Foundations — Phase plane analysis — trajectories, critical points

4.6.22 · D1 · Maths › Ordinary Differential Equations › Phase plane analysis — trajectories, critical points

Yeh page kuch bhi assume nahi karta. Agar parent note mein koi symbol use hua tha, toh hum use yahan pehle ek picture se build karte hain. Upar se neeche padho: har block agla block earn karta hai.

0. Sabse pehli picture: arrows se bhara ek plane

Plane kyun, line kyun nahi? Kyunki hamare system mein do changing quantities hain, $x$ aur $y$. Ek number ko line chahiye; do numbers ko plane chahiye. Har dot "is instant mein state" ka poora description hai.

Figure s01 dekho: ek single dot abhi hai. Jaise jaise time guzarta hai, dot sheet par slide karta hai, apne peeche ek curve khenchta hua. Hum ab un har ek ingredient ko naam dene wale hain jo dot ko move karte hain.

1. $x(t)$ aur $y(t)$ — quantities jo time par depend karti hain

Picture: $x(t)$ hamare moving dot ka horizontal coordinate hai jis moment clock $t$ read karta hai; $y(t)$ uska vertical coordinate hai. $t=0,1,2,\dots$ daalo aur dot ki jagah frame by frame milti hai.

Topic ko isko kyun chahiye: poora subject time ke saath change ke baare mein hai. Clock $t$ ke bina, kuch move nahi karta aur draw karne ke liye koi trajectory nahi hai.

2. Dot $\dot{x}$ — "rate of change" (derivative, zero se)

Newton ka shorthand: kisi letter ke upar ek dot ka matlab hai "woh letter jaise jaise time tick karta hai, kitni tezi se change karta hai".

Sirf "change" kyun nahi, derivative kyun? "x 3 upar gaya" jaisa plain change batata nahi kitni tezi se. Derivative change ko ek instant par capture karta hai — abhi $x$ ka $t$ ke against graph ka slope. Woh instant-by-instant velocity exactly wahi hai jo ek arrow ko abhi point karne ke liye chahiye.

Picture (figure s02): moving dot par, $\dot{x}$ uski rightward push ki length hai aur $\dot{y}$ uski upward push ki length. Un dono pushes ko tail-to-tail glue karo aur ek arrow milta hai — velocity arrow $(\dot{x},\dot{y})$. Woh arrow woh direction hai jis mein dot abhi travel karne wala hai.

3. $f(x,y)$ aur $g(x,y)$ — woh rule jo arrow deta hai

System likha jaata hai: $\dot{x}=f(x,y),\dot{y}=g(x,y).$

Isko zor se padhna: "meri rightward speed $\dot{x}$ equal hai jo rule $f$ mere current spot par kehta hai; meri upward speed $\dot{y}$ equal hai jo rule $g$ kehta hai." Toh $f$ aur $g$ plane ke har point par arrow paint karte hain.

Topic ko isko kyun chahiye: $f$ aur $g$ hi hawa hain. Mujhe koi bhi spot $(x,y)$ do aur ye do functions mujhe wahan ka velocity arrow turant de dete hain — koi solving required nahi. Yeh bahut saare spots par karo toh plane arrows se bhar jaata hai: direction field.

4. "Autonomous" — $f$ aur $g$ mein $t$ kyun absent hona chahiye

Picture: har spot par hawa frozen in place hai — har din same. Kal kisi spot par wapas aao aur wahan ka arrow identical hai.

Yeh poore subject ka engine kyun hai: kyunki arrow sirf position par depend karta hai, ek direction har point par rehti hai. Do alag trajectories kabhi same ordinary point se alag taraf jaate hue nahi guzar sakti — sirf ek arrow hai jisko maanna hai. Isliye trajectories cross nahi karti (sirf §6 ke special still-points ko chhodkar).

5. $(\dot{x},\dot{y})$ ek vector ki tarah, aur slope $dy/dx$

Picture: figure s02 ke velocity arrow se bilkul same. Slope $dy/dx$ sirf us arrow ka tilt hai — rise $\dot{y}$ over run $\dot{x}$ — clock chhod ke.

