Let’s work through the problem step by step for both parts (a) and (b): ## **Given** - \( \ddot{x} + f(x) = 0 \), where \( f(x) = P'(x) \). - \( P(x) \) is the potential function (see provided graphs). ## **Phase Plane Sketching** Let \( x_1 = x \), \( x_2 = \dot{x} \). Then, the system becomes: \[ \dot{x}_1 = x_2 \\ \dot{x}_2 = -f(x_1) = -P'(x_1) \] The phase plane is the \( (x_1, x_2) \) plane. --- ## **Step 1: Find Equilibria** Equilibrium points occur when \( \dot{x}_1 = 0 \) and \( \dot{x}_2 = 0 \). So, \( x_2 = 0 \) and \( P'(x_1) = 0 \). From the graphs: - **Equilibria:** all points where the slope of \( P(x) \) is zero (local minima, maxima, or inflection points). --- ## **Step 2: Stability Analysis** Use the shape of \( P(x) \): - **Local Minimum:** stable equilibrium (center) - **Local Maximum:** unstable equilibrium (saddle) --- ## **Step 3: Analyze and Sketch Each Case** ### **(a)** - **Graph:** \( P(x) \) has two minima (at \( x_1 \) and right of \( x_2 \)), one maximum (between \( x_1 \) and \( x_2 \)). - **Equilibria:** At each minimum and maximum (where \( P'(x) = 0 \)). - At \( x_1 \): local minimum → **stable** - At \( x_2 \): local maximum → **unstable** - Rightmost minimum: **stable** - **Phase Plane Sketch:** - **Around minima:** Closed orbits (energy conservation). - **Around maxima:** Separatrix (trajectories approach and then depart from the saddle). #### **Stability:** - **Minima:** Centers (stable, not asymptotically because there's no friction). - **Maxima:** Saddles (unstable). --- ### **(b)** - **Graph:** \( P(x) \) has a minimum at \( x_1 \), an inflection at \( x_2 \) (\( P'(x_2) = 0 \) and \( P''(x_2) = 0 \)), and then increases. - **Equilibria:** At \( x_1 \) (minimum, stable) and \( x_2 \) (inflection, neutral). - At \( x_1 \): **stable** - At \( x_2 \): **neutral/stability indeterminate** (higher order analysis needed). - **Phase Plane Sketch:** - **Around \( x_1 \):** Closed orbits. - **Around \( x_2 \):** Behavior determined by higher derivatives. Trajectories can pass through or be repelled/attracted depending on the nonlinear terms. #### **Stability:** - **\( x_1 \):** Center (stable). - **\( x_2 \):** *Typically unstable or semi-stable* (since it's a flat inflection, small perturbations may move the system away). --- ## **Summary Table** | Point | Type | Stability (a) | Stability (b) | |------------|------------------|---------------|---------------| | \( x_1 \) | Local minimum | Stable | Stable | | \( x_2 \) | Local maximum | Unstable | Inflection (neutral/unstable) | --- ## **Final Answers** ### **(a)** - **Equilibria:** At \( x_1 \) (min), \( x_2 \) (max), and rightmost min. - **Stability:** Minima are stable (centers), maximum is unstable (saddle). - **Phase plane:** Closed orbits around minima, separatrix through the saddle. ### **(b)** - **Equilibria:** At \( x_1 \) (min), \( x_2 \) (inflection). - **Stability:** \( x_1 \) is stable (center), \( x_2 \) is typically unstable or indeterminate (needs higher order analysis). - **Phase plane:** Closed orbits around \( x_1 \), complex behavior near \( x_2 \). --- ## **How to Sketch the Phase Plane** - Plot \( x \) vs. \( \dot{x} \). - Draw closed orbits around minima. - Draw separatrices (trajectories that approach and depart the saddle) at maxima. - At inflection, indicate possible instability or "flat" behavior. Let me know if you want explicit sketches!