D YNAMICAL S YSTEMS 2 I NSTRUCTOR : G IANNI A. D I C ARO V ECTOR - - PowerPoint PPT Presentation

LECTURE 3: DYNAMICAL SYSTEMS 2

INSTRUCTOR: GIANNI A. DI CARO

15-382 COLLECTIVE INTELLIGENCE – S18

2

VECTOR FIELDS AND ORBITS

ሶ 𝑦 = 2𝑦 = 𝑔

𝑦 (𝑦, 𝑧)

ሶ 𝑧 = −3𝑧 = 𝑔

𝑧 (𝑦, 𝑧)

• 𝒈 is a vector field in ℝ𝑜: a function

associating a vector to 𝑜-dim point 𝒚

• Solution: 𝑦0𝑓2𝑢, 𝑧0𝑓−3𝑢

Vector field Rate of change, velocity Phase portrait

• Autonomous system  no dependence from time, all information about

the solution is represented

• A fundamental theorem guarantees (under differentiability and

continuity assumptions) that two orbits corresponding to two different initial solutions never intersect with each other Orbits / Possible trajectories Flow: 𝐺(𝑢, 𝑦 𝑢0 ) Uncoupled system

𝒈 = (2𝑦, −3𝑧)

Direction and speed of solution for any (𝑦, 𝑧)

3

VECTOR FIELDS, ORBITS, FIXED POINTS

ሶ 𝑦 = 𝑧 = 𝑔

𝑦 (𝑦, 𝑧)

ሶ 𝑧 = −𝑦 − 𝑧2 = 𝑔

𝑧 (𝑦, 𝑧)

Closed (periodic) orbit Equilibrium point

• 𝒚∗ is an equilibrium (fixed) point of the ODE if 𝒈 𝒚∗ = 𝟏
• ↔ Once in 𝑦∗, the system remains there: 𝒚∗ = 𝒚 𝑢; 𝒚∗ , 𝑢 ≥ 0

Direction of increasing time

4

LINEAR MODEL FOR POPULATION GROWTH

• Linear model of population growth (Malthus model,

1798)

• Works well for small populations
• 𝑦 = size of population, 𝑏 = growth rate

Phase portrait ሶ 𝑦 = 𝑏𝑦 𝑦(0) = 𝑦0 Solution orbits / Flow:

𝑦(𝑢) = 𝑦0𝑓𝑏𝑢

(a) 𝑏>0: Exponential growth (b) 𝑏 < 0: Exponential decrease

5

LOGISTIC MODEL FOR POPULATION GROWTH

• General form for population growth:

𝑒𝑂 𝑒𝑢 = 𝑔(𝑂)

• What is a good model that captures essential aspects?

 Every living organism must have at least one parent of like kind  In a finite space, due to the limiting effect of the environment, there is an upper limit to the number of organisms that can occupy that space: resources competition constraint

•  Logistic model (1838), non-linear:

𝑒𝑂 𝑒𝑢 = 𝑠𝑂 1 − 𝑂 𝐷 𝑠 = intrinsic rate of increase [1/t] 𝐷 = max carrying capacity [# individuals] 𝑂0 = 𝑂(0)

•  Non-dimensional equation with no parameters:

𝑒𝑜 𝑒τ = 𝑜 1 − 𝑜

𝑜0 =

𝑂0

𝐷

(dimensionless time)

𝑜 = 𝑂 𝐷 ∈ [0,1] (dimensionless population)

τ = 𝑠𝑢

6

LOGISTIC MODEL FOR POPULATION GROWTH

• The logistic equation, even if not linear, can be integrated by

separation of variables:

𝑒𝑜 𝑒τ = 𝑜 1 − 𝑜 , 𝑒𝑜 𝑜 1−𝑜 = 𝑒τ,

׬

𝑒𝑜 𝑜 1−𝑜 = ׬ 𝑒τ

න 𝑒𝑜 𝑜 + න 𝑒𝑜 1 − 𝑜 = න 𝑒τ

ln 𝑜 − ln 1 − 𝑜 = τ + 𝐿 ln 1 − 𝑜 𝑜 = −τ − 𝐿 1 𝑜 − 1 = 𝑓−τ−𝐿 1 𝑜 = 1 + 𝑏𝑓−τ 𝑜(τ) = 1 1 + 𝑏𝑓−τ

The integration constant 𝑏 depends on the initial condition 𝑜0

𝑂(𝑢) = 𝐷 1 + 𝐵𝑓−𝑠𝑢 𝐵 = 𝐷 − 𝑂0 𝑂0

7

LOGISTIC MODEL FOR POPULATION GROWTH

𝑜(τ) = 1 1 + 𝑏𝑓−τ

𝑏=1

𝑔 𝑜 = 𝑜 1 − 𝑜 = 0  𝑜 =1, 𝑜 = 0 Equilibrium points: Flow, different 𝑏 values

𝑜 𝑜 𝑜 =1 𝑜 = 0

Phase portrait Flow function 𝐺(𝑢, 𝑜0) is not defined for all values of 𝑢 Asymptotic divergence

𝑜 = 0 𝑜 = 1 𝑜

8

(BASIC) LOGISTIC MODEL: DOES IT WORK?

• Population of the US in 1800: 5.3 millions
• Population of the US in 1850: 23.1 millions

 Predict population in 1900 and 1950 Answer: 76 (1900), 150.7 (1950)

• Let’s look first at what the linear (i.e., exponential growth) model would predict:

𝑂(𝑢) = 𝑂0𝑓𝑏𝑢  We need first to derive an estimate for growth parameter 𝑏: 𝑂(1850) = 𝑂(1800)𝑓𝑏𝑢  23.1 = 5.3𝑓50𝑏  𝑏 = 0.29 𝑂 1900 = 𝑂 1800 𝑓0.29∙100 = 100.7 𝑂 1950 = 𝑂 1800 𝑓0.29∙150 = 438.8

• The non-linear (i.e., logistic growth) model in the dimensional form has two

parameters  We need more information: let’s assume we know the 1900 answer:

𝑂 𝑢 = 𝐷 1 + 𝐵𝑓−𝑠𝑢 𝐵 = 𝐷 − 𝑂0 𝑂0 𝑂(1850) =

𝐷 1+ 𝐷−5.3 𝑓−50𝑠/5.3 = 23.1

𝑂(1900) =

𝐷 1+ 𝐷−5.3 𝑓−50𝑠/5.3 = 76

𝑠 = 0.031 𝐷 = 189.4

𝑂 𝑢 = 189.4 1 + 34.74𝑓−0.031𝑢

 𝑂 1950 = 144.7 (the baby boom is not accounted!)

9

LOGISTIC MODEL VS. EXPONENTIAL GROWTH

• real population values in the US

▬ Logistic model predictions

• real population values in the US

▬ Logistic model predictions Little difference for small populations Both linear and logistic model work well Logistic asymptote Exponential explosion