Let's solve **4.3.3** and **4.3.6** step by step. --- ## **Problem 4.3.3** \[ \dot{\theta} = \mu \sin\theta - \sin 2\theta \] ### **Step 1: Find Fixed Points** Fixed points satisfy \(\dot{\theta} = 0\): \[ \mu \sin\theta - \sin 2\theta = 0 \] Recall: \(\sin 2\theta = 2\sin\theta\cos\theta\): \[ \mu \sin\theta - 2\sin\theta\cos\theta = 0 \] \[ \sin\theta (\mu - 2\cos\theta) = 0 \] So, either \(\sin\theta = 0\) or \(\mu - 2\cos\theta = 0\): - \(\sin\theta = 0 \implies \theta = n\pi\) - \(\mu - 2\cos\theta = 0 \implies \cos\theta = \frac{\mu}{2}\) But \(|\cos\theta| \leq 1\), so real solutions for this only if \(|\mu| \leq 2\). ### **Step 2: Stability of Fixed Points** Linearize about fixed points: \[ f(\theta) = \mu \sin\theta - \sin 2\theta \] \[ f'(\theta) = \mu \cos\theta - 2\cos 2\theta \] #### **At \(\theta = n\pi\):** - \(\sin\theta = 0\) - \(\cos\theta = (-1)^n\) - \(\cos 2\theta = 1\) (since \(2n\pi\)) So, \[ f'(\theta = n\pi) = \mu(-1)^n - 2 \cdot 1 = \mu(-1)^n - 2 \] - For \(n\) even (\(\theta = 0, 2\pi, ...\)): \(f' = \mu - 2\) - For \(n\) odd (\(\theta = \pi, 3\pi, ...\)): \(f' = -\mu - 2\) A fixed point is **stable** if \(f'(\theta) < 0\). #### **At \(\cos\theta = \mu/2\):** Plug back into \(f'(\theta)\): - \(\cos\theta = \mu/2\) - \(\cos 2\theta = 2\cos^2\theta - 1 = 2(\mu^2/4) - 1 = \mu^2/2 - 1\) - \(f' = \mu \cos\theta - 2\cos 2\theta = \mu \cdot \frac{\mu}{2} - 2\left(\frac{\mu^2}{2} - 1\right) = \frac{\mu^2}{2} - \mu^2 + 2 = -\frac{\mu^2}{2} + 2\) So - Stable if \(-\frac{\mu^2}{2} + 2 < 0 \implies \mu^2 > 4\) But recall, these fixed points only exist for \(|\mu| \leq 2\), so they are always **unstable** or **neutral**. ### **Step 3: Bifurcations** **At \(\mu = 2\):** - For \(\theta = 0\): \(f' = 0\) (neutral stability), so a bifurcation occurs. - For \(\cos\theta = 1\): \(\theta = 0\), so the two solutions merge. **At \(\mu = -2\):** - For \(\theta = \pi\): \(f' = 0\), so another bifurcation occurs. So, bifurcations at \(\mu = \pm 2\). **Type:** These are **saddle-node bifurcations**. --- ## **Summary Table** | \(\mu\) | Fixed Points | Stability | Bifurcation Type | |------------|-----------------------------|--------------------------|------------------| | \(|\mu| > 2\) | \(\theta = 0, \pi\) | 0: stable if \(\mu>2\), \(\pi\): stable if \(\mu < -2\) | - | | \(|\mu| < 2\) | \(\theta = 0, \pi\), \(\cos\theta = \mu/2\) | 0: unstable, \(\pi\): unstable, \(\cos\theta = \mu/2\): exist, unstable | Saddle-node at \(\mu = \pm 2\) | --- ## **Problem 4.3.6** \[ \dot{\theta} = \mu + \sin\theta + \cos 2\theta \] ### **Step 1: Fixed Points** Set \(\dot{\theta} = 0\): \[ \mu + \sin\theta + \cos 2\theta = 0 \] \[ \mu + \sin\theta + 2\cos^2\theta - 1 = 0 \] \[ \sin\theta + 2\cos^2\theta = 1 - \mu \] Use \(\cos^2\theta = 1 - \sin^2\theta\): \[ \sin\theta + 2(1 - \sin^2\theta) = 1 - \mu \] \[ \sin\theta + 2 - 2\sin^2\theta = 1 - \mu \] \[ \sin\theta - 2\sin^2\theta = -1 - \mu \] \[ 2\sin^2\theta - \sin\theta + (\mu + 1) = 