Stability of ODE Solutions

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Questions and Answers

What does the definition of Lyapunov stability imply about the zero solution z = 0?

Trajectories approach z = 0 over time.

It can be unaffected by any small perturbation.

For any $ ext{ε} > 0$, a corresponding δ can ensure |z(t)| remains bounded. (correct)

It implies a fixed point at any point other than z = 0.

Based on the system defined, what are the eigenvalues of the associated matrix A?

2, -2

2i, -2i (correct)

4, -4

0, 0

What kind of stability does the zero solution y1 = y2 = 0 exhibit?

Both Lyapunov and asymptotic stability.

Only asymptotic stability.

Only Lyapunov stability. (correct)

Neither Lyapunov nor asymptotic stability.

How does the general solution of the system behave over time?

It traces out an ellipse indefinitely. Signup and view all the answers

How is the relationship between the initial conditions (a, b) and the stability of the system established?

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

What implication does the system's behavior as t approaches infinity have on its stability classification?

The solutions merely rotate but do not converge to the origin. Signup and view all the answers

What is the mathematical representation of the trajectories determined by this system?

Elliptical trajectories satisfying $y_1^2 + 4y_2^2 = a^2 + 4b^2$. Signup and view all the answers

Why is the zero solution not considered asymptotically stable?

The trajectories do not converge to the origin. Signup and view all the answers

What does the parameter s represent in the context of stability for systems of first-order linear ODEs?

The maximum real part of the eigenvalues Signup and view all the answers

When is the zero solution y = 0 considered asymptotically stable?

When the maximum real part of the eigenvalues s is less than zero Signup and view all the answers

What happens to the solutions of a system when s is greater than zero?

One or more solution components grow without bound Signup and view all the answers

What is the characteristic of the eigenvalues when s is equal to zero?

They are purely imaginary and may be complex conjugates Signup and view all the answers

For a system with eigenvalues λ1 and λ2, what can be inferred if both have a negative real part?

The system is asymptotically stable Signup and view all the answers

What conclusion can be drawn about a system with repeated eigenvalues λ1 = λ2 = λ?

The zero solution is unstable for λ > 0 and stable for λ = 0 Signup and view all the answers

What does it indicate if the modulus of the solution approaches zero as time tends to infinity?

The system is asymptotically stable Signup and view all the answers

What describes the trajectories when the eigenvalues are purely imaginary in a linear ODE system?

They follow elliptical paths at a constant distance from the origin Signup and view all the answers

What is required for a solution to be considered Lyapunov stable?

There exists a delta such that initial conditions remain within a tube around the solution. Signup and view all the answers

What additional condition defines asymptotic stability beyond Lyapunov stability?

The condition causes the solution to converge to zero as time approaches infinity. Signup and view all the answers

Which of the following statements about the definitions of stability is true?

Conditions of Lyapunov and asymptotic stability are independent of each other. Signup and view all the answers

Which notation is used to succinctly convey the conditions for Lyapunov stability?

∀ $ heta$ > 0 ∃ ε > 0 s.t. ∀ t > 0 |y(0) − y∗(0)| < ε ⇒ |y(t) − y∗(t)| < θ. Signup and view all the answers

What happens to the stability investigation when changing variables to z(t) = y(t) - y∗(t)?

It reduces the problem to studying the stability of the zero solution. Signup and view all the answers

What does the condition |y(0) − y∗(0)| < δ imply in the context of asymptotic stability?

The solution will eventually converge to the equilibrium solution. Signup and view all the answers

Which factor does not contribute directly to establishing stability in the context of the definitions provided?

The existence of periodic solutions. Signup and view all the answers

Which of the following describes the implications of the condition |y(t) − y∗(t)| < ε in stability theory?

The difference between the actual and the solution remains bounded. Signup and view all the answers

What are the two conditions that V(y) must satisfy to be a valid Lyapunov function?

V(y) = 0 for y = 0 and V(y) > 0 for y ≠ 0 Signup and view all the answers

What indicates that the zero solution is asymptotically stable?

The orbital derivative is negative. Signup and view all the answers

In the context of this system, what does the term 'gradient flow' refer to?

A system where the vector field is derived from a potential function. Signup and view all the answers

What is the implication if V(y) is a Lyapunov function for the gradient flow?

The equilibrium solution is Lyapunov stable. Signup and view all the answers

What must occur for y = 0 to be categorized as an equilibrium solution?

Both partial derivatives of V must equal zero at y = 0. Signup and view all the answers

What does the equation ∂V/∂y = 0 imply if y = 0 is an equilibrium solution?

