Dynamical Systems MCQs

\[ \textbf{Difficult MCQs on Dynamical Systems with Answers (Mathematics)} \] \[ \textbf{Q1: The eigenvalues of the Jacobian matrix at a fixed point determine:} \] \[ \text{(A) } \text{The stability and nature of the fixed point} \] \[ \text{(B) } \text{The system’s energy dissipation} \] \[ \text{(C) } \text{The system’s time period} \] \[ \text{(D) } \text{The divergence of trajectories in a chaotic system} \] \[ \textbf{Answer: (A) The stability and nature of the fixed point} \] \[ \textbf{Q2: Consider the dynamical system \( \dot{x} = x(1 – x) \). The fixed points are:} \] \[ \text{(A) } x = 0, x = 1 \] \[ \text{(B) } x = 0 \text{ only} \] \[ \text{(C) } x = -1, x = 1 \] \[ \text{(D) } x = 1 \text{ only} \] \[ \textbf{Answer: (A) \(x = 0, x = 1\)} \] \[ \textbf{Q3: The Hartman-Grobman theorem states that:} \] \[ \text{(A) } \text{Linear and nonlinear systems behave the same near a hyperbolic fixed point} \] \[ \text{(B) } \text{Every system has a periodic solution} \] \[ \text{(C) } \text{Chaos occurs in systems with more than two dimensions} \] \[ \text{(D) } \text{The system is globally linearizable} \] \[ \textbf{Answer: (A) Linear and nonlinear systems behave the same near a hyperbolic fixed point} \] \[ \textbf{Q4: A two-dimensional system \( \dot{x} = f(x), \dot{y} = g(y) \) is classified as conservative if:} \] \[ \text{(A) } \frac{\partial f}{\partial y} + \frac{\partial g}{\partial x} = 0 \] \[ \text{(B) } \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} = 0 \] \[ \text{(C) } \frac{\partial f}{\partial x} – \frac{\partial g}{\partial y} = 0 \] \[ \text{(D) } \frac{\partial f}{\partial x} + \frac{\partial g}{\partial x} = 0 \] \[ \textbf{Answer: (B) \( \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} = 0 \)} \] \[ \textbf{Q5: The stability of a limit cycle is determined using:} \] \[ \text{(A) } \text{Lyapunov exponents} \] \[ \text{(B) } \text{Floquet multipliers} \] \[ \text{(C) } \text{The Jacobian matrix eigenvalues} \] \[ \text{(D) } \text{The divergence of the vector field} \] \[ \textbf{Answer: (B) Floquet multipliers} \] \[ \textbf{Q7: The linearization of a dynamical system fails to capture behavior near a fixed point when:} \] \[ \text{(A) } \text{The fixed point is hyperbolic} \] \[ \text{(B) } \text{The Jacobian matrix has nonzero eigenvalues} \] \[ \text{(C) } \text{The Jacobian matrix has zero or purely imaginary eigenvalues} \] \[ \text{(D) } \text{The system is in \( \mathbb{R}^3 \)} \] \[ \textbf{Answer: (C) The Jacobian matrix has zero or purely imaginary eigenvalues} \] \[ \textbf{Q8: A saddle point in a two-dimensional phase space is characterized by:} \] \[ \text{(A) } \text{Two real eigenvalues of opposite signs} \] \[ \text{(B) } \text{Two complex conjugate eigenvalues} \] \[ \text{(C) } \text{Two purely imaginary eigenvalues} \] \[ \text{(D) } \text{A zero eigenvalue} \] \[ \textbf{Answer: (A) Two real eigenvalues of opposite signs} \] \[ \textbf{Q9: The Poincaré map is used to:} \] \[ \text{(A) } \text{Reduce the dimensionality of continuous systems} \] \[ \text{(B) } \text{Determine the stability of a limit cycle} \] \[ \text{(C) } \text{Linearize a nonlinear system} \] \[ \text{(D) } \text{Evaluate the divergence of a vector field} \] \[ \textbf{Answer: (B) Determine the stability of a limit cycle} \] \[ \textbf{Q10: A Hopf bifurcation occurs when:} \] \[ \text{(A) } \text{A fixed point becomes unstable and gives rise to a periodic orbit} \] \[ \text{(B) } \text{Two fixed points merge and annihilate each other} \] \[ \text{(C) } \text{A system transitions to chaos} \] \[ \text{(D) } \text{Eigenvalues transition from the negative to positive real axis} \] \[ \textbf{Answer: (A) A fixed point becomes unstable and gives rise to a periodic orbit} \]

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