Brief Summary of Modelling and Dynamical Systems

Modelling and Dynamical Systems
1st Order Linear ODEs
𝑑𝑥
linear equations: = 𝑎𝑥 + 𝑏 with 𝑥(𝑡) the state variable, 𝑎 ≠ 0 and 𝑏 parameters of the model
𝑑𝑡

𝑏 𝑏
time dep soln: 𝑥(𝑡) = − 𝑎 + (𝑥0 + 𝑎) 𝑒 𝑎𝑡

st.sts & stability:

• no st.st: 𝑎 > 0 then 𝑥(𝑡) → ±∞ as 𝑡 → ∞
𝑏
• st.st: 𝑎 < 0 then 𝑥(𝑡) → 𝑥 ∗ = − 𝑎 as 𝑡 → ∞ and as long as 𝑏 ≥ 0, st.st is asymptotically,
globally stable
• invariant sets: 𝑏, 𝑥0 both positive, then ℝ+ is a positively invariant set
𝑑𝑥 𝑥
logistic equation: = 𝑓(𝑥) = 𝑟𝑥 (1 − 𝑘) : 𝑟 the intrinsic growth rate, 𝑘 the carrying capacity
𝑑𝑡

• st.st: 𝑓(𝑥 ∗ ) = 0 gives 𝑥 ∗ = 0 or 𝑥 ∗ = 𝑘

1st Order Non-Linear ODEs
𝑑𝑥
non-linear equations: 𝑓(𝑥) non-linear, 𝐼. 𝐶: 𝑥(0) = 𝑥0 , 𝑑𝑡 = 𝑓(𝑥)

st.sts & stability: any solution of 𝑓(𝑥 ∗ ) = 0
𝑥(𝑡) 1
• time dep: 𝑡 = ∫𝑥 𝑑𝑠
0 𝑓(𝑠)
• 𝒙(𝒕) not a st.st: tends to a st.st or to ±∞
• well-posed: either 𝑥(𝑡) a st.st or strictly monotonic

Bifurcations
𝑑𝑥
= 𝑓(𝑥; 𝝁), with 𝝁 a vector of model parameters
𝑑𝑡

bifurcation: a small change in 𝝁 results in a qualitative change in the solution
1. Carry out stability analysis to find positions and stabilities of st.st (look for dependence on
parameters)
2. Draw curves which show how the positions/stabilities of st.sts depend on the model
parameters. Stable st.sts have full lines and unstable st.sts have dashed lines
3. Where stabilities or existence of st.sts change, add a cross on the x-axis to indicate a
bifurcation point

Book-Keeping
A region R with boundary 𝜕𝑅, with 𝑥(𝑡) the amount of something in R at time 𝑡, then for 𝑡 > 𝑡0:
𝑥(𝑡) = 𝑥(𝑡0 ) + 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑥 𝑖𝑛 𝑅 𝑖𝑛 (𝑡0 , 𝑡)
− 𝑑𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑥 𝑖𝑛 𝑅 𝑖𝑛 (𝑡0 , 𝑡)
+ 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑥 𝑐𝑟𝑜𝑠𝑠𝑖𝑛𝑔 𝑖𝑛𝑡𝑜 𝜕𝑅 𝑖𝑛 𝑅 𝑖𝑛 (𝑡0 , 𝑡)
− 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑥 𝑐𝑟𝑜𝑠𝑠𝑖𝑛𝑔 𝜕𝑅 𝑜𝑢𝑡 𝑜𝑓 𝑅 𝑖𝑛 (𝑡0 , 𝑡)
closed system: no movement across 𝜕𝑅, 𝑥(𝑡) = 𝑥(𝑡0 ) + 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 − 𝑑𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑜𝑛

, conservative system: no production/destruction in , 𝑥(𝑡) = 𝑥(𝑡0 ) + 𝑖𝑛𝑡𝑜 − 𝑜𝑢𝑡 𝑜𝑓
𝑑 𝑡
FTC: ∫ 𝑓(𝑠)𝑑𝑠 = 𝑓(𝑡)
𝑑𝑡 0

