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