Classifying Critical Points - Differential Equations

1. Consider the system $\vec{x}' = \begin{pmatrix} 3 & 5 \\ -2 & -3 \end{pmatrix} \vec{x}$. Classify the critical point at the origin and determine the direction of rotation of the trajectories, if applicable.
   Answer: Center, clockwise rotation When analyzing linear systems of differential equations, you need to examine the eigenvalues of the coefficient matrix to classify the critical point's behavior and stability. For the matrix $A = \begin{pmatrix} 3 & 5 \\ -2 & -3 \end{pmatrix}$, find the eigenvalues using the characteristic equation $\det(A - \lambda I) = 0$: $\det\begin{pmatrix} 3-\lambda & 5 \\ -2 & -3-\lambda \end{pmatrix} = (3-\lambda)(-3-\lambda) - (5)(-2) = \lambda^2 + 1 = 0$ This gives eigenvalues $\lambda = \pm i$, which are purely imaginary with zero real part. Since the eigenvalues are purely imaginary, the critical point is a center with closed, elliptical trajectories around the origin. The system is neutrally stable—trajectories neither spiral inward nor outward. To determine rotation direction, examine how the vector field moves a test point. At $(1,0)$, the system gives $\vec{x}' = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$. This vector points right and down, indicating clockwise rotation. Looking at the wrong answers: (A) incorrectly identifies this as a stable spiral—that would require eigenvalues with negative real parts, not purely imaginary ones. (B) correctly identifies the center but gets the rotation direction wrong. (D) confuses this with a saddle point, which requires one positive and one negative real eigenvalue. Study tip: Remember that purely imaginary eigenvalues always create centers with periodic orbits. Check rotation direction by evaluating the vector field at a simple test point like $(1,0)$ or $(0,1)$.
2. The origin is a critical point for the system $\vec{x}' = A\vec{x}$. The trace of the $2 \times 2$ matrix $A$ is $-5$. It is also known that one of the eigenvalues of $A$ is $1$. How is the origin classified?
   Answer: Saddle point The trace of a matrix is the sum of its eigenvalues. Let the eigenvalues be $\lambda_1$ and $\lambda_2$. We are given $\tau = \lambda_1 + \lambda_2 = -5$. We are also given that one eigenvalue is $1$, so let $\lambda_1 = 1$. Substituting this into the trace equation gives $1 + \lambda_2 = -5$, which means $\lambda_2 = -6$. The eigenvalues are $1$ and $-6$. Since the eigenvalues are real, distinct, and have opposite signs, the critical point at the origin is a saddle point.
3. For the system $\frac{dx}{dt} = 3x - 2y$, $\frac{dy}{dt} = 5x - 3y$, the characteristic polynomial of the coefficient matrix is $\lambda^2 - 1 = 0$. What additional information is needed to distinguish between the possible critical point types?
   Answer: The sign of the trace, since $\lambda_1 = 1$ and $\lambda_2 = -1$ already determine this is a saddle point From λ² - 1 = 0, we get λ₁ = 1 and λ₂ = -1. These are real eigenvalues with opposite signs (one positive, one negative), which uniquely determines that the critical point is a saddle point. No additional information is needed for classification - the eigenvalues alone are sufficient. The trace confirmation (τ = 1 + (-1) = 0) and determinant (Δ = 1×(-1) = -1 < 0) are consistent with a saddle. Choice B is incorrect because distinct eigenvalues automatically have geometric multiplicity 1. Choice C is unnecessary since we already know the signs differ. Choice D is wrong because real eigenvalues of opposite signs indicate a saddle, not a node.
