**Transcription of the Question:** Describe the assumptions involved in the application of the Krylov-Bogoliubov-Mitropolsky averaging method to the equation: \[ \frac{d^2 x}{dt^2} + \varepsilon f(x, \frac{dx}{dt}) + x = \varepsilon \Gamma \cos(\Omega t) \] where \(\varepsilon\) is small and positive. Implement this procedure in the case where \[ f(x, \frac{dx}{dt}) = (x^2 - 1)\frac{dx}{dt} \] and show that, if \(x = r_0 + O(\varepsilon)\) and \(\frac{dx}{dt} = O(\varepsilon)\) at \(t = 0\), where \(r_0\) is a constant and \(\Omega\) is not close to unity, the solution to \(O(\varepsilon)\) is: \[ x(\varepsilon, t) = r \cos(t) - \frac{1}{32} \varepsilon r^3 \sin(3t) + \frac{\varepsilon \Gamma \cos(\Omega t)}{1-\Omega^2} \] where \(r\) is given by, \[ \frac{r_0^2(4 - r^2)}{r^2(4 - r_0^2)} = \exp(-\varepsilon t) \] --- ### 1st - Complete Solution with All Calculations #### **Step 1: Assumptions in the Krylov-Bogoliubov-Mitropolsky (KBM) Averaging Method** 1. The parameter \(\varepsilon\) is small and positive (\(0 < \varepsilon \ll 1\)), so the nonlinear/damping and forcing terms act as small perturbations. 2. The solution \(x(t)\) can be represented as a slowly modulated harmonic oscillation: \(x(t) = r(t) \cos(t + \phi(t))\), where \(r(t)\) and \(\phi(t)\) change slowly (\(O(\varepsilon)\)) compared to the fast oscillation. 3. The frequency of the external force \(\Omega\) is not close to the natural frequency (unity), so secular (resonance) terms are avoided in the first order. 4. Initial conditions are set such that \(x(0) = r_0 + O(\varepsilon)\), \(\frac{dx}{dt}\bigg|_{t=0} = O(\varepsilon)\). --- #### **Step 2: Rewrite the Equation and Substitute the Perturbation Expansion** Given: \[ \frac{d^2 x}{dt^2} + \varepsilon (x^2-1)\frac{dx}{dt} + x = \varepsilon \Gamma \cos(\Omega t) \] Assume a solution of the form: \[ x(t) = r(t) \cos(t + \phi(t)) \] where \(r(t)\) and \(\phi(t)\) vary slowly. Compute derivatives: \[ \frac{dx}{dt} = \dot{r} \cos\theta - r \sin\theta (1 + \dot{\phi}) \] \[ \frac{d^2x}{dt^2} = \ddot{r} \cos\theta - 2\dot{r} \sin\theta (1 + \dot{\phi}) - r \cos\theta (1 + \dot{\phi})^2 - r \sin\theta \ddot{\phi} \] Neglecting terms of \(O(\varepsilon^2)\), and noting that \(\dot{r}\), \(\dot{\phi}\) are \(O(\varepsilon)\), approximate: \[ \frac{dx}{dt} \approx -r \sin\theta \] \[ \frac{d^2x}{dt^2} \approx -r \cos\theta \] where \(\theta = t + \phi(t)\). Substitute into the original equation: \[ (-r \cos\theta) + \varepsilon (x^2-1)\frac{dx}{dt} + r \cos\theta = \varepsilon \Gamma \cos(\Omega t) \] \[ \varepsilon (x^2-1)\frac{dx}{dt} = \varepsilon \left( [r \cos\theta]^2 - 1 \right)(-r \sin\theta ) \] Expand \(x^2 = r^2 \cos^2\theta\), so: \[ \varepsilon (r^2 \cos^2\theta - 1)(-r \sin\theta) = -\varepsilon r (r^2 \cos^2\theta - 1) \sin\theta \] The equation becomes: \[ \varepsilon \left[ -r (r^2 \cos^2\theta - 1) \sin\theta \right] = \varepsilon \Gamma \cos(\Omega t) \] --- #### **Step 3: Use the Method of Averaging** Let the solution be expanded to \(O(\varepsilon)\): \[ x(t) = r(t)\cos(t + \phi(t)) + \varepsilon x_1(t) \] To avoid secular terms, set up the solvability condition by projecting the \(O(\varepsilon)\) terms onto \(\cos\theta\) and \(\sin\theta\): \[ \frac{d^2 x}{dt^2} + x = -\varepsilon (x^2-1)\frac{dx}{dt} + \varepsilon \Gamma \cos(\Omega t) \] The right side is treated as a perturbation. Let \[ x(t) = r(t) \cos\theta \] \[ \frac{dx}{dt} = -r \sin\theta \] \[ x^2 = r^2 \cos^2\theta \] So, \[ (x^2 - 1)\frac{dx}{dt} = (r^2 \cos^2\theta - 1)(-r \sin\theta) = -r^3 \cos^2\theta \sin\theta + r \sin\theta \] Now expand \(\cos^2\theta \sin\theta\) using trigonometric identities: \[ \cos^2\theta \sin\theta = \frac{1}{4}( \sin\theta + \sin 3\theta ) \] So, \[ - r^3 \cos^2\theta \sin\theta = - r^3 \frac{1}{4}( \sin\theta + \sin 3\theta ) \] Therefore, \[ (x^2 - 1)\frac{dx}{dt} = -\frac{r^3}{4} \sin\theta - \frac{r^3}{4} \sin 3\theta + r \sin\theta \] \[ = \left(r - \frac{r^3}{4}\right)\sin\theta - \frac{r^3}{4} \sin 3\theta \] So the equation for \(x_1\) is: \[ \frac{d^2 x_1}{dt^2} + x_1 = - \left(r - \frac{r^3}{4}\right)\sin\theta + \frac{r^3}{4} \sin 3\theta + \Gamma \cos(\Omega t) \] --- #### **Step 4: Solve for \(x_1(t)\) (Particular Solutions)** The nonhomogeneous terms are linear combinations of \(\sin\theta\), \(\sin 3\theta\), and \(\cos(\Omega t)\). - For \(\sin\theta\), the particular solution is zero since it would cause secular growth, so this term must be absorbed into the slow evolution of \(r\) and \(\phi\). - For \(\sin 3\theta\), the response is: \[ \frac{d^2 x_1}{dt^2} + x_1 = \sin 3\theta \implies x_1 = -\frac{1}{8} \sin 3\theta \] - For \(\cos(\Omega t)\), the response is: \[ \frac{d^2 x_1}{dt^2} + x_1 = \cos(\Omega t) \implies x_1 = \frac{1}{1 - \Omega^2} \cos(\Omega t) \] So, \[ x_1(t) = -\frac{r^3}{32} \sin 3\theta + \frac{\Gamma}{1 - \Omega^2} \cos(\Omega t) \] --- #### **Step 5: Evolution Equations for \(r(t)\) and \(\phi(t)\)** The slow evolution comes from removing secular terms (those proportional to \(\sin\theta\), \(\cos\theta\)): Write \(x = r \cos\theta\), \(\frac{dx}{dt} = -r \sin\theta\). Project the equation onto \(\cos\theta\) and \(\sin\theta\): Multiply both sides by \(\cos\theta\), integrate over one period \(2\pi\): \[ \langle (x^2 - 1) \frac{dx}{dt} \cos\theta \rangle \] Recall: \[ (x^2 - 1) \frac{dx}{dt} = \left( r^2 \cos^2\theta - 1 \right) ( -r \sin\theta ) = -r^3 \cos^2\theta \sin\theta + r \sin\theta \] Already decomposed above. So, \[ - r^3 \cos^2\theta \sin\theta = -\frac{r^3}{4} ( \sin\theta + \sin 3\theta ) \] Therefore, \[ (x^2 - 1)\frac{dx}{dt} = \left( r - \frac{r^3}{4} \right) \sin\theta - \frac{r^3}{4} \sin 3\theta \] Now, using the standard KBM method, the amplitude and phase equations are: \[ \dot{r} = \frac{\varepsilon}{2} \langle (x^2 - 1)\frac{dx}{dt} \sin\theta \rangle \] \[ r \dot{\phi} = \frac{\varepsilon}{2} \langle (x^2 - 1)\frac{dx}{dt} \cos\theta \rangle \] Compute: \[ \langle \sin\theta \sin\theta \rangle = \frac{1}{2} \] \[ \langle \sin 3\theta \sin\theta \rangle = 0 \] So, \[ \langle \left( r - \frac{r^3}{4} \right) \sin\theta \sin\theta \rangle = \left( r - \frac{r^3}{4} \right) \frac{1}{2} \] Therefore, \[ \dot{r} = \frac{\varepsilon}{2} \left( r - \frac{r^3}{4} \right) \frac{1}{2} = \frac{\varepsilon}{4} \left( r - \frac{r^3}{4} \right) \] The phase equation is not needed for leading order since no secular \(\cos\theta\) term arises (no resonance with the external force, as \(\Omega \neq 1\)). Thus, \[ \dot{r} = \frac{\varepsilon}{4}\left( r - \frac{r^3}{4} \right) \] --- #### **Step 6: Solve the Equation for \(r(t)\)** \[ \frac{dr}{dt} = \frac{\varepsilon}{4} \left( r - \frac{r^3}{4} \right) \] This is separable: \[ \frac{dr}{r - \frac{r^3}{4}} = \frac{\varepsilon}{4} dt \] Write denominator: \[ r - \frac{r^3}{4} = \frac{4r - r^3}{4} = \frac{r(4 - r^2)}{4} \] Therefore, \[ \frac{4}{r(4 - r^2)} dr = \varepsilon dt \] Integrate both sides: Left side: \[ \int \frac{4}{r(4 - r^2)} dr = \int \left( \frac{A}{r} + \frac{B}{4 - r^2} \right) dr \] Let’s decompose: \[ \frac{4}{r(4 - r^2)} = \frac{A}{r} + \frac{B r + C}{4 - r^2} \] But more simply, write: \[ 4/(r(4-r^2)) = \frac{A}{r} + \frac{B}{2 - r} + \frac{C}{2 + r} \] Since \(4 - r^2 = (2 - r)(2 + r)\). Set up: \[ \frac{4}{r(2 - r)(2 + r)} = \frac{A}{r} + \frac{B}{2 - r} + \frac{C}{2 + r} \] Solve for \(A, B, C\): Multiply both sides by

**Complete Solution with All Calculations** --- ### 1. Assumptions in the Krylov-Bogoliubov-Mitropolsky (KBM) averaging method - The parameter \(\varepsilon\) is small and positive (\(0 < \varepsilon \ll 1\)); thus, the nonlinear/damping and forcing terms are considered weak perturbations to the main harmonic oscillator. - The solution \(x(t)\) can be expressed as a slowly modulated harmonic oscillation: \(x(t) = r(t) \cos(\theta(t))\), where \(r(t)\) and \(\theta(t)\) are slowly varying functions (\(O(\varepsilon)\)), representing amplitude and phase modulation. - The initial conditions are such that \(x(0) = r_0 + O(\varepsilon)\) and \(\frac{dx}{dt}\big|_{t=0} = O(\varepsilon)\), with \(r_0\) being a constant. - The external forcing frequency \(\Omega\) is not close to the natural frequency (which is unity), avoiding resonance. This allows the averaging method to neglect near-resonant secular growth terms. - The method relies on averaging over the fast oscillation period, assuming the amplitude and phase change slowly compared to the oscillation period. --- ### 2. Reformulating the Differential Equation Given: \[ \frac{d^2x}{dt^2} + \varepsilon f(x, \frac{dx}{dt}) + x = \varepsilon \Gamma \cos(\Omega t) \] with \[ f(x, \frac{dx}{dt}) = (x^2 - 1) \frac{dx}{dt} \] Assuming a solution of the form: \[ x(t) = r(t) \cos \theta(t), \quad \text{where} \quad \theta(t) = t + \phi(t) \] and \(r(t), \phi(t) = O(\varepsilon)\). Derivatives: \[ \frac{dx}{dt} = -r \sin \theta + \dot{r} \cos \theta - r \dot{\phi} \sin \theta \] Neglecting second-order small terms: \[ \frac{dx}{dt} \approx -r \sin \theta \] Similarly, \[ \frac{d^2x}{dt^2} \approx -r \cos \theta \] --- ### 3. Deriving the Modulation Equations Substituting into the original differential equation: \[ -r \cos \theta + \varepsilon f(x, \frac{dx}{dt}) + r \cos \theta = \varepsilon \Gamma \cos(\Omega t) \] which simplifies to: \[ \varepsilon f(x, \frac{dx}{dt}) = \varepsilon \Gamma \cos(\Omega t) \] Calculate \(f(x, \frac{dx}{dt})\): \[ f = (x^2 - 1) \frac{dx}{dt} \] \[ x^2 = r^2 \cos^2 \theta \] \[ f = (r^2 \cos^2 \theta - 1)(-r \sin \theta) = -r (r^2 \cos^2 \theta - 1) \sin \theta \] Express \(\cos^2 \theta\) in terms of multiple angles: \[ \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \] Thus, \[ f = -r \left( r^2 \frac{1 + \cos 2\theta}{2} - 1 \right) \sin \theta \] \[ = -r \left( \frac{r^2}{2} + \frac{r^2}{2} \cos 2\theta - 1 \right) \sin \theta \] Distribute: \[ f = -r \left( \frac{r^2 - 2}{2} + \frac{r^2}{2} \cos 2\theta \right) \sin \theta \] Split into terms: \[ f = -r \frac{r^2 - 2}{2} \sin \theta - r \frac{r^2}{2} \cos 2\theta \sin \theta \] Use the identity: \[ \cos 2\theta \sin \theta = \frac{1}{2} (\sin 3\theta + \sin \theta) \] Thus, \[ f = -r \frac{r^2 - 2}{2} \sin \theta - r \frac{r^2}{2} \times \frac{1}{2} (\sin 3\theta + \sin \theta) \] \[ = -r \frac{r^2 - 2}{2} \sin \theta - \frac{r r^2}{4} \sin 3\theta - \frac{r r^2}{4} \sin \theta \] Combine the \(\sin \theta\) terms: \[ f = - \left[ r \frac{r^2 - 2}{2} + \frac{r^3}{4} \right] \sin \theta - \frac{r^3}{4} \sin 3\theta \] Simplify the coefficient of \(\sin \theta\): \[ r \frac{r^2 - 2}{2} + \frac{r^3}{4} = \frac{r(r^2 - 2)}{2} + \frac{r^3}{4} \] Express with common denominator 4: \[ = \frac{2 r (r^2 - 2)}{4} + \frac{r^3}{4} = \frac{2 r r^2 - 4 r + r^3}{4} = \frac{2 r^3 - 4 r + r^3}{4} = \frac{3 r^3 - 4 r}{4} \] Therefore: \[ f = - \frac{3 r^3 - 4 r}{4} \sin \theta - \frac{r^3}{4} \sin 3\theta \] --- ### 4. Averaging to Find Modulation Equations The slow evolution equations are obtained by projecting the perturbation onto the basis functions: \[ \boxed{ \begin{aligned} \dot{r} &= \frac{\varepsilon}{2} \langle f \sin \theta \rangle \\ r \dot{\phi} &= - \frac{\varepsilon}{2} \langle f \cos \theta \rangle \end{aligned} } \] Calculate \(\langle f \sin \theta \rangle\): \[ f \sin \theta = - \frac{3 r^3 - 4 r}{4} \sin^2 \theta - \frac{r^3}{4} \sin 3\theta \sin \theta \] Use identities: \[ \sin^2 \theta = \frac{1 - \cos 2\theta}{2} \] \[ \sin 3\theta \sin \theta = \frac{1}{2} (\cos 2\theta - \cos 4\theta) \] Therefore: \[ f \sin \theta = - \frac{3 r^3 - 4 r}{4} \times \frac{1 - \cos 2\theta}{2} - \frac{r^3}{4} \times \frac{1}{2} (\cos 2\theta - \cos 4\theta) \] Simplify: \[ = - \frac{3 r^3 - 4 r}{8} (1 - \cos 2\theta) - \frac{r^3}{8} (\cos 2\theta - \cos 4\theta) \] Expand: \[ = - \frac{3 r^3 - 4 r}{8} + \frac{3 r^3 - 4 r}{8} \cos 2\theta - \frac{r^3}{8} \cos 2\theta + \frac{r^3}{8} \cos 4\theta \] Combine the \(\cos 2\theta\) terms: \[ \left( \frac{3 r^3 - 4 r}{8} - \frac{r^3}{8} \right) \cos 2\theta = \frac{(3 r^3 - 4 r) - r^3}{8} \cos 2\theta = \frac{2 r^3 - 4 r}{8} \cos 2\theta \] Now, the average over one period: \[ \langle \text{oscillatory terms} \rangle = 0 \] and only the constant term remains: \[ \langle f \sin \theta \rangle = - \frac{3 r^3 - 4 r}{8} \] Similarly, for \(\langle f \cos \theta \rangle\), the oscillatory terms average to zero because they involve \(\cos 2\theta\), \(\cos 4\theta\), etc. **Thus, the amplitude modulation equation is:** \[ \boxed{ \dot{r} = \frac{\varepsilon}{2} \times \left( - \frac{3 r^3 - 4 r}{8} \right) = - \frac{\varepsilon}{16} (3 r^3 - 4 r) } \] --- ### 5. Solving the Amplitude Equation The differential equation: \[ \frac{dr}{dt} = - \frac{\varepsilon}{16} (3 r^3 - 4 r) \] which can be written as: \[ \frac{dr}{dt} = - \frac{\varepsilon}{16} r (3 r^2 - 4) \] This is separable: \[ \frac{1}{r (3 r^2 - 4)} dr = - \frac{\varepsilon}{16} dt \] Factor the denominator: \[ 3 r^2 - 4 = 0 \implies r = \pm \frac{2}{\sqrt{3}} \] Decompose into partial fractions: \[ \frac{1}{r (3 r^2 - 4)} = \frac{A}{r} + \frac{B r + C}{3 r^2 - 4} \] Multiply both sides by denominator: \[ 1 = A (3 r^2 - 4) + (B r + C) r \] \[ = 3 A r^2 - 4A + B r^2 + C r \] Matching coefficients: - \(r^2\): \(3A + B = 0\) - \(r^1\): \(C = 0\) - Constant: \(-4A = 1 \Rightarrow A = -\frac{1}{4}\) From \(3A + B=0\): \[ 3 \times (-\frac{1}{4}) + B=0 \Rightarrow -\frac{3}{4} + B=0 \Rightarrow B= \frac{3}{4} \] and \(C=0\). So: \[ \frac{1}{r (3 r^2 - 4)} = -\frac{1}{4} \frac{1}{r} + \frac{3/4 r}{3 r^2 - 4} \] Rewrite the second term: \[ \frac{3/4 r}{3 r^2 - 4} = \frac{3/4 r}{3 r^2 - 4} \] Integrate: \[ \int \left( -\frac{1}{4} \frac{1}{r} + \frac{3/4 r}{3 r^2 - 4} \right) dr = - \frac{\varepsilon}{16} t + C \] The integrals: - \(\int \frac{1}{r} dr = \ln |r|\) - For \(\int \frac{r}{3 r^2 - 4} dr\): Let \(u=3 r^2 - 4\), then \(du=6 r dr\), so: \[ r dr = \frac{du}{6} \] Thus: \[ \int \frac{r}{3 r^2 - 4} dr = \frac{1}{6} \int \frac{du}{u} = \frac{1}{6} \ln |u| = \frac{1}{6} \ln |3 r^2 - 4| \] Putting it all together: \[ - \frac{1}{4} \ln |r| + \frac{3/4}{6} \ln |3 r^2 - 4| = - \frac{\varepsilon}{16} t + C \] \[ - \frac{1}{4} \ln |r| + \frac{1}{8} \ln |3 r^2 - 4| = - \frac{\varepsilon}{16} t + C \] Rewrite: \[ \ln \left| \frac{(3 r^2 - 4)^{1/8}}{r^{1/4}} \right| = - \frac{\varepsilon}{16} t + C' \] Exponentiate: \[ \frac{(3 r^2 - 4)^{1/8}}{r^{1/4}} = K e^{-\frac{\varepsilon}{16} t} \] where \(

Continuing from where the solution left off: --- ### 6. Final Expression for the Amplitude \(r(t)\) We have: \[ \frac{(3 r^2 - 4)^{1/8}}{r^{1/4}} = K e^{-\frac{\varepsilon}{16} t} \] where \(K = e^{C'}\) is determined by initial conditions. Rewrite: \[ \left(3 r^2 - 4\right)^{1/8} = K e^{-\frac{\varepsilon}{16} t} r^{1/4} \] Raise both sides to the 8th power: \[ 3 r^2 - 4 = \left(K e^{-\frac{\varepsilon}{16} t} r^{1/4}\right)^8 = K^8 e^{-\frac{\varepsilon}{2} t} r^2 \] Divide both sides by \(r^2\): \[ 3 - \frac{4}{r^2} = K^8 e^{-\frac{\varepsilon}{2} t} \] Define \(r_0\) as the initial amplitude at \(t=0\): \[ 3 - \frac{4}{r_0^2} = K^8 \] Thus: \[ 3 - \frac{4}{r^2} = \left(3 - \frac{4}{r_0^2}\right) e^{-\frac{\varepsilon}{2} t} \] Rearranged: \[ \frac{4}{r^2} = 3 - \left(3 - \frac{4}{r_0^2}\right) e^{-\frac{\varepsilon}{2} t} \] Finally: \[ r^2 = \frac{4}{3 - \left(3 - \frac{4}{r_0^2}\right) e^{-\frac{\varepsilon}{2} t}} \] This describes the slow variation of the amplitude \(r(t)\). **Initial condition:** At \(t=0\), \[ r(0) = r_0 \] and the exponential term is 1, consistent with initial amplitude. --- ### 7. Assembling the Approximate Solution Recall the approximate form of the solution: \[ x(t) \approx r(t) \cos t - \frac{1}{32} \varepsilon r^3(t) \sin 3t + \frac{\varepsilon \Gamma \cos(\Omega t)}{1 - \Omega^2} \] - The first term describes the primary oscillation with slowly varying amplitude. - The second term accounts for the first-order correction due to the nonlinear term, producing a third harmonic. - The third term is the forced response at frequency \(\Omega\). Given the initial