Cobweb Diagram
cobweb diagram:
1. Plot 𝑥𝑛+1 as a function of 𝑥𝑛 and plot the straight line 𝑥𝑛+1 = 𝑥𝑛 , the intersections are the
st.sts
2. Choose a starting point on the 𝑥𝑛 axis and label it 𝑥0 , then determine 𝑥1 from the line for
the linear map
3. Marking 𝑥1 on 𝑥𝑛+1 axis, move horizontally across to the line 𝑥𝑛+1 = 𝑥𝑛 and mark the point.
Then move vertically up to the line 𝑥𝑛+1 = 𝜆𝑥𝑛 + 𝜇 to get 𝑥2
4. Repeat to see how the system evolves

Discrete Maps
𝑥𝑛+1 = 𝑓(𝑥𝑛 ; 𝝁 ) with 𝑥𝑛 the state variable (may be a vector), 𝑛 ∈ 𝑇 ⊂ ℤ+ the counter, 𝝁 ∈ ℝ𝑚 the
set of model parameters and 𝑓 the mapping from 𝑥𝑛 to 𝑥𝑛+1
well-posedness: unique, 𝑓 𝑛 ∈ 𝑋 for all n, continuous dependence
st.sts: any 𝑥 ∗ such that if 𝑥0 = 𝑥 ∗ then 𝑥𝑛 = 𝑥 ∗ for all n, 𝑓(𝑥 ∗ ) = 𝑥 ∗

• stable: if for any 𝜀 > 0, ∃ 𝛿 > 0 st |𝑥𝑛 − 𝑥 ∗ | < 𝜀 for all 𝑛 whenever |𝑥0 − 𝑥 ∗ | < 𝛿
• asym. stable: if above true and |𝑥𝑛 − 𝑥 ∗ | → 0 as 𝑛 → ∞
• unstable: if it isn’t stable
positively invariant sets: 𝑌 ⊂ 𝑋 if 𝑥0 ∈ 𝑌 implies 𝑥𝑛 ∈ 𝑌 for all 𝑛 > 0, 𝑓: 𝑌 → 𝑌
Linear Maps: 𝑥𝑛+1 = 𝜆𝑥𝑛 + 𝜇
𝑐𝑓
• find a complementary function, 𝑥𝑛 , which satisfies the homo. eqn
• then find a particular solution, 𝑥𝑛𝑝𝑠
• use initial conditions
𝜇
• general solution from 4.2.1 is 𝑥𝑛 = 𝑥0 𝜆𝑛 + 1−𝜆(1−𝜆𝑛 )
𝜇
st.st solutions: 𝑥𝑛+1 = 𝑥𝑛 = 𝑥 ∗ so 𝑥 ∗ = 1−𝜆

• 𝜆 > 1: then |𝑥𝑛 | → ∞ and 𝑥𝑛 changes monotonically as 𝑛 → ∞
• 𝜆 < −1: then |𝑥𝑛 | → ∞ and 𝑥𝑛 changes in an oscillatory way as 𝑛 → ∞
• 0 < 𝜆 < 1: then 𝑥𝑛 → 𝑥 ∗ monotonically as 𝑛 → ∞
• −1 < 𝜆 < 1: then 𝑥𝑛 → 𝑥 ∗ in an oscillatory way as 𝑛 → ∞
Non-Linear Maps
𝑥𝑛+1 = 𝑓(𝑥𝑛 )
st.sts: 𝑥 ∗ = 𝑓(𝑥 ∗ )
Linear Stability Analysis
𝑥𝑛+1 = 𝑥 ∗ + 𝑦𝑛+1 = 𝑓(𝑥 ∗ + 𝑦𝑛 ) and Taylor Expand to get 𝑦𝑛+1 = 𝑓′(𝑥 ∗ )𝑦𝑛

• 𝑓 ′ (𝑥 ∗ ) > 1 then |𝑥𝑛 | → ∞ and 𝑥𝑛 changes monotonically as 𝑛 → ∞
• 𝑓 ′ (𝑥 ∗ ) < −1 then |𝑥𝑛 | → ∞ and 𝑥𝑛 changes in an oscillatory way as 𝑛 → ∞