Flashcards
Non Linear Dynamic Systems
Cards in this deck (39)
- 1Taylor Expansion Linearized System
- 2Conclusions based upon Eigenvalues The fixed point is stableλ1, λ2 < 0
- 3Conclusions based upon Eigenvalues The fixed point is unstableλ1, λ2 > 0
- 4Conclusions based upon Eigenvalues The fixed point is a saddle pointλ1 < 0
λ2 > 0 - 5If λ1, λ2 are in CThe fixed point is a spiral or centre point
- 6Conclusions based upon Eigenvalues The fixed point is stable spiralRe(λ1), Re(λ2) < 0
- 7Conclusions based upon Eigenvalues The fixed point is unstable spiralRe(λ1), Re(λ2) > 0
- 8Conclusions based upon Eigenvalues If Re(λ) = 0It is a centre point
- 9Trace of J τ =τ = (∂f/∂x) + (∂g/∂y)
- 10Discriminant of J ∆ =
- 11If the fixed point is a saddle point τ = ∆ = τ^(2) - 4∆ =If the fixed point is a saddle point
∆ < 0 - 12What is the nature of the fixed point: τ < 0 ∆ > 0 τ^(2) - 4∆ > 0Stable Node
- 13What is the nature of the fixed point: τ < 0 ∆ < 0 τ^(2) - 4∆ > 0Saddle Point
- 14What is the nature of the fixed point: τ > 0 ∆ > 0 τ^(2) - 4∆ < 0Unstable Spiral
- 15What is the nature of the fixed point: τ < 0 ∆ > 0 τ^(2) - 4∆ < 0Stable Spiral
- 16What is the nature of the fixed point: τ > 0 ∆ < 0 τ^(2) - 4∆ > 0Saddle Point
- 17What is the nature of the fixed point: τ < 0 ∆ > 0 τ^(2) - 4∆ = 0Star or degenerate node
- 18What is the nature of the fixed point: τ > 0 ∆ > 0 τ^(2) - 4∆ > 0Unstable Node
- 19What is the nature of the fixed point: τ < 0Stable
- 20What is the nature of the fixed point: τ > 0Unstable
- 21What is the nature of the fixed point: τ = 0Centre Point
- 22What is the nature of the fixed point: ∆ = 0It is a non-isolated fixed point
- 23—
- 24Polar Coordinates x = y = x' = y' =x= r cos θ
y = r sin θ
x' = r' cos θ - r θ' sin θ
y' = r' sin θ - r θ' cos θ - 25Definition of a smooth autonomous systemsConsider the system x = Ax , x = (x(t), y(t))
The system is called a smooth system, if the function x(t) has continuous derivatives
When the change of the output is not proportional to the change of the input, the system is nonlinear.
The smooth nonlinear system is called autonomous when A is independent from time t. - 26Definition for the periodical solution in the form of a closed orbit. What is the difference of the closed orbit with limit cycle (LC)?If x(t) is a periodical solution for the linear system x=Ax, then so is cx(t) for any constant c=/0. Hence x(t) is surrounded by a one-parameter family of closed orbits. On the other hand, limit cycles are inherently nonlinear phenomena and they are isolated closed trajectories. i.e. their neighbouring trajectories are not closed
- 27Give a definition of a Lyapunov function and explain how it can be used for proving the stability of critical pointsA Liapunov function is a continuous, differentiable, real-valued function V(𝑥⃗)>0 with the following
properties:
a) 𝑉(𝑥⃗) > 0 for all 𝑥⃗ ≠ 𝑥∗⃗, and 𝑉(𝑥∗⃗) = 0. (We say V is positive definite.)
b) 𝑉̇ < 0 for all 𝑥⃗ ≠ 𝑥∗⃗. (All trajectories flow “downhill” toward 𝑥∗⃗.)