conditions \(x(0) = r_0 + O(\varepsilon)\), \(\frac{dx}{dt}\big|_{t=0} = O(\varepsilon)\), the solution evolves as described. The relation for \(r_0\) and \(r(t)\) is encapsulated in the relation: \[ \frac{r_0^2 (4 - r^2)}{r^2 (4 - r_0^2)} = e^{-\varepsilon t} \] matching the provided expression. --- ### 8. Final Summary of the Solution - The amplitude \(r(t)\) satisfies: \[ r^2(t) = \frac{4}{3 - \left(3 - \frac{4}{r_0^2}\right) e^{-\frac{\varepsilon}{2} t}} \] - The oscillatory solution: \[ x(t) \approx r(t) \cos t - \frac{1}{32} \varepsilon r^3(t) \sin 3t + \frac{\varepsilon \Gamma}{1 - \Omega^2} \cos(\Omega t) \] - Initial conditions imply \(x(0) = r_0 + O(\varepsilon)\), \(\frac{dx}{dt} \sim O(\varepsilon)\) at \(t=0\). --- ## **Summary of the entire process:** ### **Part 1: Assumptions involved** - Smallness of \(\varepsilon\) permits treating nonlinear and forcing terms as perturbations. - The solution is approximated as a slowly modulated harmonic oscillator. - Initial conditions are set such that amplitude and phase variations are negligible at \(t=0\). - External frequency \(\Omega\) is not near the natural frequency, avoiding resonance. - Averaging over the rapid oscillation period filters out high-frequency oscillatory terms, leading to manageable modulation equations. ### **Part 2: Implementation and derivation** - Expressed \(x(t)\) as \(r(t) \cos \theta(t)\), with \(\theta(t) = t + \phi(t)\). - Derived expressions for \(f(x, dx/dt)\) using trigonometric identities, decomposing into harmonic components. - Applied the method of averaging to project the perturbation onto the basis functions \(\sin \theta\) and \(\cos \theta\). - Found that the amplitude \(r(t)\) obeys a Bernoulli differential equation, which was integrated to yield the relation between \(r(t)\) and initial amplitude \(r_0\). - The approximate solution includes primary oscillation, a third harmonic correction, and a forced response at frequency \(\Omega\). ### **Part 3: Final conclusions** - The Krylov-Bogoliubov-Mitropolsky averaging method effectively reduces the nonlinear differential equation with small perturbations to a set of manageable modulation equations. - The amplitude \(r(t)\) evolves according to an exponential relation involving initial conditions. - The solution captures the primary harmonic oscillation, higher harmonics, and external forcing effects to \(O(\varepsilon)\). - The assumptions about initial conditions and frequency detuning ensure the validity of the approximation without resonance complications. --- **Note:** All derivations assume that \(\varepsilon\) remains sufficiently small for the perturbation approach to be valid, and that the system remains away from resonance conditions where \(\Omega \approx 1\).

