2.1 First Order non-Linear ODE Flashcards
State the two possibilities of the ODEs we study
Deterministic (continuous) and Stochastic
- 1 degree of freedom
State the determinstic assumption
Given a q(t(0)), we have an equation that uniquely determines q(t) for all subsequent t
- Only need to look at 1 trajectory
State the stochastic assumption
When the equation contains terms that are only known statistically (ie due to random noise)
Then we need to look at multiple trajectories
State the general equation for population growth and define all the terms
dq/dt = αq - βq^3
q(t) - population density
α - Initial growth rate
β - Saturation term at high population
What are the three steps to solving an ODE
- Find the fixed points - time independent solutions
- Examine the stability of the fixed points
- Phase plane analysis
Describe the solutions for q for a q-alpha graph
q = 0 is a solution for all alpha
Then q = +- sqrt(alpha/beta)
Describe the process used in examining the stability of the fixed points of q
Look in the vicinity of q
q(t) = q bar + 𝛿q(t), and sub into dq/dt
q bar is a constant, 𝛿q is small
How do we linearize the q equation?
Ignore terms larger than (𝛿q)^2
State the solution for 𝛿q(t)
𝛿q(t) = 𝛿q(0)e^ηt
What are the two possible forms of 𝛿q(t) = 𝛿q(0)e^ηt ?
η < 0: Exponential type decay
𝛿q(t) -> e^-ηt -> 0 as t -> Infinity
q(t) -> q bar STABLE
η < 0: Exponential type growth
𝛿q(t) -> e^+ηt -> Infinity as t -> infinity
q(t) -> infinity UNSTABLE
What behviour do real systems exhibit for 𝛿q(t) at large t?
They are stable as t -> infinity
How would we describe the fixed points for q and q bar?
Stable: If there is a small perturbation, q will remain at q bar
Unstable: A small perturbation will move the system away from q bar
