1
Romeo and Juliet
Time delays Stochastic excitation
Oscillations
Nullcline angles
2
JJ = Love that Juliet professes for RomeoR = Love that Romeo professes for Juliet
Both profess love
Both profess hate Romeo professes love, but Juliet professes hate
Juliet professes love, but Romeo professes hate
Romeo and Juliet’s affections form a state space
R
3
R
J
𝑑 𝐽𝑑𝑡
=𝑅
Juliet professes more and more lovewhen Romeo expresses love
Juliet’s dynamics
4
J
𝑑 𝐽𝑑𝑡
=𝑅
𝑑𝑅𝑑𝑡
=− 𝐽
Romeo professes less and less love when Juliet professes love
R
Romeo’s dynamics
5
J
𝑑 𝐽𝑑𝑡
=𝑅
𝑑𝑅𝑑𝑡
=− 𝐽
Romeo and Juliet chase each other around in a perpetual soap opera
Trajectory of Romeo and Juliet’s dynamics form a closed loop
R
6
J
𝑑 𝐽𝑑𝑡
=𝑅
𝑑𝑅𝑑𝑡
=− 𝐽
𝑑2𝑅𝑑𝑡2
=− 𝑑 𝐽𝑑𝑡
𝑑2𝑅𝑑𝑡2
=−𝑅
𝑅=𝐴0 cos (𝑡 )
𝐽=𝐴0 sin (𝑡 )
Romeo and Juliet’s dynamics are sinusoidal
R
7
Romeo and Juliet
Time delays Stochastic excitation
Oscillations
Nullcline angles
8
R
J
𝑑𝑅𝑑𝑡
=0
Derivatives can change sign at nullclines
9
J
𝑑 𝐽𝑑𝑡
=0
R
Derivatives can change sign at nullclines
10
J
Crossed nullclines can support stable stars
R
J
R11
Crossed nullclines can support stable spirals
12
J
Crossed nullclines can support closed cycles
R
13
J
Crossed nullclines can support unstable spirals
R
14
Romeo and Juliet
Time delays Stochastic excitation
Oscillations
Nullcline angles
15
Time delay
J
R
16
Time delay
J
Current state
Previous state
R
17
Romeo and Juliet
Time delays Stochastic excitation
Oscillations
Nullcline angles
18
Stochastic excitation
J
R
19
J
Stochastic excitation
R
20
J
Stochastic excitation
R