\[
\textbf{Difficult MCQs on Dynamical Systems with Answers (Mathematics)}
\]
\[
\textbf{Q1: The eigenvalues of the Jacobian matrix at a fixed point determine:}
\]
\[
\text{(A) } \text{The stability and nature of the fixed point}
\]
\[
\text{(B) } \text{The system’s energy dissipation}
\]
\[
\text{(C) } \text{The system’s time period}
\]
\[
\text{(D) } \text{The divergence of trajectories in a chaotic system}
\]
\[
\textbf{Answer: (A) The stability and nature of the fixed point}
\]
\[
\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 \( \mathbb{R}^3 \)}
\]
\[
\textbf{Answer: (C) The Jacobian matrix has zero or purely imaginary eigenvalues}
\]
\[
\textbf{Q8: A saddle point in a two-dimensional phase space is characterized by:}
\]
\[
\text{(A) } \text{Two real eigenvalues of opposite signs}
\]
\[
\text{(B) } \text{Two complex conjugate eigenvalues}
\]
\[
\text{(C) } \text{Two purely imaginary eigenvalues}
\]
\[
\text{(D) } \text{A zero eigenvalue}
\]
\[
\textbf{Answer: (A) Two real eigenvalues of opposite signs}
\]
\[
\textbf{Q9: The Poincaré map is used to:}
\]
\[
\text{(A) } \text{Reduce the dimensionality of continuous systems}
\]
\[
\text{(B) } \text{Determine the stability of a limit cycle}
\]
\[
\text{(C) } \text{Linearize a nonlinear system}
\]
\[
\text{(D) } \text{Evaluate the divergence of a vector field}
\]
\[
\textbf{Answer: (B) Determine the stability of a limit cycle}
\]
\[
\textbf{Q10: A Hopf bifurcation occurs when:}
\]
\[
\text{(A) } \text{A fixed point becomes unstable and gives rise to a periodic orbit}
\]
\[
\text{(B) } \text{Two fixed points merge and annihilate each other}
\]
\[
\text{(C) } \text{A system transitions to chaos}
\]
\[
\text{(D) } \text{Eigenvalues transition from the negative to positive real axis}
\]
\[
\textbf{Answer: (A) A fixed point becomes unstable and gives rise to a periodic orbit}
\]