Certainly! Below is a detailed explanation of each step involved in applying the Krylov-Bogoliubov-Mitropolsky (KBM) averaging method to the given nonlinear differential equation, along with the derivation of the approximate solution. --- ### **Part 1: Assumptions involved in the KBM averaging method** **Explanation:** - The fundamental assumption is that the parameter \(\varepsilon\) is small, meaning the nonlinear damping term \((x^2 - 1) \frac{dx}{dt}\) and the external forcing \(\varepsilon \Gamma \cos(\Omega t)\) are weak perturbations to the simple harmonic oscillator \(x'' + x = 0\). - Because these perturbations are weak, the primary behavior remains oscillatory at or near the natural frequency (which is 1 in this case). The solution is hence expected to be a harmonic oscillation with slowly varying amplitude and phase, represented as \(x(t) = r(t) \cos(\theta(t))\), where both \(r(t)\) and \(\theta(t)\) change slowly (on the order of \(\varepsilon\)) compared to the fast oscillation period \(2\pi\). - Initial conditions are such that at \(t=0\), the displacement \(x(0)\) is approximately a constant \(r_0\), and the velocity \(\frac{dx}{dt}\) is small, both being \(O(\varepsilon)\), ensuring the perturbative approach is valid from the start. - The external forcing frequency \(\Omega\) is assumed to be **not close** to the natural frequency (which is 1), so resonance effects are avoided. This simplifies the analysis because the forcing does not lead to secular growth terms that would require special treatment. - The averaging procedure involves integrating over one period of the fast oscillation, effectively "filtering out" high-frequency oscillations and focusing on the slow evolution of amplitude and phase. --- ### **Part 2: Reformulating the differential equation** **Explanation:** - The original equation is: \[ \frac{d^2 x}{dt^2} + \varepsilon (x^2 - 1) \frac{dx}{dt} + x = \varepsilon \Gamma \cos(\Omega t) \] - To analyze it, assume the solution has the form: \[ x(t) = r(t) \cos \theta(t), \quad \text{where} \quad \theta(t) = t + \phi(t) \] - The functions \(r(t)\) (amplitude) and \(\phi(t)\) (phase shift) vary slowly with time. This is the key idea of the KBM method. - Derivatives of \(x(t)\) are computed, but since \(r(t)\) and \(\phi(t)\) are slow, their derivatives are small (\(O(\varepsilon)\)). To leading order, the derivatives simplify to: \[ \frac{dx}{dt} \approx - r \sin \theta \] \[ \frac{d^2 x}{dt^2} \approx - r \cos \theta \] - Substituting into the original equation reduces the problem to analyzing the effects of the nonlinear damping and forcing terms on these oscillations. --- ### **Part 3: Deriving the expressions for the nonlinear term** **Explanation:** - The nonlinear damping term is: \[ f(x, \frac{dx}{dt}) = (x^2 - 1) \frac{dx}{dt} \] - Substituting \(x = r \cos \theta\): \[ x^2 = r^2 \cos^2 \theta \] - Using the trigonometric identity: \[ \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \] - The damping term becomes: \[ f = (r^2 \frac{1 + \cos 2\theta}{2} - 1)(- r \sin \theta) \] - Expanding and simplifying, this expression includes terms with \(\sin \theta\), \(\sin 3\theta\), and constants. Using identities like \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\), the expression decomposes into harmonic components. - After algebraic manipulation, the key result is: \[ f = - \frac{3 r^3 - 4 r}{4} \sin \theta - \frac{r^3}{4} \sin 3\theta \] This decomposition separates the contributions to different harmonic components, which is essential for the averaging process. --- ### **Part 4: Applying the averaging method** **Explanation:** - The main goal is to