Dono viewpoints kyun: vector $(\dot{x},\dot{y})$ speed aur direction rakhta hai; slope $dy/dx$ sirf direction rakhta hai. Subject vector use karta hai arrows se flow dekhne ke liye, aur slope bare curve shapes describe karne ke liye.

6. Critical point — jahan arrow ki length zero ho

Picture (figure s04): windy field mein dead calm ka ek spot. Patta wahan exactly drop karo aur woh kabhi nahi hilega. Plane mein baaki har arrow inhi still-points ke around arrange hota hai, isliye inhe dhundna drains aur fountains dhundne jaisa hai jo saara flow shape karte hain.

"Critical" kyun: ye wahi jagahein hain jahan trajectory start, end, ya meet kar sakti hai, aur ye poori portrait organize karte hain. Inhe $f=0$ aur $g=0$ ke do equations simultaneously solve karke dhundho (ordinary algebra, no calculus).

7. Partial derivatives $f_x,f_y,g_x,g_y$ — ek direction mein ek baar mein slope

Ek still-point ke paas zoom karne ke liye hume chahiye "arrow kaise change hota hai jab main thoda step karta hun?" — lekin do directions mein. Yahi ek partial derivative measure karta hai.

Picture: surface height $f(x,y)$ par khade ho. $f_x$ steepness hai agar aap east (along $x$) chalo; $f_y$ steepness hai agar aap north (along $y$) chalo. Do alag slopes kyunki aap do independent directions mein tilt kar sakte ho.

Topic ko isko kyun chahiye: ek critical point ke paas curved wind field straight dikhta hai — aur chaar partials $f_x,f_y,g_x,g_y$ exactly woh chaar numbers hain jo us straight (linear) approximation ko describe karte hain.

8. Jacobian $J$ — chaar partials ek box mein packed

Isko padhna: top row batati hai $\dot{x}$ $x$ (left) aur $y$ (right) mein nudge ke response mein kaise respond karta hai; bottom row wahi $\dot{y}$ ke liye karta hai. Toh $J$ still-point ke tiny neighborhood mein arrows ka local instruction sheet hai.

Still-point ke tiny neighborhood mein arrows ke liye $J$ local instruction sheet hai. Full Taylor-expansion story ke liye Linearization and the Jacobian matrix dekho; yahan hume sirf $J$ ko is chaar slopes ke box ke roop mein recognize karna hai.

9. $\mathrm{tr}J$ aur $\det J$ — box ke do summary numbers

Do single numbers $2\times 2$ box ka essence squeeze karte hain:

Picture: $\det J$ ko socho "local arrows se patto ka ek tiny square kitna stretch ya squash hota hai aur flip hota hai", aur $\mathrm{tr}J$ ko socho "patte net outward kitna puff karte hain". Sirf unke signs (positive/negative) already still-point ka type decide kar dete hain. Full map ke liye Trace–determinant plane dekho.

Kyun: tumhe poora box kum hi chahiye — ye do numbers verdict rakhte hain, jaisa §11 dikhata hai.

10. $\lambda$, eigenvalues, aur $e^{\lambda t}$ — growth/rotation dials

$e^{\lambda t}$ kyun aata hai: $\mathbf{v}{e}^{\lambda t}$ shaped solution guess karo (ek arrow jiska length growth factor $e^{\lambda t}$ se scale hota hai). Plug in karo aur equation force karti hai $J\mathbf{v}=\lambda \mathbf{v}$ — toh $\lambda$ zaroor ek eigenvalue hona chahiye. Eigenvalues and eigenvectors dekho.

Do eigenvalues aate hain: ${\lambda }^2-(\mathrm{tr}J)\lambda +\det J=0.$ Toh §9 ke do summary numbers seedha do dials ${\lambda }_{1,2}$ mein feed hote hain jo sab kuch decide karte hain.

11. Dials ko saath mein rakhna — chaar portraits

Linearization se aage deeper stability tools ke liye Stability and Lyapunov functions dekho; outward-spiral-to-a-ring behavior ke liye Limit cycles and Poincaré–Bendixson theorem dekho.

Foundations topic ko kaise feed karte hain

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