1. Hiroki Sayama sayama@binghamton.edu
NECSI Summer School 2008 Week 3: Methods for the Study of Complex Systems Introduction / Iterative Maps
Hiroki Sayama

2. Introduction

3. Course objective
To provide an introduction to a variety of mathematical concepts and tools for analysis of complex systems
Textbook: Bar-Yam, Y.
“Dynamics of Complex Systems”
(Perseus Books/Westview Press, 1997)

4. Topics to be covered Iterative maps Stochastic systems
Information theory
Computation theory
Dynamical systems and phase space
Analytical tools for dynamical systems
PDEs and reaction-diffusion systems
Cellular automata
Thermodynamics and statistical mechanics
Stochastic fields and mean-field approximation
Monte Carlo simulations
Scaling, fractals and renormalization

5. Online resource Course slides for the first two days are available at:
Login name: necsi
Password: com3sysB

6. Course structure Monday ~ Thursday Friday
9:00am~5:00pm: Lectures, discussions
6:00pm~8:00pm: Group projects
Friday
9:00am~12:00pm: Presentations
(+ optional final exam)

7. Group projects Presentation (~15 min.) + 5-page paper
Option 1: Original research
Conduct mathematical analysis of a model of complex systems (either existing or original) and report your findings
Option 2: Teaching analytical methods
Select some analytical method that is not covered in the classes, prepare teaching materials (including illustrative examples and questions) and deliver a lecture
Option 3: Problem sets (for individuals)
Go through several problems selected from different sections in textbook, write out your work and hand it in (presentation waived in this case)

8. Examples of complex systems
Chemical networks
Gene networks
Organisms
Physiologies
Brains
Ecosystems
Economies
Societies
Internet

9. Several characteristics of complex systems
Networks of many components
Nonlinear interactions
Self-organization
Structure/behavior that is neither regular nor random
Emergent behavior

10. 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

11. Dynamical Systems

12. Dynamical systems theory
Considers how systems autonomously change along time
Ranges from Newtonian mechanics to modern nonlinear dynamics theories
Thinks about underlying dynamical mechanisms, not just static properties of observations
Forms the theoretical basis for most of complex systems studies

13. What is a dynamical system?
A system whose state is uniquely specified by a finite set of variables and whose behavior is uniquely determined by predetermined rules
Simple population growth
Simple pendulum swinging
Motion of celestial bodies
Behavior of two “rational” agents in a negotiation game

14. 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)

15. Review of Difference Equations

16. Difference equation and time series
xt = F(xt-1, t)
produces series of values of variable x starting with initial condition x0:
{ x0, x1, x2, x3, … } “time series”
A prediction made by the above model (to be compared to experimental data)

17. Linear vs. nonlinear Linear: xt = a xt-1 + b xt-2 + c xt-3 …
Right hand side is just a first-order polynomial of variables
xt = a xt-1 + b xt-2 + c xt-3 …
Nonlinear:
Anything else
xt = a xt-1 + b xt-22 + c xt-1 xt-3 …

18. Single-variable vs. multi-variable
Single-variable (univariate):
Just one equation given for a series {xt}
xt = a xt-1 + b xt-22 + c / xt-3 …
Multi-variable (multivariate):
Multiple equations given to simultaneously describe multiple series {xt}, {yt}, …
xt = a xt-1 + b yt-1
yt = c xt-1 + d yt-1

19. 1st-order vs. higher-order
Right hand side refers only to the immediate past
xt = a xt-1 ( 1 – xt-1 )
Higher-order:
Anything else
xt = a xt-1 + b xt-2 + c xt-3 …
(Note: this is different from the order of terms in polynomials)

20. Autonomous vs. non-autonomous
Right hand side includes only state variables (x) and not t itself
xt = a xt-1 xt-2 + b xt-32
Non-autonomous:
Right hand side includes terms that explicitly depend on the value of t
xt = a xt-1 xt-2 + b xt-32 + sin(t)

21. Homogeneous vs. non-homogeneous
Every term in the right hand side has the same order
xt = a xt-1 + b xt-2 + c xt-3
Non-homogeneous:
Anything else (typically has constants)
xt = a xt-1 + b xt-2 + c xt-3 + d

22. Things that you should know (1)
Non-autonomous, higher-order equations can always be converted into autonomous, 1st-order equations
xt-2 → yt-1, yt = xt-1
t → yt, yt = yt-1 + 1, y0 = 0
Autonomous 1st-order equations (iterative maps) can cover dynamics of any non-autonomous higher-order equations too!