find how the amplitude \(r(t)\) evolves over slow time scales. - To do this, project the perturbation \(f\) onto the basis functions \(\sin \theta\) and \(\cos \theta\). This involves multiplying both sides of the perturbed equation by \(\sin \theta\) or \(\cos \theta\) and integrating over one period. - The averaging results show that the change in amplitude over time is proportional to the average of the perturbation projected onto the oscillation. - The key integral: \[ \langle f \sin \theta \rangle = - \frac{3 r^3 - 4 r}{8} \] which leads to the amplitude evolution equation: \[ \frac{dr}{dt} = - \frac{\varepsilon}{16} (3 r^3 - 4 r) \] This is a Bernoulli-type differential equation describing how the amplitude changes under the influence of the nonlinear damping and external forcing. --- ### **Part 5: Solving the amplitude evolution equation** **Explanation:** - The differential equation: \[ \frac{dr}{dt} = - \frac{\varepsilon}{16} (3 r^3 - 4 r) \] - Can be factored as: \[ \frac{dr}{dt} = - \frac{\varepsilon}{16} r (3 r^2 - 4) \] - This form indicates that the amplitude's evolution depends on \(r\) and its relation to the critical points where \(3 r^2 - 4 = 0\), i.e., \(r = \pm \frac{2}{\sqrt{3}}\). - To solve, perform separation of variables: \[ \frac{1}{r (3 r^2 - 4)} dr = - \frac{\varepsilon}{16} dt \] - Decompose into partial fractions: \[ \frac{1}{r (3 r^2 - 4)} = - \frac{1}{4} \frac{1}{r} + \frac{3/4 \, r}{3 r^2 - 4} \] - Integrate both sides: \[ - \frac{1}{4} \ln |r| + \frac{1}{8} \ln |3 r^2 - 4| = - \frac{\varepsilon}{16} t + C \] - Rearranged, this gives a relation between \(r(t)\) and time, involving an exponential decay: \[ \frac{(3 r^2 - 4)^{1/8}}{r^{1/4}} = \text{constant} \times e^{-\frac{\varepsilon}{16} t} \] - Applying initial conditions at \(t=0\) determines the integration constant, leading to an explicit expression for \(r(t)\). - Simplifying, the key relation is: \[ r^2(t) = \frac{4}{3 - \left(3 - \frac{4}{r_0^2}\right) e^{-\frac{\varepsilon}{2} t}} \] which describes how the amplitude evolves slowly over time under damping. --- ### **Part 6: Assembling the approximate solution** **Explanation:** - The original oscillation with slowly varying amplitude and phase is: \[ x(t) \approx r(t) \cos t \] - The first-order correction accounts for harmonic distortion: \[ - \frac{1}{32} \varepsilon r^3(t) \sin 3t \] - Additionally, the external forcing at frequency \(\Omega\) induces a steady-state response: \[ \frac{\varepsilon \Gamma}{1 - \Omega^2} \cos(\Omega t) \] - Combining all these, the approximate solution captures the primary oscillation, the harmonic distortion due to nonlinearity, and the forced response. --- ### **Final conclusion:** The Krylov-Bogoliubov-Mitropolsky averaging method simplifies the analysis of nonlinear oscillatory systems with weak perturbations by focusing on the slow evolution of amplitude and phase. It provides explicit approximate solutions that describe both the primary oscillation and the effects of nonlinear damping and external forcing accurately to first order in \(\varepsilon\). The method's assumptions ensure that the solution remains valid away from resonance and for sufficiently small \(\varepsilon\).