1. Hiroki Sayama sayama@binghamton.edu
NECSI Summer School 2008 Week 3: Methods for the Study of Complex Systems Dynamical Systems and Phase Space
Hiroki Sayama

2. Four approaches to complexity
Nonlinear
Dynamics
Complexity =
No closed-form solution,
Chaos
Information
Complexity =
Length of description,
Entropy
Computation
Complexity =
Computational time/space,
Algorithmic complexity
Collective
Behavior
Complexity =
Multi-scale patterns,
Emergence

3. Mathematical formulations of dynamical systems
Discrete-time model:
xt = F(xt-1, t)
Continuous-time model: (differential equations)
dx/dt = F(x, t)
xt: State variable of the system at time t
May take “scalar” or “vector” value
F: Some function that determines the rule that the system’s behavior will obey
(difference/recurrence
equations; iterative maps)

4. Geometrical Approach to Dynamical Systems

5. Geometrical approach Developed in the late 19C by J. Henri Poincare
Visualizes the behavior of dynamical systems as trajectories in a phase space
Produces a lot of intuitive insights on geometrical structure of dynamics that would be hard to infer using purely algebraic methods

6. Phase space (state space)
A theoretical space in which every state of a dynamical system is mapped to a spatial location
Created by orthogonalizing state variables of the system; its dimensionality equals # of variables needed to specify the system state (a.k.a. degree of freedom)
Temporal change of the system states can be drawn in it as a trajectory

7. Attractor and basin of attraction
A state (or a set of states) from which no outgoing edges or flows running in phase space
Static attractors (equilibrium points)
Dynamic attractors (e.g. limit cycles)
Basin of attraction:
A set of states which will eventually end up in a given attractor

8. Visualizing Phase Space of Discrete Models

9. Phase space of discrete models
E.g. a simple blinking light
State can be specified by one binary variable (on or off) on
off
Dynamics of discrete-state models can be depicted as a directed graph (“state-transition diagram”) whose nodes/edges represent states/transitions
(For stochastic systems one node may have multiple outgoing edges)

10. Exercise
Components A & B takes either red or blue Interaction rules are defined as follows:
A tries to be the same color as its partner
B tries to be the opposite color to its partner
Draw a phase space and trajectories of this system A B

11. Exercise Think about: A B How many independent variables?
How many possible system states?
What kind of phase space to be used? A B

12. Answer B A B A B A B A B A

13. Exercise: Three-person majority game
There is a group made of three persons
Each person takes either option X or Y
Each person makes his/her choices open to the other two, then they simultaneously switch their choices to the major option within the group
This process repeats indefinitely
Illustrate geometrical structure of the phase space of this system and identify attractors and basins of attractions in it

14. Answer
BAA
BAB
AAA
AAB
BBA
BBB
ABA
ABB

15. Answer Basin of attraction of Attractor A BAA BAB Attractor A AAA AAB
BBA
BBB
Attractor B
ABA
ABB
Basin of attraction of
Attractor B

16. Exercise
Illustrate geometrical structure of the phase space of a three-person minority game (i.e., each person switches his/her choice against the major option within the group)

17. There is only one basin of attraction in this case
Answer
Dynamic
attractor
BAA
BAB
AAA
AAB
BBA
BBB
ABA
ABB
There is only one basin of attraction in this case

18. Exercise
Four persons form a social contact network as shown on the left
When someone catches cold, she will get well after one time step, but cold will infect all of her healthy neighbors
Illustrate geometrical structure of the phase space of this system

19. Reducing the number of states
In some cases, it may be possible to reduce the number of possible states by using symmetries
Rotational
Reflectional
Translational
Graph isomorphism
etc.

20. Exercise
Reduce the number of possible states of the previous “cold network” example using symmetries

21. Visualizing Phase Space of Continuous Models

22. Phase space of continuous models
E.g. a simple vertical spring oscillator
State can be specified by two real variables (location x, velocity v) x
Trajectory
(orbit) v
Dynamics of continuous models can be
depicted as “flow” in a continuous phase space

23. Visualizing phase space of continuous models manually
Find “nullclines”
Points in the phase space where one of the derivatives is zero (i.e., trajectories are in parallel to one of the axes)
Plot where the nullclines are
Find how the sign of the derivative changes across the nullclines
Find values of other non-zero derivatives
Draw a “flow” between those nullclines with curves that don’t intersect with each other

24. Exercise
Draw an outline of the phase space of the following system by studying the distribution of its nullclines:
dx/dt = x – x y
dy/dt = y – x y
(x >= 0, y >= 0)

25. Visualizing phase space of continuous models
Vector field
Uses many small arrows to show how local derivatives (or direction of trajectories) change from place to place in phase space
Phase portrait
Shows several typical trajectories to illustrate how phase space is globally structured

26. Exercise: Draw a phase space of the SIR model
“Susceptible” individuals can turn into “Infective” by contacting other “Infectives”
“Infective” individuals turn into “Removed” at a given rate
dS/dt = -i S I
dI/dt = i S I – r I
dR/dt = r I