V(y) has a relative minimum at y = 0. Signup and view all the answers

Which statement about the function V(y) is incorrect when it is considered a Lyapunov function?

V(y) can be negative for certain y values. Signup and view all the answers

What can be deduced if Df(V) is non-positive?

The trajectories are converging to the equilibrium. Signup and view all the answers

What is the expression for the time derivative of a continuously differentiable function V(y) along a solution y(t)?

$old{v} = rac{ ext{d}V}{ ext{d}t} = rac{ ext{d}V}{ ext{d}y_1} y_1' + rac{ ext{d}V}{ ext{d}y_2} y_2'$ Signup and view all the answers

What condition regarding the Lyapunov function V(y) indicates that the zero solution y(t) = 0 is stable?

D_f(V) ≤ 0 for all (y1, y2) ≠ (0, 0) Signup and view all the answers

What is the relationship between the third condition of the Lyapunov Stability Theorem and asymptotic stability?

The third condition must be replaced by D_f(V) < 0. Signup and view all the answers

Which of the following properties must a Lyapunov function V(y) possess at y = 0?

V(y) = 0 for y = 0 Signup and view all the answers

What can be said about the values of the Lyapunov function V(y) for y ≠ 0 within the radius R?

V(y) > 0 when |y| < R Signup and view all the answers

What does the notation D_f(V) represent in the context of Lyapunov functions?

The directional derivative of V along the function f Signup and view all the answers

Which characteristic of V(y) allows determining stability without needing the explicit solution of the differential equation system?

V(y) must be continuously differentiable. Signup and view all the answers

What implication does a strictly negative derivative D_f(V) < 0 have on solutions y(t)?

The solutions will converge to the stability point. Signup and view all the answers

What condition must be met for the zero solution to be asymptotically stable?

Both Reλ1 and Reλ2 less than 0 Signup and view all the answers

Under what conditions is the zero solution unstable?

At least one of Reλ1 or Reλ2 is positive Signup and view all the answers

What does the condition max{Reλ1, Reλ2} = 0 imply regarding the stability of the zero solution?

Stability depends on the nonlinear terms present in the system Signup and view all the answers

For the given system, what values of 'a' lead to instability?

-1 < a < 3 Signup and view all the answers

What factor influences the stability of the linear part of the nonlinear system?

The eigenvalues of the matrix A Signup and view all the answers

What is the characteristic equation derived from the matrix A for the system?

λ^2 + 2λ + (a^2 - 2a - 3) = 0 Signup and view all the answers

What happens to the zero solution when both eigenvalues are complex conjugates?

The zero solution is asymptotically stable Signup and view all the answers

If the roots of the characteristic equation are real and positive, what can be concluded about the zero solution?

The zero solution is unstable Signup and view all the answers

Study Notes

Stability of Solutions of ODEs

The study of stability examines how changes in initial conditions or parameters affect dynamical system solutions (e.g., coefficients in front of derivatives). The main goal is to establish criteria ensuring solution changes are minimal with minor changes to initial conditions or parameters.
Lyapunov Stability: A solution y*(t) is Lyapunov stable (or simply stable) if, for any ε > 0, there exists a δ > 0 such that if another initial condition y(0) is within δ of y*(0), then the solution y(t) corresponding to y(0) remains within ε of y*(t) for all t > 0. This means the solution stays close to the initial solution.
Asymptotic Stability: A solution y*(t) is asymptotically stable if it is Lyapunov stable, and for any initial condition y(0) sufficiently close to y*(0), the solution y(t) approaches y*(t) as t tends to infinity.
Stability of the zero solution is often examined. Changing variables (z(t) = y(t)-y*(t) in the system equation) transforms the problem to analyzing the stability of the zero solution (z(t)=0).
Stability Criteria for Linear Systems: For systems of two first-order linear ODEs with constant coefficients, stability depends on the eigenvalues of the matrix A in the system. If the real parts of eigenvalues are all negative, the zero solution is asymptotically stable; if they are all zero, it is stable; and if any are positive, it is unstable.
Lyapunov Function Method: Instead of directly analyzing the system, a function V(y) is sought, satisfying certain conditions (V(0) = 0, V(y) > 0 for y ≠ 0, and ∂V/∂t ≤ 0). If such a Lyapunov function exists, the equilibrium point is stable (or asymptotically stable if ∂V/∂t < 0).

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Description

This quiz focuses on the stability of solutions to ordinary differential equations (ODEs). It covers concepts such as Lyapunov stability and asymptotic stability, providing criteria that ensure minimal solution changes in response to variations in initial conditions or parameters.