4. Which of the following systems has a critical point at the origin that is stable but not asymptotically stable?
   Answer: $\vec{x}' = \begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix} \vec{x}$ A critical point is stable but not asymptotically stable if it is a center. A center occurs for a linear system when the eigenvalues are purely imaginary, which corresponds to having trace $\tau = 0$ and determinant $\Delta > 0$. Let's check the options: A) $\tau = 0+0=0$, $\Delta = (0)(0) - (2)(-2) = 4 > 0$. This is a center. Correct. B) $\tau = -1-1=-2$, $\Delta = 1 > 0$. This is a stable node, which is asymptotically stable. C) $\tau = 1+1=2$, $\Delta = 1-4=-3 < 0$. This is a saddle point, which is unstable. D) $\tau = -2+0=-2$, $\Delta = 0$. The eigenvalues are $-2$ and $0$. The system has a line of critical points ($x=0$). A trajectory starting at $(\epsilon, \delta)$ goes to $(0, \delta)$ as $t \to \infty$. Thus, the origin is unstable.
5. For the system $\frac{dx}{dt} = ax + by$, $\frac{dy}{dt} = cx + dy$ where $a, b, c, d$ are constants, the coefficient matrix has determinant $\Delta = 8$ and trace $\tau = -2$. What type of critical point occurs at the origin?
   Answer: Stable spiral because the discriminant is negative and the trace indicates complex eigenvalues with negative real parts The characteristic equation is λ² - τλ + Δ = λ² + 2λ + 8 = 0. The discriminant is τ² - 4Δ = 4 - 32 = -28 < 0, indicating complex eigenvalues. Using the quadratic formula: λ = (-2 ± i√28)/2 = -1 ± i√7. Since the real part is -1 < 0, this is a stable spiral. Choice A is wrong because the eigenvalues are complex, not real. Choice B is wrong for the same reason. Choice D is incorrect because saddle points require real eigenvalues of opposite signs, which would need Δ < 0.
6. A linear system has coefficient matrix with repeated eigenvalue $\lambda = -3$ but geometric multiplicity 1 (only one linearly independent eigenvector). How should this critical point be classified?
   Answer: Stable improper node, with trajectories approaching the origin tangentially to the eigenspace direction When analyzing critical points in linear systems, the key is understanding how eigenvalue properties determine the behavior of solution trajectories near equilibrium. The algebraic multiplicity (how many times an eigenvalue appears) versus geometric multiplicity (number of linearly independent eigenvectors) tells you everything about trajectory structure. With repeated eigenvalue $\lambda = -3$ having geometric multiplicity 1, you have a deficient eigenspace—fewer eigenvectors than the algebraic multiplicity would suggest. Since $\lambda < 0$, the critical point is stable (trajectories approach the origin). The deficient eigenspace creates an improper node, where most trajectories approach tangentially to the single eigenvector direction rather than along multiple straight-line paths. Choice A incorrectly describes a proper node, which would require geometric multiplicity equal to algebraic multiplicity, giving multiple linearly independent eigenvectors and thus multiple straight-line approach directions. Choice C misidentifies this as a spiral node—spirals require complex eigenvalues, not repeated real ones. Choice D uses the non-standard term "degenerate node" and incorrectly emphasizes curved approaches, when the key feature is tangential approach to the eigenspace direction. The Jordan canonical form is indeed needed to analyze this system completely, but that mathematical requirement doesn't change the classification—it's still an improper node. Study tip: Remember the classification hierarchy: negative real eigenvalues mean stability, while geometric multiplicity deficiency (fewer eigenvectors than expected) always creates "improper" nodes with tangential approach patterns, never spirals or multiple straight-line trajectories.
