Chaotic systems — sensitivity to initial conditions, Lyapunov exponents

2.1.25 · Physics › Analytical Mechanics

Chaos KYA hai?

"Deterministic" + "unpredictable" contradict kyun nahi karte? Equations future ko exactly fix karti hain. Lekin hum present ko kabhi exactly nahi jaante — hamare paas ek error ${\delta }_0$ hoti hai. Non-chaotic system mein woh error slowly (linearly) badhti hai. Chaotic system mein woh $e^{\lambda t}$ ki tarah badhti hai, toh ek microscopic ${\delta }_0$ bhi finite time mein order-1 tak pahunch jaati hai. Determinism rules ke baare mein hai; predictability hamare knowledge ke baare mein hai.

Trajectories kitni tezi se alag hoti hain? — Lyapunov exponent derive karna

Hume ek number chahiye jo separation rate measure kare. Ise scratch se banate hain.

Step 1 — Do nearby trajectories set up karo. Maano $x(t)$ follow karta hai $\dot{x}=f(x)$ ko. Ek neighbour ko $x(0)+{\delta }_0$ par start karo. Separation define karo: $\delta (t)=x^{\prime }(t)-x(t).$ Yeh step kyun? Hume literally yahi care hai ki do runs ke beech ka gap kaise evolve hota hai — wahi gap hai jo chaos mein badhta hai.

Step 2 — Chhoti separation ke liye linearise karo. $\dot{\delta }=f(x+\delta )-f(x)\approx f^{\prime }(x)\delta .$ Yeh step kyun? Jab tak $\delta$ chhota hai, sirf first derivative (local stretching rate) matter karta hai. Hum Taylor-expand karte hain aur leading term rakhte hain.

Step 3 — Linear growth solve karo. Agar $f^{\prime }(x)$ ek constant $\lambda$ hota, toh $\dot{\delta }=\lambda \delta \Rightarrow \delta (t)={\delta }_0{e}^{\lambda t}$. Generally local rate vary karta hai, isliye hum ise orbit ke saath average karte hain: $|\delta (t)|\sim |{\delta }_0|e^{\lambda t},\lambda \equiv {\lim }_{t\to \infty }{\lim }_{|{\delta }_0|\to 0}\frac{1}{t}\ln \frac{|\delta (t)|}{|{\delta }_0|}.$ Yeh step kyun? Exponential ek hi function hai jo constant relative growth rate ko ek clean number $\lambda$ mein convert karta hai. $\ln$ lekar phir $t$ se divide karne se woh rate nikal aati hai.

Do limits kyun, isi order mein? Pehle ${\delta }_0\to 0$ taaki linearisation sahi rahe (hum sirf infinitesimal separation chahte hain). Phir $t\to \infty$ taaki poore attractor par average ho aur $\lambda$ system ki property rahe, kisi ek lucky starting point ki nahi.

Ek $n$-dimensional system mein $n$ exponents hote hain (the Lyapunov spectrum) ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n$, jo har independent direction mein stretching describe karte hain. Maximal wala ${\lambda }_1$ chaos decide karta hai.

Predictability horizon (80/20 takeaway)

Worked Example 1 — Logistic map (discrete chaos)

Map: $x_{n+1}=r{x}_n(1-x_n)$. Maps ke liye, $\lambda ={\lim }_{N\to \infty }\frac{1}{N}{\sum }_{n=0}^{N-1}\ln |f^{\prime }(x_n)|$ jahan $f^{\prime }(x)=r(1-2x)$.

Yeh formula kyun? Har iteration ek chhoti si separation ko $|f^{\prime }(x_n)|$ se multiply karta hai. $N$ steps ke baad in factors ka product $\prod |f^{\prime }(x_n)|$ hota hai; $\frac{1}{N}\ln$ lene se product ek average log-stretch mein badal jaata hai — exactly $\lambda$.

• $r=2$ par map ka ek stable fixed point hai $x^{\ast }=0.5$, $f^{\prime }(x^{\ast })=2(1-2\cdot 0.5)=0$, toh $\lambda =-\infty$ direction → strongly stable, koi chaos nahi.
• $r=4$ par yeh fully chaotic hai exact value $\lambda =\ln 2\approx 0.693$ ke saath.

$\ln 2$ kyun? $r=4$ par map tent map ke conjugate hai jo har step mein distances double karta hai → factor 2 → $\lambda =\ln 2$.

Worked Example 2 — Weather ke liye predictability time

Maano $\lambda \approx 0.4{\text{day}}^{-1}$, initial error ${\delta }_0=10^{-3}$ (fractional), tolerable error $\Delta =1$.

$t_{\text{pred}}=\frac{1}{0.4}\ln \frac{1}{10^{-3}}=\frac{\ln 10^3}{0.4}=\frac{6.9}{0.4}\approx 17\text{days}.$

Yeh step kyun? Hum bas $\Delta ={\delta }_0{e}^{\lambda t}$ ko invert kar rahe hain. Ab sensors 1000× improve karo, ${\delta }_0=10^{-6}$: $t_{\text{pred}}=\frac{\ln 10^6}{0.4}=\frac{13.8}{0.4}\approx 34\text{days}.$ Million-fold better data ne horizon sirf double kiya. Yahi hai woh brutal logarithm.

Worked Example 3 — $\lambda$ ka sign dynamics batata hai

Damped pendulum jo rest mein spiral karta hai: saare ${\lambda }_i<0$ (errors khatam ho jaati hain). Frictionless pendulum: $\lambda =0$ (periodic, errors bounded rehti hain). Driven damped pendulum apne chaotic regime mein: ${\lambda }_1>0$ jabki ${\sum }_i{\lambda }_i<0$ (volume ek strange attractor par shrink karta hai, phir bhi usके andar stretch karta hai).

Volume shrink kaise kar sakta hai jab ${\lambda }_1>0$ hai? Dissipation phase-space volume contract karti hai ($\sum {\lambda }_i<0$), lekin flow saath hi saath ek direction mein stretch bhi karta hai (${\lambda }_1>0$) aur fold bhi — stretch-and-fold hi chaos ka engine hai.

Flashcards

Connections

• Phase space and flows
• Fixed points and linear stability analysis
• Strange attractors and fractal dimension
• The logistic map and period-doubling route to chaos
• Hamiltonian chaos and the KAM theorem
• Driven damped pendulum
• Liouville's theorem (phase-space volume)

Concept Map