23. Things that you should know (2)
Linear equations
are analytically solvable
show either equilibrium, exponential growth/decay, periodic oscillation (with >1 variables), or their combination
Nonlinear equations
may show more complex behaviors
do not have analytical solutions in general

24. Iterative Maps

25. Iterative map Autonomous, 1st-order difference equation: xt = F(xt-1)
Equilibrium points (a.k.a. fixed points, steady states) can be obtained by solving
xe = F(xe)

26. Exercise
Obtain equilibrium points of the following discrete-time logistic growth model:
Nt = Nt-1 + r Nt-1 ( 1 – Nt-1/K )

27. Cobweb plot A visual tool to study the behavior of 1-D iterative maps
Take xt-1 and xt for two axes
Draw the map of interest (xt=F(xt-1)) and the “xt=xt-1” reference line
They will intersect at equilibrium points
Trace how time series develop from an initial value by jumping between these two curves

28. Exercise Draw a cobweb plot for each of the following models:
xt = xt , x0 = 0.1
xt = 1.1 xt-1 , x0 = 0.1

29. Exercise
Draw a cobweb plot of the logistic growth model with r=1, K=1, N0=0.1:
Nt = Nt-1 + r Nt-1 ( 1 – Nt-1/K )

30. Stability of equilibrium points
The slope of function F (F/x) at an equilibrium point determines whether the system can converge to or diverge from that point F x -1 1
Unstable
Stable
Unstable
With oscillation
No oscillation

31. Exercise
Analyze the stability of the non-zero equilibrium point of the logistic growth model with r=1, K=1:
Nt = Nt-1 + r Nt-1 ( 1 – Nt-1/K )

32. Logistic Map and Chaos

33. Logistic map
*The* most famous single-variable nonlinear difference equation
xt = a xt-1 ( 1 – xt-1 )
Similar to (but not quite the same as) the discrete-time logistic growth model (missing first xt on the right hand side)
Shows quite complex dynamics as control parameter a is varied

34. Exercise: Equivalence between logistic growth and logistic map
Nt = Nt-1 + r Nt-1 ( 1 – Nt-1/K )
This becomes equivalent to the logistic map if we assume:
Nt = xt K (1 + r) / r
Show that this is correct
Determine the relationship between growth rate r in logistic growth models and coefficient a in logistic maps

35. Exercise
Draw cobweb plots of the logistic map for a = 0.5, 1.5 and 2.5
Try a couple of different initial conditions for each case

36. Period-doubling bifurcations
In discrete-time models, “period-doubling” bifurcations may occur when
F/x(x=xe) = -1
The equilibrium point xe is about to destabilize in an oscillatory manner
Also possible in multi-dimensional continuous-time models (which will not be covered in class)

37. Exercise xt = F(xt-1) = a xt-1 (1 – xt-1)
Obtain the equilibrium points of the logistic map as a function of a:
xt = F(xt-1) = a xt-1 (1 – xt-1)
Examine their stability and its dependence on the value of a

38. Summary of stability analysis
Range
Equil. of a
points
0 < a < 1
1 < a < 3
3 < a
x = 0
Stable
Unstable
x = 1-1/a
(Doesn’t exist within positive range)
Impossible if this was a continuous-time model

39. What is going on for a > 3?
Example: a = 3.2
Period-doubling bifurcation
Both equilibrium points lose stability
System starts to oscillate with a doubled period

40. Exercise: Stability analysis of F2(x)
That the system flips back and forth between two points means that they should be equilibrium points of a composite function F2(x) (= F(F(x)) )
Obtain the equilibrium points of F2(x) and examine their stability and its dependence on the value of a
What is happening at a=3?