0 \] Which is a quadratic in \(x = \sin\theta\): \[ 2x^2 - x + (\mu + 1) = 0 \] ### **Step 2: Discriminant and Number of Fixed Points** The discriminant of the quadratic is \[ D = (-1)^2 - 8(\mu + 1) = 1 - 8(\mu + 1) \] - **Two real solutions** if \(D > 0\) (and \(|x| \leq 1\)) - **One real solution** if \(D = 0\) - **No real solution** if \(D < 0\) So, \[ 1 - 8(\mu + 1) > 0 \implies \mu < -\frac{7}{8} \] \[ 1 - 8(\mu + 1) = 0 \implies \mu = -\frac{7}{8} \] Thus, at \(\mu = -\frac{7}{8}\) a **saddle-node bifurcation** occurs. ### **Step 3: Stability** Let \(x = \sin\theta\): \[ f(\theta) = \mu + \sin\theta + \cos 2\theta \] \[ f'(\theta) = \cos\theta - 2\sin 2\theta \] Evaluate at the fixed points as needed. --- ## **Final Answers** ### **4.3.3**: - **Bifurcation values**: \(\mu = \pm 2\) - **Type**: Saddle-node bifurcation ### **4.3.6**: - **Bifurcation value**: \(\mu = -\frac{7}{8}\) - **Type**: Saddle-node bifurcation #### **Phase portraits**: - For \(\mu\) above the bifurcation value: Fewer or no fixed points. - At the bifurcation value: Fixed points merge. - Below the bifurcation value: Two fixed points (one stable, one unstable). If you'd like, I can sketch the phase portraits or elaborate further!

Certainly! Let me provide a detailed step-by-step solution for each problem, including all calculations, stability analysis, bifurcation classification, and final answers. --- # **Problem 4.3.3** \[ \boxed{ \dot{\theta} = \mu \sin\theta - \sin 2\theta } \] ## **Step 1: Find Fixed Points** Set \(\dot{\theta} = 0\): \[ \mu \sin\theta - \sin 2\theta = 0 \] Recall the double angle identity: \[ \sin 2\theta = 2 \sin\theta \cos\theta \] Substitute: \[ \mu \sin\theta - 2 \sin\theta \cos\theta = 0 \] Factor out \(\sin\theta\): \[ \sin\theta (\mu - 2 \cos\theta) = 0 \] This yields two cases: ### **Case 1: \(\sin\theta = 0\)** \[ \Rightarrow \theta = n\pi, \quad n \in \mathbb{Z} \] ### **Case 2: \(\mu - 2 \cos\theta = 0\)** \[ \Rightarrow \cos\theta = \frac{\mu}{2} \] Since \(|\cos\theta| \leq 1\), the solutions exist only if: \[ \left| \frac{\mu}{2} \right| \leq 1 \quad \Rightarrow \quad |\mu| \leq 2 \] --- ## **Step 2: Stability Analysis** To analyze the stability of these fixed points, compute the derivative of the right-hand side: \[ f(\theta) = \mu \sin\theta - 2 \sin\theta \cos\theta \] \[ f'(\theta) = \frac{d}{d\theta} \left( \mu \sin\theta - 2 \sin\theta \cos\theta \right) \] Calculate term-by-term: \[ f'(\theta) = \mu \cos\theta - 2 \left( \cos^2\theta - \sin^2\theta \right) \] because: \[ \frac{d}{d\theta} (\sin\theta) = \cos\theta \] \[ \frac{d}{d\theta} (\sin\theta \cos\theta) = \cos^2\theta - \sin^2\theta = \cos 2\theta \] Thus, \[ f'(\theta) = \mu \cos\theta - 2 \cos 2\theta \] --- ### **At \(\theta = n\pi\):** - \(\sin(n\pi) = 0\) - \(\cos(n\pi) = (-1)^n\) - \(\cos 2n\pi = 1\) Evaluate: \[ f'(\theta) = \mu (-1)^n - 2 \times 1 = \mu (-1)^n - 2 \] - For **even \(n\)** (\(\theta = 0, 2\pi, ...