Question bank — Chaotic systems — sensitivity to initial conditions, Lyapunov exponents

False — rules 100% deterministic hote hain; identical starts se identical futures milte hain. Apparent randomness sirf hamare imperfect knowledge ki wajah se aati hai jo exponentially amplify hoti hai.

False — $e^{\lambda t}$ do nearby points ke beech ke local gap ko describe karta hai, trajectory khud ko nahi. Path ek bounded attractor par rehta hai aur baar baar fold back karta rehta hai.

False — $\dot{x}=x$ jaisa pure blow-up sensitive hai lekin unbounded aur non-chaotic hai. Chaos ko SDIC plus boundedness plus mixing chahiye.

False — $\lambda <0$ matlab nearby trajectories converge karti hain, toh errors khatam ho jaati hain. Yeh calm, predictable behaviour hai: ek stable fixed point ya limit cycle.

False — kyunki $t_{\text{pred}}\approx \frac{1}{\lambda }\ln (\Delta /{\delta }_0)$ hai, accuracy sirf ek logarithm ke through enter hoti hai, toh $1000\times$ better ${\delta }_0$ horizon mein ek fixed constant $\frac{1}{\lambda }\ln 1000$ add karta hai, 1000 ka factor nahi.

False — identical starts identical deterministic rules follow karte hain aur hamesha saath locked rehte hain. Divergence ke liye nonzero initial gap ${\delta }_0$ chahiye.

True — ek dissipative flow mein ${\sum }_i{\lambda }_i<0$ hota hai (volume contract hota hai, Liouville-type reasoning se) jabki phir bhi ${\lambda }_1>0$ ho sakta hai (ek stretching direction). Strange attractor par stretch-and-fold.

Principle mein True lekin yeh $\pm$ pairs mein aate hain jinka sum zero hota hai; volume conserved hota hai toh ek direction mein contraction bilkul exactly doosri direction mein expansion se match hota hai — net volume loss nahi hota.

False — $t\to \infty$ limit poore attractor par local stretching ko average karta hai, toh $\lambda$ system ko characterise karta hai, kisi lucky initial condition ko nahi.

Local rate $f^{\prime }(x)$ vary karta hai jab $x$ orbit ke saath move karta hai; tumhe ise trajectory ke saath average karna padega, starting point par freeze nahi karna. Ek value use karna baad ki har stretch aur fold ko ignore karta hai.

Yeh cosmetic nahi hai — linearisation $f(x+\delta )-f(x)\approx f^{\prime }(x)\delta$ sirf infinitesimal $\delta$ ke liye valid hai. ${\delta }_0\to 0$ lena us approximation ko averaging se pehle honest rakhta hai.

Nahi — woh $2$ do nearby trajectories ke beech ka distance-stretching factor hai (map tent map ke conjugate hai, jo gaps double karta hai), population value khud ki doubling nahi.

Order matter karta hai: ${\delta }_0\to 0$ pehle separation ko infinitesimal rakhta hai toh linear theory valid rahti hai; agar tum finite ${\delta }_0$ ke saath pehle $t\to \infty$ lete ho, gap bounded attractor par saturate ho jaata hai aur log ratio true exponential rate nahi measure karta.

SDIC stable periodic behaviour ko forbid karta hai, lekin chaotic systems unstable periodic orbits se bhare hote hain (dense periodic orbits) — trajectory sirf unpar kabhi settle nahi hoti.

$\lambda =0$ regular, quasi-periodic, ya periodic motion ka marginal case hai — nearby trajectories ek bounded, non-growing distance par saath rehti hain. Bahut kuch hota hai; yeh sirf predictable hota hai.

Galat conclusion — better models aur data ${\delta }_0$ ko neeche push karte hain aur $\lambda$ ko refine karte hain, kuch extra days kharidkar. Point sirf yeh hai ki logarithmic payoff indefinite prediction impossible banata hai, yeh nahi ki improvement worthless hai.

Determinism rules ke baare mein ek statement hai (woh future bilkul fix kar dete hain); predictability hamare present ke knowledge ke baare mein hai (hamesha uncertain). Chaos us unavoidable uncertainty ko exponentially magnify karta hai.

Ek constant relative growth rate — gap ka har unit time mein ek fixed fraction se badhna — bilkul exponential ki defining property hai; phir $\ln$ lena us single rate $\lambda$ ko reveal kar deta hai.

Ek finite region ke bina fold back karne ke liye, exponential stretching trajectories ko infinity tak bhej deti (jaise $\dot{x}=x$). Boundedness woh stretch-and-fold force karta hai jo phase space ko mix karta hai aur genuine chaos create karta hai.

Koi bhi ek single positive direction nearby trajectories ko exponentially diverge karne ke liye kaafi hai; sabse bada exponent ${\lambda }_1$ growth par dominate karta hai, toh ${\lambda }_1>0$ on/off switch hai.

Dissipation total volume contract karta hai ($\sum {\lambda }_i<0$) lekin woh contraction unevenly share ho sakta hai: flow abhi bhi ek direction mein stretch karta hai (${\lambda }_1>0$) jabki doosri jagah zyada squeeze karta hai, strange attractor par chaos ke liye jagah chhodkar.

Kyunki $t_{\text{pred}}\approx \frac{1}{\lambda }\ln (\Delta /{\delta }_0)$ hai: horizon double karne ke liye initial error ko $e^{\lambda t}$ factor se shrink karna padta hai, ek exponential cost jo kisi bhi conceivable sensor se jaldi aage nikal jaati hai.

Iska matlab hai ki trajectory hamesha kisi periodic pattern ke paas hoti hai lekin kisi par bhi lock nahi ho sakti (woh unstable hain), toh motion lagaataar almost-regular dikhta rehta hai jabki endlessly wander karta rehta hai — chaos ki pehchaani texture.

$\lambda =0$ — nearby trajectories na converge karti hain na exponentially diverge; gap bounded aur roughly constant rehti hai, regular (non-chaotic) motion ko mark karte hue.

Wahan $f^{\prime }(x^{\ast })=r(1-2x^{\ast })=0$ hai, toh $\ln |f^{\prime }|\to -\infty$: separations instantly crush ho jaati hain, sabse strongest possible convergence, aur koi chaos nahi hai.

Nahi — yeh sensitive hai lekin unbounded aur non-mixing hai; ek single trajectory sirf infinity ki taraf bhaag jaati hai. Boundedness aur mixing missing hone se yeh disqualify ho jaata hai.

Nahi — ek continuous variable ke saath trajectory sirf monotonically fixed points ki taraf ya unke beech move kar sakti hai; stretch aur fold karne ki koi jagah nahi hai, toh continuous 1D flows chaotic nahi ho sakte (tumhe kam se kam teen dimensions chahiye, ya ek discrete map).

Yeh undefined hai — $\ln (|\delta (t)|/|{\delta }_0|)$ zero se divide karta hai. Yahi wajah hai ki definition ${\delta }_0\to 0$ limit leti hai instead of zero set karne ke, infinitesimal separation capture karte hue bina kabhi zero reach kiye.

Us idealized point ke liye nahi — bilkul uspar baithkar, yeh hamesha wahan rehta hai. Lekin koi bhi infinitesimal nudge exponentially door grow karti hai, jo exactly woh SDIC hai jo iske aas paas har jagah chaos seed karta hai.