Foundations — Linearization of nonlinear systems

4.6.24 · D1 · Maths › Ordinary Differential Equations › Linearization of nonlinear systems

Is page par kuch bhi assumed nahi hai. Agar parent note Linearization mein koi symbol use hua hai, toh hum use yahan ground up se build karenge, us order mein jo har idea ko pehle wale par lean karne deta hai.

1. Dot, plane, aur ek position

Sab kuch ek single point se shuru hota hai jo kaagaz ki flat sheet par move karta hai.

Poora map — har possible $(x,y)$ — phase plane kehlata hai (properly banaya gaya hai Phase Plane Analysis mein). Ek dot = system ka ek complete snapshot.

2. Time mein dot aur dot-notation $\dot{x}$

Ab dot ko move karne do. Jaise time $t$ aage badhta hai, uska address bhi badalti hai.

$dx/dt$ ki jagah overdot kyun? Pure laziness jo kaafi kaam aati hai: hamaare equations mein rates ki bhaarmaar hogi, aur $\dot{x}$ unhe readable rakhta hai. Dot ka matlab hamesha "time ke saath rate of change" hota hai, aur kuch nahi.

3. Position ke functions: $f(x,y)$ aur $g(x,y)$

Velocity arrow aata kahan se hai? Ek rule se.

Toh map ke har point par, $f$ aur $g$ milkar ek arrow paint karte hain. Arrows se bhara map ek vector field hai — system ki rulebook.

Nonlinear simply matlab hai ki in machines mein products ya curves hain — jaise $x^2$, $xy$, ya $\sin x$ — toh arrows bend karte hain. Agar $f,g$ sirf $ax+by$ hote (straight sums), toh system linear hota aur already solvable hota.

4. Resting spot: equilibrium $(x^{\ast },y^{\ast })$

Kuch special addresses par zero arrow hota hai.

5. Chhote displacements $u$ aur $v$

Hum resting spot se door nahi dekhte; hum ekdum paas dekhte hain.

Hum $u,v$ ko tiny rakhenge. Woh tininess is poori method ka engine hai (agla section).

6. Slopes: partial derivatives $f_x,f_y$

Curved arrow-field par zoom in karne ke liye, hume measure karna hoga ki har push kitni steeply change hoti hai jab hum $x$ ya $y$ ko nudge karte hain.

Partial (ordinary nahi) derivatives kyun? Kyunki $f$ do inputs par depend karta hai. Hume ek waqt mein ek nudge ka effect isolate karna hoga — doosre ko freeze karo, pucho "ground kidhar tilt karta hai?", phir repeat karo. Poori tilt do answers milke capture hoti hai.

7. Taylor's approximation: curve ko seedha karna

Yeh ek woh single move hai jo linearization ko power deti hai: ek curve ko uske tangent line se replace karo kisi point ke paas.

Yeh allowed kyun hai, aur yahan kyun rukein? Kyunki equilibrium ke paas steps $u,v$ tiny hain. Drop kiye gaye terms $u^2,uv,v^2$ carry karte hain — tiny number ka square bahut zyada chhota hota hai ($0.01^2=0.0001$). Toh straight-line part dominate karta hai; curvature negligible hai. Yeh "ek smooth curve seedhi dikhti hai agar kaafi zoom in karo" ka ground-up version hai — poori machinery Taylor Series mein hai.

Equilibrium par pehla term zero hota hai (wahi ${}^{\ast }$ ki definition hai!), toh approximation ek pure straight-line rule mein $u,v$ collapse ho jaati hai: $\dot{u}\approx f_{x}u+f_{y}v.$

8. Jacobian matrix $J$

Section 7 dono $f$ aur $g$ ke liye karo, aur chaaon slopes ko ek grid mein stack karo.

Ek matrix sirf ek aisi machine hai jo ek arrow ko doosre mein turn karti hai. $J$ hamare zoomed-in world ke liye wahi machine hai.

9. Eigen-cheezein: $\lambda$ aur $\mathbf{v}$

Linear machine $J$ ke favourite directions hain jinmein woh pure stretching se kaam karta hai.

Poori construction Eigenvalues and Eigenvectors mein hai. Yahan hume kyun care hai? Kyunki ek eigen-direction ke along future bilkul simple hai: displacement $e^{\lambda t}$ follow karta hai.

Symbols $\tau =f_x+g_y$ (trace, diagonal ka sum) aur $\Delta =f_x{g}_y-f_y{g}_x$ (determinant) sirf $\lambda$ dhundhne ke quick handles hain $\lambda =\frac{\tau \pm \sqrt{{\tau }^2-4\Delta }}{2}$ ke through.

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