\)): \[ f' = \mu - 2 \] - For **odd \(n\)** (\(\theta = \pi, 3\pi, ...\)): \[ f' = -\mu - 2 \] **Stability:** - Stable if \(f' < 0\) \[ \Rightarrow \text{At } \theta = 2k\pi: \quad \mu - 2 < 0 \Rightarrow \mu < 2 \] \[ \Rightarrow \text{At } \theta = (2k+1)\pi: \quad -\mu - 2 < 0 \Rightarrow -\mu < 2 \Rightarrow \mu > -2 \] --- ### **At \(\cos \theta = \frac{\mu}{2}\):** - Fixed points exist only if \(|\mu/2| \leq 1\). - Compute the derivative at these points: Recall: \[ f'(\theta) = \mu \cos\theta - 2 \cos 2\theta \] But at \(\cos\theta = \frac{\mu}{2}\), find \(\cos 2\theta\): \[ \cos 2\theta = 2 \cos^2 \theta - 1 = 2 \left(\frac{\mu}{2}\right)^2 - 1 = 2 \frac{\mu^2}{4} - 1 = \frac{\mu^2}{2} - 1 \] Substitute into \(f'\): \[ f' = \mu \left(\frac{\mu}{2}\right) - 2 \left(\frac{\mu^2}{2} - 1 \right) = \frac{\mu^2}{2} - \left(\mu^2 - 2\right) = \frac{\mu^2}{2} - \mu^2 + 2 = -\frac{\mu^2}{2} + 2 \] **Stability:** \[ f' = -\frac{\mu^2}{2} + 2 \] - Fixed point is **stable** if \(f' < 0\): \[ -\frac{\mu^2}{2} + 2 < 0 \Rightarrow \frac{\mu^2}{2} > 2 \Rightarrow \mu^2 > 4 \] But since \(\left| \frac{\mu}{2} \right| \leq 1 \Rightarrow |\mu| \leq 2\), the only possible values for \(\mu\) are within \([-2, 2]\). - For \(\mu\) strictly between \(-2\) and \(2\), the fixed points on \(\cos\theta = \mu/2\) exist but are **unstable** (since \(f' > 0\) when \(\mu^2 < 4\)). --- ## **Step 3: Bifurcation Analysis** - **At \(\mu = \pm 2\):** The solutions \(\cos\theta = \pm 1\) collide with the boundary \(|\cos\theta| \leq 1\). - **For \(\mu > 2\):** No solutions for \(\cos\theta = \mu/2\) because \(|\mu/2| > 1\). Fixed points are only at \(\theta = 0, \pi, ...\). - **At \(\mu=2\):** \(\cos\theta = 1\), i.e., \(\theta=0\). The fixed point at \(\theta=0\) merges with the boundary. - **Similarly at \(\mu = -2\):** \(\cos\theta = -1\), i.e., \(\theta=\pi\). This indicates **saddle-node bifurcations** occur at \(\mu = \pm 2\). --- # **Final Summary for 4.3.3** | \(\mu\) | Fixed points | Stability | Bifurcation Type | Comments | |---------|----------------|--------------|-------------------|----------| | \(\mu < -2\) | \(\theta = \pi + 2k\pi\) (stable), others unstable | \(\theta=\pi\): stable if \(\mu < -2\) | None | Only one fixed point \(\theta=\pi\) stable | | \(-2 < \mu < 2\) | \(\theta = n\pi\) (unstable or stable depending), plus \(\cos\theta = \mu/2\) solutions | \(\theta = 0\): unstable, \(\theta=\pi\): unstable, \(\cos\theta = \mu/2\): unstable | Saddle-node at \(\mu = \pm 2\) | Fixed points appear/disappear at \(\pm 2\) | | \(\mu > 2\) | Only \(\theta = n\pi\) | \(\theta=0\): unstable | None | No fixed point \(\cos\theta=\mu/2\) solutions | --- # **Problem 4.3.6** \[ \boxed{ \dot{\theta} = \mu + \sin\theta + \cos 2\theta } \] ## **Step 1: Find Fixed Points** Set \(\dot{\theta} = 0\): \[ \mu + \sin\theta + \cos 2\theta = 0 \] Recall: \[ \cos 2\theta = 2 \cos^2 \theta - 1 \] Substitute: \[ \mu + \sin\theta + 2 \cos^2 \theta - 1 = 0 \] Express \(\cos^2 \theta\): \[ \cos^2 \theta = 1 - \sin^2 \theta \] So, \[ \mu + \sin\theta + 2(1 - \sin^2 \theta) - 1 = 0 \] \[ \mu + \sin\theta + 2 - 2 \sin^2 \theta - 1 = 0 \] \[ \mu + \sin\theta - 2 \sin^2 \theta + 1 = 0 \] Rearranged: \[ -2 \sin^2 \theta + \sin \theta + (\mu + 1) = 0 \] Let \(x = \sin \theta\). Then: \[ -2 x^2 + x + (\mu + 1) = 0 \] Multiply through by \(-1\): \[ 2 x^2 - x - (\mu + 1) = 0 \] --- ## **Step 2: Discriminant for solutions** Quadratic in \(x\): \[ 2 x^2 - x - (\mu + 1) = 0 \] Discriminant: \[ D = (-1)^2 - 4 \times 2 \times [-(\mu + 1)] = 1 + 8 (\mu + 1) \] Solutions for \(x\): \[ x = \frac{1 \pm \sqrt{D}}{2 \times 2} = \frac{1 \pm \sqrt{1+8(\mu + 1)}}{4} \] For real solutions, require: \[ D \geq 0 \Rightarrow 1 + 8 (\mu + 1) \geq 0 \] \[ 8 (\mu + 1) \geq -1 \Rightarrow \mu + 1 \geq -\frac{1}{8} \] \[ \Rightarrow \mu \geq -1 - \frac{1}{8} = -\frac{9}{8} \] --- ## **Step 3: Validity of solutions** Since \(x = \sin\theta\), solutions are only valid if: \[ |x| \leq 1 \] Calculate \(x\): \[ x_{\pm} = \frac{1 \pm \sqrt{1 + 8 (\mu + 1)}}{4} \] - For fixed points to exist, both solutions need to satisfy \(|x_{\pm}| \leq 1\). --- ## **Step 4: Bifurcation points** At the boundary where \(\sqrt{D} = 0\): \[ 1 + 8 (\mu + 1) = 0 \] \[ 8 (\mu + 1) = -1 \] \[ \mu + 1 = -\frac{1}{8} \] \[ \boxed{ \textbf{Bifurcation at } \mu = -\frac{9}{8} } \] This is a **saddle-node bifurcation**, where two fixed points merge. --- ## **Final classification:** - For \(\mu > -\frac{9}{8}\), **two fixed points** (solutions for \(x\)), with their stability depending on the derivative. - At \(\mu = -\frac{9}{8}\), a saddle-node bifurcation occurs. - For \(\mu < -\frac{9}{8}\), **no fixed points**. --- # **Summary of Final Answers** | **Problem** | **Bifurcation value(s)** | **Type** | **Comments** | |--------------|------------------------|-----------|--------------| | **4.3.3** | \(\boxed{\mu = \pm 2}\) | Saddle-node | Fixed points merge at these values | | **4.3.6** | \(\boxed{\mu= -\frac{9}{8}}\) | Saddle-node | Two fixed points merge here | --- # **Phase Portraits Overview** - For \(\mu\) less than bifurcation points: fewer fixed points, possibly none. - At bifurcation points: fixed points merge and annihilate. - For \(\mu\) greater than bifurcation points: the number of fixed points changes, leading to different dynamical regimes. --- I hope this detailed explanation clarifies each step! If you'd like, I can also help sketch the phase portraits or analyze the stability of each fixed point further.

Certainly! Here are the final answers: **Problem 4.3.3:** - **Bifurcation values:** \(\boxed{\mu = \pm 2}\) - **Type of bifurcation:** Saddle-node bifurcation occurring at \(\mu = 2\) and \(\mu = -2\). --- **Problem 4.3.6:** - **Bifurcation value:** \(\boxed{\mu = -\frac{9}{8}}\) - **Type of bifurcation:** Saddle-node bifurcation at \(\mu = -\frac{9}{8}\). These are the critical control parameter values where fixed points appear, merge, or annihilate.