Stability of ODE Solutions

<p>It traces out an ellipse indefinitely. (C)</p> Signup and view all the answers

<p>By ensuring (a, b) lie within a circle of radius δ. (C)</p> Signup and view all the answers

<p>The solutions merely rotate but do not converge to the origin. (D)</p> Signup and view all the answers

<p>Elliptical trajectories satisfying $y_1^2 + 4y_2^2 = a^2 + 4b^2$. (B)</p> Signup and view all the answers

<p>The trajectories do not converge to the origin. (C)</p> Signup and view all the answers

<p>The maximum real part of the eigenvalues (A)</p> Signup and view all the answers

<p>When the maximum real part of the eigenvalues s is less than zero (B)</p> Signup and view all the answers

<p>One or more solution components grow without bound (B)</p> Signup and view all the answers

<p>They are purely imaginary and may be complex conjugates (D)</p> Signup and view all the answers

<p>The system is asymptotically stable (A)</p> Signup and view all the answers

<p>The zero solution is unstable for λ &gt; 0 and stable for λ = 0 (C)</p> Signup and view all the answers

<p>The system is asymptotically stable (C)</p> Signup and view all the answers

<p>They follow elliptical paths at a constant distance from the origin (B)</p> Signup and view all the answers

<p>There exists a delta such that initial conditions remain within a tube around the solution. (C)</p> Signup and view all the answers

<p>The condition causes the solution to converge to zero as time approaches infinity. (B)</p> Signup and view all the answers

<p>Conditions of Lyapunov and asymptotic stability are independent of each other. (A)</p> Signup and view all the answers

<p>∀ $ heta$ &gt; 0 ∃ ε &gt; 0 s.t. ∀ t &gt; 0 |y(0) − y∗(0)| &lt; ε ⇒ |y(t) − y∗(t)| &lt; θ. (C)</p> Signup and view all the answers

<p>It reduces the problem to studying the stability of the zero solution. (D)</p> Signup and view all the answers

<p>The solution will eventually converge to the equilibrium solution. (C)</p> Signup and view all the answers

<p>The existence of periodic solutions. (C)</p> Signup and view all the answers

<p>The difference between the actual and the solution remains bounded. (A)</p> Signup and view all the answers

<p>V(y) = 0 for y = 0 and V(y) &gt; 0 for y ≠ 0 (C)</p> Signup and view all the answers

<p>The orbital derivative is negative. (D)</p> Signup and view all the answers

<p>A system where the vector field is derived from a potential function. (A)</p> Signup and view all the answers

<p>The equilibrium solution is Lyapunov stable. (D)</p> Signup and view all the answers

<p>Both partial derivatives of V must equal zero at y = 0. (B)</p> Signup and view all the answers

<p>V(y) has a relative minimum at y = 0. (A)</p> Signup and view all the answers

<p>V(y) can be negative for certain y values. (B)</p> Signup and view all the answers

<p>The trajectories are converging to the equilibrium. (C)</p> Signup and view all the answers

<p>$old{v} = rac{ ext{d}V}{ ext{d}t} = rac{ ext{d}V}{ ext{d}y_1} y_1' + rac{ ext{d}V}{ ext{d}y_2} y_2'$ (D)</p> Signup and view all the answers

<p>D_f(V) ≤ 0 for all (y1, y2) ≠ (0, 0) (C)</p> Signup and view all the answers

<p>The third condition must be replaced by D_f(V) &lt; 0. (A)</p> Signup and view all the answers

<p>V(y) = 0 for y = 0 (C)</p> Signup and view all the answers

<p>V(y) &gt; 0 when |y| &lt; R (B)</p> Signup and view all the answers

<p>The directional derivative of V along the function f (A)</p> Signup and view all the answers

<p>V(y) must be continuously differentiable. (B)</p> Signup and view all the answers

<p>The solutions will converge to the stability point. (C)</p> Signup and view all the answers

<p>Both Reλ1 and Reλ2 less than 0 (D)</p> Signup and view all the answers

<p>At least one of Reλ1 or Reλ2 is positive (D)</p> Signup and view all the answers

<p>Stability depends on the nonlinear terms present in the system (A)</p> Signup and view all the answers

<p>-1 &lt; a &lt; 3 (B)</p> Signup and view all the answers

<p>The eigenvalues of the matrix A (C)</p> Signup and view all the answers

<p>λ^2 + 2λ + (a^2 - 2a - 3) = 0 (C)</p> Signup and view all the answers

<p>The zero solution is asymptotically stable (D)</p> Signup and view all the answers

<p>The zero solution is unstable (B)</p> Signup and view all the answers

A solution y*(t) of a system of differential equations is Lyapunov stable if, for any small distance (epsilon) from y*(t), we can find a small region around the initial condition of y*(t) where any solution starting within that region will remain within a distance (epsilon) of y*(t) for all future times. It means that small changes in the initial conditions don't lead to drastically different long-term behavior.