7. The origin is an unstable node for the system $\vec{x}' = A\vec{x}$. Which of the following could be the matrix $A$?
   Answer: $\begin{pmatrix} 3 & 1 \\ -1 & 1 \end{pmatrix}$ When analyzing the stability of a linear system $\vec{x}' = A\vec{x}$, you need to examine the eigenvalues of matrix $A$. An unstable node occurs when both eigenvalues are real and positive, causing solutions to grow exponentially away from the origin along straight-line trajectories. For an unstable node, we need both eigenvalues to be real and positive. Let's check each matrix by finding their eigenvalues using the characteristic equation $\det(A - \lambda I) = 0$. Option C gives us the correct answer. For $\begin{pmatrix} 3 & 1 \\ -1 & 1 \end{pmatrix}$, the characteristic equation is $(3-\lambda)(1-\lambda) - (1)(-1) = \lambda^2 - 4\lambda + 4 = (\lambda - 2)^2 = 0$. This yields a repeated eigenvalue $\lambda = 2 > 0$, creating an unstable node. Option A has eigenvalues that are real but include negative values (you can verify the trace and determinant suggest mixed signs). Option B produces complex eigenvalues since the discriminant of its characteristic equation is negative - this would create a spiral, not a node. Option D has negative eigenvalues (notice the negative diagonal entries and positive determinant), making it a stable node rather than unstable. Remember this key pattern: for a 2×2 system to have an unstable node, both eigenvalues must be real and positive. This happens when the determinant is positive (ensuring real eigenvalues of the same sign) and the trace is positive (ensuring that common sign is positive). Always check both conditions when classifying equilibrium points.
8. Consider the linear system $\vec{x}' = \begin{pmatrix} k & 2 \\ -4 & -k \end{pmatrix} \vec{x}$. For which values of the real parameter $k$ is the critical point at the origin a saddle point?
   Answer: $k > 2\sqrt{2}$ or $k < -2\sqrt{2}$ Let the matrix be $A$. The classification of the critical point depends on the trace $\tau = \mathrm{tr}(A)$ and the determinant $\Delta = \det(A)$. For this system, $\tau = k + (-k) = 0$ and $\Delta = (k)(-k) - (2)(-4) = 8 - k^2$. A critical point is a saddle point if and only if $\Delta < 0$. Therefore, we must have $8 - k^2 < 0$, which simplifies to $k^2 > 8$. Taking the square root of both sides gives $|k| > \sqrt{8} = 2\sqrt{2}$. This corresponds to the range $k > 2\sqrt{2}$ or $k < -2\sqrt{2}$.
9. The motion of a mechanical system is described by the second-order differential equation $x'' + 6x' + kx = 0$. When converted to a first-order system, for what range of values of $k$ is the equilibrium point $(0,0)$ a stable node?
   Answer: $0 < k \le 9$ Let $y=x'$. The second-order equation can be written as the first-order system $\vec{z}' = \begin{pmatrix} 0 & 1 \\ -k & -6 \end{pmatrix} \vec{z}$, where $\vec{z} = \begin{pmatrix} x \\ y \end{pmatrix}$. The trace is $\tau = -6$ and the determinant is $\Delta = k$. For a stable node, we need real, negative eigenvalues. The characteristic equation is $\lambda^2 + 6\lambda + k = 0$. The eigenvalues are real if the discriminant $D = (6)^2 - 4k = 36 - 4k \ge 0$, which implies $k \le 9$. The eigenvalues have the same sign if $\Delta = k > 0$. The eigenvalues are negative if $\tau = -6 < 0$, which is always true. Combining the conditions, we need $0 < k \le 9$. The case $k=9$ corresponds to a repeated eigenvalue (a degenerate or improper stable node), which is still a type of stable node.
10. Consider the nonlinear system $x' = x(2 - x - y)$ and $y' = y(3 - 2x - y)$. This system has a critical point in the first quadrant where $x>0$ and $y>0$. How is this critical point classified?
    Answer: Saddle point First, find the critical point by setting $x'=0$ and $y'=0$. For $x>0, y>0$, we must solve the system $2-x-y=0$ and $3-2x-y=0$. Subtracting the first equation from the second gives $(3-2x-y) - (2-x-y) = 0$, which simplifies to $1-x=0$, so $x=1$. Substituting $x=1$ into the first equation gives $2-1-y=0$, so $y=1$. The critical point is $(1,1)$. Next, find the Jacobian matrix: $J(x,y) = \begin{pmatrix} 2-2x-y & -x \\ -2y & 3-2x-2y \end{pmatrix}$. Evaluate the Jacobian at $(1,1)$: $J(1,1) = \begin{pmatrix} 2-2-1 & -1 \\ -2 & 3-2-2 \end{pmatrix} = \begin{pmatrix} -1 & -1 \\ -2 & -1 \end{pmatrix}$. For this linearized system, the determinant is $\Delta = (-1)(-1) - (-1)(-2) = 1 - 2 = -1$. Since $\Delta < 0$, the critical point is a saddle point.