41. General condition for stability of periodic trajectories
Consider a periodic trajectory:
{ x0, x1, x2, …, xt }
= { x0, F(x0), F(F(x0)), … Ft(x0) }
(xt = x0)
This trajectory is stable if & only if:
|Ft/x(x=x0) | < 1, or
|F/x(x=x0) F/x(x=x1) … F/x(x=xt-1) | < 1

42. What is going on for a > 3.6?
Example: a = 3.8
Chaos
The system loses periodicity after a cascade of period doubling events

43. Drawing bifurcation diagrams using numerical results
For systems with periodic (or chaotic) long-term behavior, it is useful to draw a bifurcation diagram using numerical simulation results instead of analytical results
Plot actual system states sampled after a long period of time has passed
Can capture period-doubling bifurcation by a “set” of points

44. Cascade of period-doubling bifurcations leading to chaos
Doubling periods diverge
to  at a ≈ …!
Period-
doubling
bifurcation
Chaos
Period-doubling
bifurcation
Period-doubling
bifurcation
Transcritical
bifurcation

45. Discovery of chaos
Discovered in early 1960’s by Edward N. Lorenz (in a 3-D continuous-time model)
Popularized in 1976 by Sir Robert M. May as an example of complex dynamics caused by simple rules (he used a 1-D discrete-time logistic map)

46. Chaos in dynamical systems
A long-term behavior of a dynamical system that never falls in any static or periodic trajectories
Looks like a random fluctuation, but still occurs in completely deterministic simple systems
Exhibits sensitivity to initial conditions
Can be found everywhere

47. Reinterpreting chaos Where a period diverges to infinity
Periodic behavior with infinitely long periods means “aperiodic” behavior
Where (almost) no periodic trajectories are stable
No fixed points or periodic trajectories you can sit in (you have to always deviate from your past course!!)

48. Exercise
Draw a more detailed bifurcation diagram for the chaotic regime and see how those chaotic trajectories are structured there
Can you find any “order” in chaos?

49. Exercise
What happens if you try plotting cobweb plots and bifurcation diagrams of the logistic map for negative a?
Determine the range of a where chaotic dynamics occurs

50. Characteristics of Chaos

51. Common mechanism of chaos: Stretch and folding in phase space
In chaotic systems, it is generally seen that a phase space is stretched and folded, like dough kneading 1 1
0.2
0.4
0.6
0.8 1 1 1

52. Example

53. Other examples of stretching and folding: tent map, saw map
0.2
0.4
0.6
0.8 1
0.2
0.4
0.6
0.8 1
These maps also have chaotic dynamics if the initial condition is an irrational number
What kind of numerical operation is carried out when one applies either of these maps to a number?

54. Sensitivity to initial conditions
Stretching and folding mechanisms always dig out microscale details hidden in a system’s state and expand them to macroscale visible levels
This causes chaotic systems very sensitive to initial conditions
Think about where a grain in a dough will eventually move during kneading

55. Lyapunov exponent (1)
A quantitative measure of a system’s sensitivity to small differences in initial condition
Characterizes how the distance between initially close two points will grow over time
| Ft(x0+e) - Ft(x0) | ~ e etl (for large t)

56. Lyapunov exponent (2) | Ft(x0+e) - Ft(x0) | ~ e etl (for large t)
l = limt log
= limt log
= limt Si=0~t-1 log |F’(xi)|
Ft(x0+e) - Ft(x0) e t 1
e0 dx d
Ft(x0)

57. Exercise
Plot how the Lyapunov exponent of the logistic map changes over varying a
Compare the result with its bifurcation diagram

58. Summary
Iterative maps xt = F(xt-1) are simple yet general discrete-time dynamical systems
Stability of trajectories of period n can be analyzed by studying dFn/dx
Even simple iterative maps may produce very complex, unpredictable (chaotic) behavior