Foundations — Stability of equilibria — stable, unstable, saddle, spiral, centre

4.6.23 · D1 · Maths › Ordinary Differential Equations › Stability of equilibria — stable, unstable, saddle, spiral,

Kisi "stable spiral" ko classify karne se pehle tumhare liye zaroori hai ki page par har ek mark padh sako. Neeche har ek symbol aur idea diya gaya hai jis par parent note depend karta hai, is tarah order kiya gaya hai ki har ek cheez pehle wali se build ho. Kuch bhi assume nahi kiya gaya.

1. Plane par ek point: $(x,y)$

Ek bird's-eye map ki picture karo. Mujhe $x$ do (kitna east) aur $y$ do (kitna north) aur main exactly ek jagah par apni ungli rakh sakta hoon. Vo ungli-wali jagah hi state hai.

Yeh topic kyun isko zaroori maanta hai: poora subject is baare mein hai ki time ke saath us dot ka kya hota hai — toh pehle hamen agree karna hoga ki dot ek 2D plane par exist karta hai jo do numbers se describe hoti hai.

2. Time aur dot ki motion: $t$, $\dot{x}$, $\dot{y}$

Pair $(\dot{x},\dot{y})$ velocity vector hai. $\dot{x}$ ko socho "east–west direction mein bilkul bhi nahi rukne" ke opposite ke taur par — positive $\dot{x}$ dot ko right push karta hai, negative left push karta hai.

Yeh topic kyun isko zaroori maanta hai: "stability" ek nudge ke baad motion ke baare mein ek sawaal hai. Speed ka notion nahi, toh "wapas aata hai" ya "door bhag jaata hai" ka notion bhi nahi.

3. Arrows set karne ka rule: $f$ aur $g$

Word autonomous ka matlab hai ki $f$ aur $g$ sirf is par depend karte hain ki tum kahan ho, kaunsa time hai par nahi — haawa ka pattern frozen hai, ek baar ke liye map par paint kar diya gaya hai.

Yeh topic kyun isko zaroori maanta hai: yeh pair hi differential equation hai. Baad mein aane wali har cheez — equilibria, Jacobian, eigenvalues — $f$ aur $g$ se nikalti hai.

4. Jahan haawa ruk jaati hai: equilibrium $(x^{\ast },y^{\ast })$

Yeh topic kyun isko zaroori maanta hai: equilibria long-term rest ke liye wahi candidates hain. Poora topic unke baare mein ek hi cheez poochta hai — kya shant jagah ek trap hai, ek launchpad hai, ya ek whirlpool hai?

5. Chhote nudges: $u=x-x^{\ast }$, $v=y-y^{\ast }$

Yeh sirf map ko shant jagah par origin laake dobara draw karna hai. Ek nudge ka matlab hai small $u,v$ se shuru karna (dot bilkul shant point ke paas rakha hua) aur dekhna ki kya $u,v$ zero tak shrink hote hain (wapas aata hai) ya badhte hain (bhaag jaata hai).

Yeh topic kyun isko zaroori maanta hai: stability ek local sawaal hai. Equilibrium par centre karke hum sirf chhote $u,v$ se deal karte hain, jo exactly wohi cheez hai jo agla step — linearising — legal banati hai.

6. Haawa ki slopes: partial derivatives $f_x,f_y,g_x,g_y$

Hum ek straight-line slope use kyun kar sakte hain: shant jagah ke bahut paas, curved haawa ka pattern flat dikhta hai, bilkul waise jaise tumhari khidki se Earth flat dikhti hai. Yeh ek first-order Taylor approximation hai — sach ke curved rule ko uski tangent se trade karo, jo ek slope times ek step hai.

Yeh topic kyun isko zaroori maanta hai: yeh char numbers Jacobian matrix ka raw material hain — local "wind rulebook".

7. Slopes ko ek grid mein pack karna: matrix $J$

Equilibrium ke paas flow is beautifully simple rule ka palan karta hai: $\left(\begin{array}{c}\dot{u}\\ \dot{v}\end{array}\right)=J\left(\begin{array}{c}u\\ v\end{array}\right).$ Ise padhein: "mujhe apna current offset $(u,v)$ do aur $J$ tumhari current velocity wapas karta hai." Matrix hi linearised law of motion hai.

Yeh topic kyun isko zaroori maanta hai: har point ko alag-alag study karne ki jagah, ek $2\times 2$ grid poore local behaviour ko capture karta hai. Topic ka baaki hissa sirf is grid se sawaal karna hai.

Matrices ko properly seekho Eigenvalues and eigenvectors mein.

8. Do summary numbers: trace $\tau$ aur determinant $\Delta$

Yeh topic kyun isko zaroori maanta hai: parent sirf in do numbers aur ek derived quantity se har equilibrium classify karta hai — eigenvector hunting ki zaroorat nahi. Poora map dekho Trace–determinant plane mein.

9. Growth rates: eigenvalues $\lambda$

$i$ (jahan $i^2=-1$) wohi hai jo ek akele formula ko turning describe karne deta hai, kyunki $e^{i\omega t}$ ek circle trace karta hai. Yahi mathematical reason hai ki spirals aur centres bilkul exist karte hain.

Eigenvalues $\tau ,\Delta$ se aate hain: ${\lambda }_{1,2}=\frac{\tau \pm \sqrt{{\tau }^2-4\Delta }}{2}.$ Root ke neeche wali quantity, $D={\tau }^2-4\Delta$, discriminant hai: $D<0$ square root ko imaginary force karta hai — yahi tab hota hai jab spirals/centres appear hote hain.

Yeh topic kyun isko zaroori maanta hai: har $\lambda$ ke real part ka sign stability decide karta hai; imaginary part rotation decide karta hai. Eigenvalues final verdict hain.

10. Flow ko poore tarah padhna: phase portrait

Inhe fully explore karo Phase portraits mein. Jab tum kuch bhi solve kiye bina stability prove karna chahte ho, tum ek energy-jaisi function tak pahunchte ho — Lyapunov stability — aur stable spiral ka physical archetype hai Damped harmonic oscillator.

Prerequisite map

Equipment checklist

Khud test karo — right side cover karo aur zyaazor se jawab do.