11. A system $\frac{d\mathbf{x}}{dt} = A\mathbf{x}$ has coefficient matrix $A$ with eigenvalues $\lambda_1 = -2$ and $\lambda_2 = -\frac{1}{2}$. If a small perturbation changes the matrix to $A + \epsilon B$ where $\epsilon$ is small and $B$ is a fixed matrix, which statement about the perturbed system's critical point is most accurate?
    Answer: The critical point will remain a stable node since small perturbations cannot change the fundamental character of hyperbolic critical points When analyzing how perturbations affect linear systems, the key concept is hyperbolic critical points and their structural stability. A critical point is hyperbolic when all eigenvalues have nonzero real parts, and such points are remarkably robust to small changes. Your original system has eigenvalues $\lambda_1 = -2$ and $\lambda_2 = -\frac{1}{2}$, both negative and real, making the origin a stable node and the critical point hyperbolic. The fundamental theorem of structural stability tells us that hyperbolic critical points maintain their essential character under small perturbations - the eigenvalues will shift slightly, but they cannot cross the imaginary axis or change the overall stability picture when $\epsilon$ is sufficiently small. Answer (A) incorrectly assumes perturbations typically create complex eigenvalues. While possible, this isn't guaranteed, and the stability would be preserved regardless. Answer (B) suggests a saddle point could form, but this would require an eigenvalue to become positive, which contradicts the "small perturbation" condition - you'd need the perturbation to overcome the substantial negative real parts. Answer (C) overcomplicates the situation by suggesting the classification depends on preserving specific eigenvalue properties, when structural stability theory already guarantees this for hyperbolic points. Answer (D) correctly identifies that small perturbations cannot fundamentally alter hyperbolic critical points - the stable node character persists. Study tip: Remember that hyperbolic critical points are "structurally stable" - when all eigenvalues have nonzero real parts, small perturbations preserve the qualitative behavior. Focus on whether critical points are hyperbolic when analyzing perturbation effects.
12. For the system $\vec{x}' = \begin{pmatrix} -2 & 1 \\ -5 & c \end{pmatrix} \vec{x}$, the critical point at the origin transitions between being a stable spiral and a stable node at a certain value of the parameter $c$. What is this value of $c$?
    Answer: $c = -2 - 2\sqrt{5}$ The transition between a spiral and a node occurs when the eigenvalues switch from complex to real, which is when the discriminant of the characteristic equation is zero. The trace is $\tau = c-2$ and the determinant is $\Delta = -2c+5$. The discriminant is $D = \tau^2 - 4\Delta = (c-2)^2 - 4(5-2c) = c^2 - 4c + 4 - 20 + 8c = c^2 + 4c - 16$. Setting $D=0$ and solving the quadratic equation $c^2 + 4c - 16 = 0$ gives $c = \frac{-4 \pm \sqrt{16 - 4(1)(-16)}}{2} = \frac{-4 \pm \sqrt{80}}{2} = -2 \pm 2\sqrt{5}$. For the critical point to be stable, we need $\tau = c-2 < 0$ (so $c<2$) and $\Delta = 5-2c > 0$ (so $c<2.5$). The root $c = -2 + 2\sqrt{5} \approx -2 + 2(2.236) = 2.472$ violates the condition $c<2$. The root $c = -2 - 2\sqrt{5} \approx -2 - 4.472 = -6.472$ satisfies both stability conditions. Thus, the transition occurs at $c = -2 - 2\sqrt{5}$.
13. The phase portrait near a critical point shows trajectories that approach the origin along two distinct straight-line directions, while all other trajectories curve away from the origin. Based on the trace-determinant analysis, which conditions on $\tau$ (trace) and $\Delta$ (determinant) are consistent with this behavior?