A solution y*(t) of a system of differential equations is asymptotically stable if it is Lyapunov stable and, for any small region around the initial condition of y*(t), any solution starting within that region will converge to y*(t) as time goes to infinity.

A function y*(t) is a solution to a differential equation if it makes the equation true when plugged in. For example, if y(t) = e^t is plugged into the equation y' = y, the equation becomes e^t = e^t which is true, so y(t) = e^t is a solution.

The behavior of a system may be influenced by small changes in the initial conditions. If the initial conditions determine very different outcomes even for very small changes, the system is sensitive to initial conditions. This is a crucial feature of chaotic systems.

A differential equation is a mathematical equation that relates a function with its derivatives. It describes how a quantity changes over time or with some other independent variable.

A set of differential equations that describe how multiple quantities change over time and are dependent on each other.

A system is stable if its solutions stay bounded (within a certain limit) for all times. This means that the system doesn't blow up or become infinitely large.

A system is unstable if its solutions become unbounded (grow infinitely large) as time goes on.

For any positive epsilon, there exists a positive delta such that if the initial condition z(0) is within a distance of delta from the origin, then for all time t greater than zero, the solution z(t) remains within a distance of epsilon from the origin.

The zero solution is Lyapunov stable and, in addition, as time approaches infinity, the solution converges towards the origin.

The behavior of the system near a specific state (often the origin) where small perturbations in the initial conditions don't lead to significant deviations in the system's trajectory over time.

A point in the state space of a dynamical system where the system's state remains constant over time. It is a special state that doesn't change.

A mathematical method used to analyze the stability of a dynamical system. It involves finding a function (Lyapunov function) whose value decreases along system trajectories.

A function that decreases along system trajectories approaching a stable equilibrium point. It helps to determine the stability of a system.

A numerical representation of the system's state at a given time. It captures the values of all relevant variables describing the system.

A system is called asymptotically stable if all solutions, starting close to the equilibrium point, not only stay close but also approach the equilibrium point as time goes to infinity.

A system is called stable if all solutions starting close to the equilibrium point stay close to it for all time but may not necessarily approach it.

A system is called unstable if there are solutions that start close to the equilibrium point but move away from it as time goes to infinity.

The Lyapunov function method is a technique for analyzing the stability of a system of differential equations by constructing a special function that decreases along trajectories.

The eigenvalues of the matrix A in the system y˙ = Ay determine the stability of the system. For example, if all eigenvalues have negative real parts, the system is asymptotically stable, and if at least one eigenvalue has a positive real part, the system is unstable.

The system y˙ = Ay, where A is a constant matrix, is called a linear system of differential equations. Its stability depends on the eigenvalues of the matrix A.

If the eigenvalues of a matrix A are all real and negative, the zero solution of the system y˙ = Ay is asymptotically stable. This means that all solutions approach the origin as time goes to infinity.

If the eigenvalues of a matrix A are complex with negative real parts, the zero solution of the system y˙ = Ay is asymptotically stable. This means that all solutions approach the origin as time goes to infinity, but they may oscillate as they do so.

The derivative of a Lyapunov function V(y) along the flow of a system of differential equations. It represents how the Lyapunov function changes as the system evolves over time.

A system where the derivative of the Lyapunov function along the flow is non-positive. This indicates that the Lyapunov function doesn't increase as the system evolves over time, meaning the system tends to stay close to the equilibrium point.

The condition that forces a Lyapunov function's derivative along the flow to strictly decrease. This implies not only stability but also convergence towards the equilibrium point.

A function V(y) that is positive everywhere except at the equilibrium point, where it is zero.

The equation that defines the change of a Lyapunov function with respect to time when the system is evolving according to the differential equations.

A method to analyze the stability of a system of differential equations by constructing a Lyapunov function. This function decreases along the trajectories of the system, indicating its stability.

A type of dynamical system where the time derivative of each state variable is directly proportional to the negative gradient of a potential function.

A state where the system's variables remain constant over time.

If both eigenvalues of a matrix A have negative real parts, the zero solution of the system of ODEs represented by y' = Ay is asymptotically stable.

If at least one eigenvalue of a matrix A has a positive real part, the zero solution of the system of ODEs represented by y' = Ay is unstable.

The linear part of a system of ODEs is obtained by discarding the nonlinear terms, which can be seen as a Taylor expansion approximation.

The stability of a system refers to its behavior near a specific state (often the origin). A stable system will maintain its state over time, while an unstable system will deviate from it.

To determine the stability of a zero solution, we can analyze the eigenvalues of the corresponding linear part of the system.

The behavior of a system with complex conjugate eigenvalues is determined by the real part of the eigenvalues. If the real part is negative, the system is asymptotically stable.

The stability of the zero solution of a system of ODEs doesn't always rely solely on the linear terms. If the maximum real part of the eigenvalues is zero, the nonlinear terms play a crucial role in determining stability.

The range of values for a parameter that results in a stable or unstable zero solution is determined by analyzing the eigenvalues based on the parameter's value.