\dot{x} = x - y^2 \\ \dot{y} = x^2 - y \\ \dot{z} = x - z \end{cases}$$ **Equilibrium points** occur when the derivatives are zero: 1. From $$\dot{x} = x - y^2 = 0$$: $$x = y^2$$ 2. From $$\dot{y} = x^2 - y = 0$$: $$y = x^2$$ 3. From $$\dot{z} = x - z = 0$$: $$z = x$$ ### Substituting $$x = y^2$$ into $$y = x^2$$: - Substitute $$x = y^2$$ into $$y = x^2$$: $$y = (y^2)^2 = y^4$$ - Solve $$y = y^4$$: Factorize: $$y(y^3 - 1) = 0$$ Solutions are: $$y = 0 \quad \text{or} \quad y^3 = 1 \implies y = 1$$ ### Corresponding $$x$$ and $$z$$ values: 1. For $$y = 0$$: $$x = y^2 = 0, \quad z = x = 0$$ Equilibrium point: $$(x, y, z) = (0, 0, 0)$$. 2. For $$y = 1$$: $$x = y^2 = 1, \quad z = x = 1$$ Equilibrium point: $$(x, y, z) = (1, 1, 1)$$. ### Final Answer: The equilibrium points are: $$(0, 0, 0) \quad \text{and} \quad (1, 1, 1).$$ Would you like a detailed explanation of stability for these points? Here are some related questions to deepen your understanding: 1. What are the stability properties of the equilibrium points? 2. How do we linearize the system around an equilibrium point? 3. Can we classify the type of equilibrium points (e.g., saddle, node)? 4. What happens to the system's behavior near these equilibrium points? 5. How does the coupling between the equations affect the dynamics? **Tip**: To analyze equilibrium stability, you can use the Jacobian matrix of the system.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Dynamical Systems
Equilibrium Points
Nonlinear Differential Equations
Formulas
\(\dot{x} = x - y^2\)
\(\dot{y} = x^2 - y\)
\(\dot{z} = x - z\)
Theorems
Equilibrium Condition: \(\dot{x} = \dot{y} = \dot{z} = 0\)
Suitable Grade Level
Undergraduate Level (Calculus and Differential Equations)
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