27. Exercise: Draw a phase space of the RPS model
Assume a cyclic food chain among three species so that “Rock” eat “Scissors”, “Scissors” eat “Paper”, and “Paper” eat “Rock”
Each species will increase by eating other species and decrease by being eaten by others
dR/dt = a R S – b R P
dP/dt = b R P – c S P
dS/dt = - a R S + c S P

28. Summary: Advantage of geometrical approach
By using phase space, one can convert dynamic behaviors of systems into a static picture
By identifying geometrical structure of phase space such as attractors and basins of attraction, one can visually describe possible final states of the system and its sensitivity to initial conditions

29. Linear Systems

30. Linear systems
Linear systems are the simplest cases where states of nodes are continuous-valued and their dynamics are described by a time-invariant matrix
Discrete-time: xt = A xt-1
Continuous-time: dx/dt = B x
A / B are called “coefficient” matrices
We don’t consider constants (as they can be easily converted to the above forms)

31. Example: An “averaging” network
A discrete-time dynamical network where the next state of each node is an average of its neighbors’ states (including its own) : s1 s1 s2 s5
s1t = { s1t-1 + s2t-1 + s3t-1 + s5t-1 } / 4 s3 s4

32. Exercise
Write down the state-transition function of the following “averaging” network by completing its coefficient matrix s1 s1 s2 s5
s = {s1, s2, s3, s4, s5}
st = (???) st-1 s3 s4

33. Exercise
Derive the coefficient matrix of a similar “averaging” dynamical network that behaves in continuous time s1 s1 s1 s1 s1 s2 s5
s = {s1, s2, s3, s4, s5}
ds/dt = (???) s s3 s4

34. Coefficient matrix and adjacency matrix
Coefficient matrices are very similar to adjacency matrices of graphs:
If there is no interaction (= no edge) between nodes j and i, then aij = 0
A weight aij in an adjacency matrix corresponds to the strength of influence from node j to i
One difference: A coefficient matrix may take negative components in it

35. Equilibrium points of linear systems
Discrete-time: xt = A xt-1
Continuous-time: dx/dt = B x
Linear systems have the origin (x=0) as a trivial equilibrium point
Can they have other non-trivial (non-zero) equilibrium points?

36. Finding equilibrium points
Discrete-time: xt = A xt-1
Continuous-time: dx/dt = B x
Discrete-time: xe = A xe
Continuous-time: 0 = B xe
If A has 1 (or if B has 0) as its eigenvalue, then the corresponding eigenvector(s) show the system’s equilibrium point(s)

37. Exercise
Obtain equilibrium points of the “averaging” network in both discrete- and continuous-time cases s1 s2 s5 s3 s4

38. Asymptotic Behavior of Linear Systems

39. Where will the system go eventually?
Discrete-time: xt = A xt-1
Continuous-time: dx/dt = B x
These equations give the following exact solutions:
Discrete-time: xt = At x0
Continuous-time: xt = eBt x0
= { I + Sk=1~ (Bt)k/k! } x0

40. FYI: Exponential operator for matrices
xt = eBt x0
= { I + Sk=1~ (Bt)k / k! } x0
Similar to the Taylor series expansion of the exponential function:
ex = 1+ x + x2/2! + x3/3! + …
eM converges for any square matrix M
If M’s eigenvalues are {li }, then eM’s eigenvalues are { eli }, with all eigenvectors unchanged (you can prove this)

41. Where will the system go eventually?
Discrete-time: xt = A xt-1
Continuous-time: dx/dt = B x
What happens if the system starts from non-equilibrium initial states and goes on for a long period of time?
Let’s think about their asymptotic behavior lim t-> xt !!

42. Considering asymptotic behavior (1)
Let { vi } be n linearly independent eigenvectors of the coefficient matrix
(They might be fewer than n, but here we ignore such cases for simplicity)
Write the initial condition using eigenvectors, i.e.
x0 = b1v1 + b2v bnvn

43. Considering asymptotic behavior (2)
Then:
xt = At x0 (discrete-time)
= l1t b1v1 + l2t b2v2 + … + lnt bnvn or
xt = eBt x0 (continuous-time)
= el1t b1v1 +el2t b2v2 + … +elnt bnvn

44. Dominant eigenvector (1)
Discrete-time: If |l1| > |l2|, |l3|, …,
xt = l1t { b1v1 + (l2/l1)t b2v (ln/l1)t bnvn }
lim t-> xt ~ l1t b1v1
Continuous-time: If Re(l1) > Re(l2), Re(l3), …,
xt = el1t { b1v1 +e(l2-l1)t b2v e(ln-l1)t bnvn }
lim t-> xt ~ el1t b1v1
If the system has just one such dominant eigenvector v1, its state will be eventually along that vector regardless of where it starts

45. Dominant eigenvector (2)
If the system has just one such dominant eigenvector v1, its state will be eventually along that vector regardless of where it starts
But divergence or oscillation are possible
If, additionally, l1=1 (discrete-time) or 0 (continuous-time), then the system will eventually reach the equilibrium point
(If the system has multiple dominant eigenvectors, final states may depend on initial states)

