1. Understanding Chaos Theory:
A Presentation
MariJoy G. Tiongson
2009-34871
COMPUTATIONAL PHYSICS LABORATORY
BSAP, CAS,IMSP,UPLB
Chaos: When the present determines the future, but
the approximate present does not approximately
determine the future.
- Edward Lorenz
2. ChAOs is...
Beginner's thoughts
● A great disorder
● No specific pattern is
foreseen
● Seems to be related
to randomness
A Scientist's
Point of View
● order in disorder
● Not readily
predicted
● deterministic
3. ChAos Defined
● study of complex nonlinear dynamic
systems (catch: necessarily nonlinear but
not all nonlinear systems are chaotic)
● Unstable & Deterministic in a sense that
initial conditions necessarily predict the
future with no randomness involved
● Qualitative in knowing the system's
long-term behaviour, not seeking
predictions
4. Features/ Characteristics
● Strong dependence and sensitivity to initial
conditions and changes in system parameters
● If it is linear, its not chaotic
● Sustained irregularity in system's behaviour
● Impossible to predict
● Presence of strong harmonics and stretch
direction* (positive Lyapunov exponent)
● Fractional dimension of space state trajectories
● Density of periodic orbits (every point is
approached by periodic orbits) and topological
mixing (eventual overlapping of phase space)
---as summarized from references
5. Attractors
● Defined as set of states (points in the phase space)
towards which other states tend to approach or evolve
– Point – only one outcome for the system
– Limit cycle – system settles into a cycle
– Strange attractor – a double spiral which never
repeats itself (or it would be periodic attractor), but
the values always move towardsa certain range of
values.
6. Chaos Application:
Population Dynamics
● Chaos Theory as applied in the prediction of
biological populations
● Robert May's experiment of fluctuating values of
growth rate
● Related Literature's [Simulation of Chaotic
Behaviour in Population Dynamics] experiment
in increasing the Verhulst factor [by competing
reproductive growth rate and birth rate] produce
a limit cycle and chaotic attractors; they use
Penna model to simulate these data
7. – at low values of the growth rate, the population
would settle down to a single number. As the
growth rate increases, the final population would
increase as well but Instead of settling down to a
single population, it would jump between two
different populations. Raising the value a little, it
results to 4 different values. Past a certain growth
rate, it becomes impossible to predict the behavior
of the equation
Robert May's
experiment of
fluctuating
values of
growth rate
8. RRL: Simulation of Chaotic Behaviour in
Population Dynamics
– Simulate chaotic behaviours – limit cycles and chaotic
regime – using Penna model
– Penna model can exhibit the three attractors, namely
fixed point, limit cycles and chaotic regime
– Chaos are found in species with high reproductive rate
and timely/cyclic breeding strategy
– Fluctuating Verhulst and birth rate [B] values,
time-dependent λ, and T>R case proves to show chaos
– Intrinsic relative growth rate is not constant but a
time-changing one following the period of the attractor
Generalized logistic equationfor the evolution of population
where
9. RRL: Simulation of Chaotic Behaviour in
Population Dynamics
Penna works by dividing life into 32 time intervals and by
representing the genome (DNA) through a string of 32 bits,
each of which can be zero or one.
A zero bit means health, a bit set to one means a dangerous
inherited disease starts to act from that age on which
corresponds to the position of this bit in the bit-string.
If T (typically, T = 6, T>R case) bits are active, their
combined effect kills the individual.
Each individual which has reached the minimum
reproduction age of R (typically, R = 4) gets B (typically, 20
≤ B ≤ 35) children at each time step, where 32 time steps
give the maximum life span.
The child inherits the mother’s genome except
for M (typically, M = 1) mutations of randomly selected bits
where a zero bit becomes a one bit.
11. The logistic map is a
polynomial mapping of
degree 2 showing how
complex, chaotic behaviour
can arise from very simple
non-lineardynamical
equations.
Logistic Map
13. - when a trajectory approaches ergodically a desired
periodic orbit embedded in the attractor, one applies
small perturbations to stabilize such an orbit.
- If one switches on the stabilizing perturbations, the
trajectory moves to the neighbourhood of the desired
periodic orbit that can now be stabilized.
– This fact has suggested the idea that the critical sensitivity of a chaotic system
to changes (perturbations) in its initial conditions may be, in fact, very desirable in
practical experimental situations. (This is known as Ott, Grebogi, and Yorke
(OGY) approach of controlling chaos.)
- There are three ways to control chaos:
1. Alter organizational parameters so that the range
of fluctuations is limited.
2. Apply small perturbations to the chaotic system to
try and cause it to organize.
3. Change the relationship between the organization
and the environment.
Controlling Chaos
