Download AKTU B-Tech 8th Sem 2016-17 Non Linear Dynamics System Question Paper

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B. TECH.

THEORY EXAMINATION (SEM–VIII) 2016-17

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NON-LINEAR DYNAMICS SYSTEM

Max. Marks : 100

Note : Be precise in your answer. In case of numerical problem assume data wherever not provided.

SECTION-A

1. Attempt all parts of the following- (10×2=20)

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1. What is a dynamical system?
2. What is a Strange Attractor?
3. What are simple experiments to demonstrate chaos?
4. What is a Cantor set?
5. What is an attractor?

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6. How do I know if my data are deterministic?
7. What is quantum chaos?
8. What are cellular automata?
9. What are solitons?
10. What is spatio-temporal chaos?

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SECTION-B

2. Attempt any five of the following: (10×5=50)

1. State and explain Liapunov's theorems on (i) stability, (ii) asymptotic stability (iii) global asymptotic stability and (iv) instability.
2. Consider the linear autonomous system
   X' =
   

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   0   1

   -1   -2

   X
   Using direct method of Lyapunov, determine the stability of the equilibrium state.
3. Explain Peano's theorem?
4. What is the normal form theory and application to non-linear system?
5. What is a Bifurcation?

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6. What is a degree of freedom? How are maps related to flows (differential equation)?
7. Explain the control of chaos?
8. Describe the different types of solutions.

SECTION-C

Attempt any two of the following: (15×2=30)

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3. For x' = x⁴ - x² + a,

1. Sketch the phase portrait for a = 0.
2. How many bifurcations are taking place in this system as a function of a.
3. In each case, determine the type of bifurcation by reducing to normal form
4. Draw the bifurcation diagram.

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4. for the nonlinear system given by: x' = sin y, y' = x(1 - x² ),

Answer the following questions:

1. How many fixed points does it have. Determine the fixed points of this system.
2. Determine the Jacobian matrix for this system for any arbitrary fixed point (x *, y* ).
3. For the fixed points on x = 0 line, determine the type of fixed points.

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4. Draw the phase portrait ONLY around the fixed points lying on x = 0 line.

5. What is Generic? What is the minimum phase space dimension for chaos?

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