    Answer: $\Delta < 0$ and $\tau^2 - 4\Delta > 0$, ensuring real eigenvalues of opposite signs with distinct eigendirections When analyzing phase portraits near critical points, you need to connect the visual behavior to the trace-determinant conditions that determine eigenvalue types. The key insight is recognizing what "trajectories approaching along two distinct straight-line directions while others curve away" tells you about the eigenvalues. This description matches a saddle point, which occurs when you have real eigenvalues of opposite signs. The straight-line approaches happen along the eigendirections—one eigenvalue is negative (creating the stable direction where trajectories approach), while the other is positive (creating the unstable direction where trajectories move away). The curved trajectories result from the competing influences of both eigenvalues. Answer D is correct because $\Delta < 0$ guarantees eigenvalues of opposite signs (since $\Delta = \lambda_1 \lambda_2$), and $\tau^2 - 4\Delta > 0$ ensures they're real and distinct, giving you the two separate eigendirections described. Answer A fails because $\tau = 0, \Delta > 0$ gives purely imaginary eigenvalues, creating a center with circular or elliptical trajectories—no straight-line approaches. Answer B produces a stable node since both eigenvalues are negative ($\Delta > 0, \tau < 0$). While trajectories approach along eigendirections, they don't curve away from the origin. Answer C creates an unstable node with a repeated eigenvalue. All trajectories move away from the origin, contradicting the "approach" behavior described. Study tip: Remember the saddle signature: $\Delta < 0$ always means opposite-signed eigenvalues, which creates the mixed approach/repel behavior characteristic of saddle points.
14. A nonlinear system has been linearized near the critical point $(2, -1)$, yielding the Jacobian matrix $J = \begin{pmatrix} -3 & 4 \\ -2 & 1 \end{pmatrix}$. The eigenvalues are $\lambda_1 = -1 + 2i$ and $\lambda_2 = -1 - 2i$. Which classification is correct for the behavior near this critical point?
    Answer: The linearization predicts a stable spiral, and this classification is reliable for the nonlinear system The eigenvalues λ₁ = -1 + 2i and λ₂ = -1 - 2i are complex conjugates with negative real part (-1) and nonzero imaginary parts (±2), indicating a stable spiral in the linearization. Since the real parts are nonzero (hyperbolic case), the linearization theorem guarantees that the nonlinear system has the same qualitative behavior near the critical point. Choice B incorrectly identifies it as unstable. Choice C incorrectly calls it a center (which would require purely imaginary eigenvalues). Choice D incorrectly identifies it as a node rather than a spiral.
15. Consider a predator-prey system that has been linearized around an equilibrium point, yielding the Jacobian $J = \begin{pmatrix} 0 & -a \\ b & 0 \end{pmatrix}$ where $a, b > 0$. What can be concluded about the critical point classification and its biological interpretation?
    Answer: It's a center with eigenvalues $\pm i\sqrt{ab}$, suggesting neutrally stable oscillations, but higher-order terms determine actual stability When analyzing linearized predator-prey systems, you need to find the eigenvalues of the Jacobian matrix to classify the equilibrium point and understand the system's local behavior. To find the eigenvalues of $J = \begin{pmatrix} 0 & -a \\ b & 0 \end{pmatrix}$, solve the characteristic equation $\det(J - \lambda I) = 0$. This gives us $\lambda^2 + ab = 0$, so $\lambda = \pm i\sqrt{ab}$. These are purely imaginary eigenvalues, which classify the equilibrium as a center. In the linearized system, this indicates neutrally stable oscillations around the equilibrium. However, since linearization only captures first-order behavior, the actual stability of the nonlinear system depends on higher-order terms that were neglected during linearization. Looking at the wrong answers: Choice A incorrectly suggests complex eigenvalues with negative real parts, but our eigenvalues have zero real parts. Choice C mistakes the eigenvalues as real ($\pm\sqrt{ab}$), but they're purely imaginary ($\pm i\sqrt{ab}$). Choice D claims positive real eigenvalues, which would occur if we had $\lambda^2 - ab = 0$, but our characteristic equation has $\lambda^2 + ab = 0$. Study tip: For predator-prey Jacobians with this zero-diagonal, opposite-sign off-diagonal structure, expect purely imaginary eigenvalues and a center classification. Remember that linearization around centers doesn't determine true stability—you'd need to analyze the full nonlinear system or use methods like Lyapunov functions to determine whether oscillations grow, decay, or remain constant.