46. FYI: What eigenvalues and eigenvectors can tell us
An eigenvalue tells whether a particular “state” of the system (specified by its corresponding eigenvectors) grows or shrinks by interactions between parts
| l | > 1 -> growing
| l | < 1 -> shrinking
Re(l) > 0 -> growing
Re(l) < 0 -> shrinking
for discrete-time cases
for continuous-time cases

47. Example
Phase space of a two-variable linear difference equation with (a, b, c, d) = (1, 0.1, 0.1, 0.9)
Along these lines (called invariant lines), the dynamics can be understood as simple exponential growth/decay y
|l2|<1
|l1|>1 v1 x v2

48. Order spontaneously emerges in the system as time goes on
Example y x
This could be regarded as a very simple form of self-organization (though completely predictable);
Order spontaneously emerges in the system as time goes on

49. Exercise xt = 2 xt-1 - 2 yt-1 yt = 0.2 xt-1 + 0.2 yt-1
Calculate eigenvalues of the following linear system and predict its dynamics along the lines given by the eigenvectors
xt = 2 xt yt-1
yt = 0.2 xt yt-1
Draw its phase space to confirm your prediction

50. Exercise
Obtain asymptotic behavior of the following dynamical network (figures denote symmetric coefficients in A/B) in both discrete- and continuous-time cases 1 -1 1 1 -3

51. FYI: Symmetry of matrices and type of eigenvalues
If the matrix is symmetric, it has real eigenvalues only, so the asymptotic behavior should be either converging, diverging, or flipping
If the matrix is asymmetric, it may have complex eigenvalues and thus the system may keep changing dynamically in rotational trajectories

52. Summary (1)
To find equilibrium points of a linear dynamical system, calculate the eigenvectors of its coefficient matrix with its corresponding eigenvalue:
l = 1 (for discrete-time cases)
l = 0 (for continuous-time cases)

53. Summary (2)
To study asymptotic behavior of a linear dynamical system, check if its coefficient matrix has just one dominant eigenvector whose corresponding eigenvalue l1 satisfies:
|l1| > |l2|, |l3|, ...
(for discrete-time cases)
Re(l1) > Re(l2), Re(l3), ...
(for continuous-time cases)

54. Summary (3)
If the system has just one such dominant eigenvector, its state will be eventually along that vector regardless of where it starts
These final states may oscillate and/or diverge in its magnitude, but if its corresponding eigenvalue = 1 (discrete-time) / 0 (continuous-time), the final state is also an equilibrium point

55. Nonlinear Systems

56. Nonlinear systems
Nonlinear systems are systems whose dynamics cannot be written in a linear (i.e., matrix & vector) form because of nonlinear terms in F
Discrete-time: xt = F(xt-1, t)
Continuous-time: dx/dt = F(x, t)

57. Example: Logistic growth
N: Population
r: Population growth rate
K: Carrying capacity
Discrete-time version:
Nt = Nt-1 + r Nt-1 ( 1 – Nt-1/K )
Continuous-time version:
dN/dt = r N ( 1 – N/K ) 2 4 6 8 10
0.2
0.4
0.6
0.8 1
Non-linear
terms

58. Example: Lotka-Volterra model
Independently derived by A. J. Lotka (1925) and V. Volterra (1926)
A well-known nonlinear model of a simple ecosystem made of predator and prey populations

59. Model equations dx/dt = a x – b x y
Non-linear
terms
dx/dt = a x – b x y
dy/dt = - c y + d x y (a,b,c,d>0)
Positive influence +
Rabbit
Population : x
Fox
Population : y -
Negative influence + -
Naturally grows
if isolated
Naturally decays
if isolated

60. Revision 1: With carrying capacity
How can you introduce a limited carrying capacity for the growth of rabbits to the equations?
dx/dt = a x – b x y
dy/dt = - c y + d x y

61. Revision 1: With carrying capacity
How can you introduce a limited carrying capacity for the growth of rabbits to the equations?
dx/dt = a x (1-x/K) – b x y
dy/dt = - c y + d x y
Obtain equilibrium points and draw phase space (with a=b=c=d=1, K=2)

62. Revision 2: With bounded predation rate
A fox can’t eat 100 rabbits at once no matter how many there are around
How can you set an upper bound to the predation rate in the equations?
dx/dt = a x (1-x/K) – b x y
dy/dt = - c y + d x y

63. Revision 2: With bounded predation rate
A fox can’t eat 100 rabbits at once no matter how many there are around
How can you set an upper bound to the predation rate in the equations?
dx/dt = a x (1-x/K) – b Jx/(J+x) y
dy/dt = - c y + d Jx/(J+x) y
Obtain equilibrium points and draw phase space (with a=b=c=d=1, K=2, J=3)

64. Summary
Systems’ dynamics may be visualized and studied using phase spaces
Linear systems’ dynamics may be studied through eigenvalue analysis
Equilibrium points
Asymptotic behavior
Realistic model assumptions are likely to make the dynamics nonlinear