Consider a system 𝑥̇ ⃗ = 𝑓⃗(𝑥⃗) with a fixed point at 𝑥∗⃗. Suppose that we can find a Liapunov
Function. Then 𝑥∗⃗ is globally asymptotically stable: for all initial conditions, 𝑥⃗ → 𝑥∗⃗, as 𝑡 → ∞.
In particular, the system has no closed orbits. - 28Theorem for Conservative SystemsConsider the system x' = f(x), where x ∈ R^2 and f is continuously
differentiable. Let’s assume that ∃E(x) (a conserved quantity) and
x* an isolated fixed point. If x*
is a local minimum of E, then all
trajectories sufficiently close to x*
are closed (i.e. the linear centre
is a nonlinear centre too) - 29What is a Liapunov FunctionA continuously differentiable, real-valued function V(x) with the
following properties:
1) V(x) > 0, ∀x ~= x*
2) V(x*) = 0, where x* is the fixed point.
3) V' < 0, ∀x ~= x* - 30How many fixed points in a linear system? How many fixed points can have a nonlinear autonomous system?An equilibrium solution of the system 𝑥̇ ⃗ = A𝑥⃗ is a point (x1, x2) where 𝑥̇ ⃗ = 0, that is, where x1' =
x2'= 0. An equilibrium solution is a constant solution of the system, and is usually called a fixed
point. For a linear system 𝑥̇ ⃗ = 𝐴𝑥⃗, an equilibrium solution occurs at each solution of the system (of
homogeneous algebraic equations) Ax = 0. As we have seen, such a system has exactly one solution,
located at the origin, if det(A) ≠ 0. If det(A) = 0, then there are infinitely many solutions. - 31Name different types of critical points in two-dimensional nonlinear autonomous system, give their characterisation in terms of linearized system eigenvalues and eigenvectors, sketch the phase portrait for these critical points.Repellers (also called sources): both eigenvalues have positive real part.
Attractors (also called sinks): both eigenvalues have negative real part.
Saddles: one eigenvalue is positive and one is negative.
Centers: both eigenvalues are pure imaginary.
Higher-order and non-isolated fixed points: at least one eigenvalue is
zero. - 32θ' =
- 33Working out r'
- 34Saddle-node Bif'n1. If µ > 0 the fixed points exist and are two (±
õ, 0).
2. If µ = 0 the fixed points collide in one fixed point (0, 0).
3. If µ < 0 there are no fixed points (ghost).
Occurs at λ1 = 0 or λ2 = 0
- 35Transcritical Bif'n1. If µ < 0 the fixed points exist and are two (0, 0) and (µ, 0).
2. If µ = 0 the fixed points collide in one fixed point (0, 0).
3. If µ > 0 there are two fixed points again at (0, 0) and (µ, 0)
Occurs at λ1 = 0 or λ2 = 0
- 36Supercritical Pitchfork Bif’n1. If µ < 0 the fixed point exists at (0, 0).
2. If µ = 0 the fixed point is still at (0, 0).
3. If µ > 0 there are three fixed points at (0, 0), (−
õ, 0) and
(
õ, 0).
Occurs at λ1 = 0 or λ2 = 0
- 37Subcritical Pitchfork Bif’nIf µ < 0 there are three fixed points at (0, 0), (−√−µ, 0) and
(√−µ, 0).
2. If µ = 0 the fixed point is still at (0, 0).
3. If µ > 0 the fixed point exists at (0, 0).
Occurs at λ1 = 0 or λ2 = 0
- 38Supercritical Hopf Bif’nOccurs when two complex conjugate eigenvalues simultaneously cross the
imaginary axis of the Im λ vs. Re λ graph.
- 39Subcritical Hopf Bif’nOccurs when two complex conjugate eigenvalues simultaneously cross the
imaginary axis of the Im λ vs. Re λ graph.
Information
- Description
- Degree Mathematics Flashcards on Non Linear Dynamic Systems, created by Benjamin Smith on 21/03/2023.
- Created by
- Benjamin Smith
- Views
- 0
- Rating