16. A system modeling two competing species is given by $dx/dt = x(3 - x - 2y)$ and $dy/dt = y(2 - x - y)$. The system has a non-trivial equilibrium point $(x_E, y_E)$ where both species coexist ($x_E > 0, y_E > 0$). How is this equilibrium point classified?
    Answer: Saddle point, indicating unstable coexistence. To find the coexistence equilibrium, we set the growth rates to zero for $x>0, y>0$: $3-x-2y=0$ and $2-x-y=0$. Solving this system yields $x=1, y=1$. To classify this point, we compute the Jacobian matrix $J(x,y) = \begin{pmatrix} 3-2x-2y & -2x \\ -y & 2-x-2y \end{pmatrix}$. At $(1,1)$, the Jacobian is $J(1,1) = \begin{pmatrix} -1 & -2 \\ -1 & -1 \end{pmatrix}$. The determinant of this matrix is $\Delta = (-1)(-1) - (-2)(-1) = 1 - 2 = -1$. Since the determinant is negative, the equilibrium point is a saddle point. In the context of population dynamics, a saddle point is unstable, meaning that small perturbations from this equilibrium will lead to one of the species outcompeting the other.
17. For the system $\vec{x}' = \begin{pmatrix} \alpha & -2 \\ 8 & \alpha \end{pmatrix} \vec{x}$ with real parameter $\alpha$, which statement accurately describes the critical point at the origin?
    Answer: The origin is a stable spiral if $\alpha < 0$, a center if $\alpha = 0$, and an unstable spiral if $\alpha > 0$. Let $A$ be the matrix of the system. The trace is $\tau = \alpha + \alpha = 2\alpha$, and the determinant is $\Delta = (\alpha)(\alpha) - (-2)(8) = \alpha^2 + 16$. The discriminant of the characteristic equation is $D = \tau^2 - 4\Delta = (2\alpha)^2 - 4(\alpha^2 + 16) = 4\alpha^2 - 4\alpha^2 - 64 = -64$. Since $D < 0$ for all real $\alpha$, the eigenvalues are always complex conjugates: $\lambda = \frac{\tau \pm i\sqrt{|D|}}{2} = \frac{2\alpha \pm i\sqrt{64}}{2} = \alpha \pm 4i$. The classification depends on the real part of the eigenvalues, which is $\alpha$. If $\alpha < 0$, it is a stable spiral. If $\alpha > 0$, it is an unstable spiral. If $\alpha = 0$, it is a center.
18. Consider the system $\frac{dx}{dt} = -x + ky$, $\frac{dy}{dt} = -kx - y$ where $k$ is a positive parameter. As $k$ increases from 0 to large positive values, how does the classification of the critical point at the origin change?
    Answer: The critical point remains a stable spiral for all $k > 0$, since the discriminant $\tau^2 - 4\Delta = -4k^2 < 0$ The coefficient matrix is A = [[-1, k], [-k, -1]], so trace τ = -2 < 0 and determinant Δ = 1 + k² > 0 for all k > 0. The discriminant is τ² - 4Δ = 4 - 4(1 + k²) = -4k² < 0 for all k > 0. Since the discriminant is always negative (complex eigenvalues) and τ < 0 (negative real parts), the critical point is always a stable spiral for any k > 0. At k = 0, we get eigenvalue λ = -1 with multiplicity 2 (stable proper node), but for any k > 0, it's a stable spiral.

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