APPLICATIONS GALORE SCTPLS Annual Conference Cincinnati Applications Galore 1 Frictionfree introduction to NDS concepts and how they connect Stephen Guastello 2 ADAM KIEFER Physiology rehabilitation ID: 641316
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Nonlinear Dynamics Workshop, 2017APPLICATIONS GALORE
SCTPLS Annual Conference, CincinnatiSlide2
Applications Galore
1. Friction-free introduction to NDS concepts and how they connect. (Stephen Guastello)
2. ADAM KIEFER – Physiology, rehabilitation3. DAVID PINCUS – Clinical psychology, psychotherapy4. JEFFERY GOLDSTEIN – Organizational behaviorSlide3
By the end of the day it’ll all look easy…
So let’s get started…Slide4
Friction-free intro
THE PARADIGM SHIFTELEMENTARY NONLINEAR DYNAMICAL
SYSTEMSAttractors and stabilityBifurcations and instability
ChaosFractalsCOMPLEXITY – When agents and variables interact
Self-organizationSynchronizationEmergenceCatastrophe theory
Phase shifts
Entropy
Nonlinear methodsSlide5
Why Nonlinear Dynamics?
Explains changes that occur over time.
Small interventions at the right time can have a big impact.Large interventions at the wrong time can do nothing or worse.
Allows for structural comparison of models across situations, even theories that are very different by appearances.Better explanations of data
R2The qualitative
aspects of the phenomenaSlide6
Paradigm Shift
Conventional Paradigm
Linear relationshipsStatic situations
Nonlinear DynamicsNonlinear relationships
Changes over timeDifferent kinds of changes display different dynamicsSlide7
Paradigm Shift
Conventional Paradigm
Seemingly random processOutcomes are proportional in inputs
Simple cause and effectNonlinear Dynamics
Determined by simple functionsLittle
things can have big consequences, vice-versa
Control variables that behave differently
Emergent phenomenaSlide8
Paradigm Shift
Conventional response
Ignore, stifle random blipsMaintain equilibrium, stability, and controlLook for disruptions external to the system
Nonlinear response
It was no blipNavigate a repertoire of nonlinear change
processes
“Equilibrium” replaced by “
attractors
”
Look for intrinsic dynamicsSlide9
Conventional Paradigm
Nonlinear DynamicsSlide10
Paradigm Shift
According to Ian Stewart, the natural sciences made the transition more than a half century ago:So ingrained became the linear habit, that by the 1940s and 1950s many scientist[s] and engineers knew little else . . . [W]e live in a world which for centuries acted as if the only animal in existence was the elephant, which assumed that holes in the skirting-board must be made by tiny elephants, which saw the soaring eagle as a wing-eared Dumbo, [and] the tiger as an elephant with a rather short trunk and stripes (
1989, p. 83-84). Slide11
And speaking of random processes:
Gaussian distribution
Exponential
Any differential process (dy/dx) can be represented as a complex exponential distribution.The simple one:
Power Law (inverse)
Common in fractal and self-organizing processesSlide12
Attractors
Fixed pointsAn incoming point can proceed in directlyOr spiral inSlide13
AttractorsAttractor
basinArea around an attractor where the attracting force can operate.
Some attractors are stronger than othersChaos in multi-attractor systems
This point almost escaped the grip:Slide14
AttractorsRepellor --
An inverse-attractorIncoming points (objects) are deflected outward in any direction
.
System of 3 repellorsSlide15
AttractorsSaddle
Has properties of an attractor and a repellor.You can visit, but you can’t stay too long.
Perturbed pendulumSlide16
Attractors
Fixed point attractor + saddleUnfolding of a romantic relationship over time
Industry introduces new productsSlide17
Attractors
Limit cyclesOscillators, sine wavesDampened, control parameter is involved
There are many in real life –EconomicsBiology
Signal processingSlide18
Stability, instability
Principle of structural stabilityAll points are behaving according to the same mathematical rule.Attractors, usually
Repellors and saddles, no.Bifurcations, no.Chaos
YES if it’s a chaotic attractorNO if not. Not all chaos is a chaotic attractorSlide19
Order and Control Parameters
Order parameter – essentially a dependent measure. We measure the
position and movement of the order parameter in fixed pts, limit cycles, chaos, complex dynamic fields
Control parameter – essentially an independent variable that alters the dynamics of the order parameter.Unlike conventional IVs, not all control parameters in a system have the same effect on behavior.Slide20
Bifurcations
A split in a dynamical field where different dynamics exist in each local region.Represent a pattern of instabilityTypes
Simplest is a critical point – tipping pointPoints follow different trajectoriesHopf
: A fixed point turns into a limit cycleEntire dynamic fields come and go depending a
control parameterTwo dynamic fields annihilate each other producing a new dynamic field.Slide21
Bifurcations
Pitchfork with 1 critical pointSeen in cusp catastrophe modelsSlide22
Bifurcations
Logistic mapVerhulst equation
Iteration: x2 = cx1(1 – x
1)Critical points
Logistic mapSlide23
ChaosSeemingly random events that are predictable with simple but special equations.
First discovered by Henri Poincaré
while trying to solve the 3-body problem.Strange attractor discovered by E. Lorenz, meteorologist.Slide24
ChaosProperties of chaos
Expanding and contracting trajectoriesSensitivity to initial conditionsTwo points start arbitrarily close together, but become far apart as iteration progresses.
Non-repeating sequencesBoundedLatter 2 are matters of degreeSlide25
ChaosA chaotic time seriesSlide26
Chaos
Lorenz attractorOrder parameters: X, Y, ZControl parameters: A, B, CSlide27
Pathways to Chaos
Attractor fields with 2 or more attractors
Coupled oscillatorsNewhouse et al: 3 is sufficientNot all combinations produce chaosSlide28
Pathways to Chaos
Logistic map when the control parameter > 3.56Other bifurcations structures tooSlide29
Chaos
At least 60 chaotic systems are currently knownAssess generically for level of complexityLyapunov Exponent
Fractal DimensionEntropySlide30
Lyapunov Exponent
Properties
Based on successive differences in y valuesSpectrum of
valuesIndicates level of turbulenceGeneralizes as ||y||=el
tIf largest l > 0, system is
chaotic
Converts to a fractal dimension:
D
L
= e
l
(Kaplan-
Yorke
)
Interpretation
Large
negative = fixed point
Small
negative = dampened oscillator
0
= oscillator
Small
positive = aperiodic,
self-organized criticality
Large
positive = chaos
Turbulent air flowSlide31
Fractals
Geometric items in fractional dimensionsProduced by iterative processes
or affine transformations e.g. z
lz
(1-z)Self-similarity at different levels of magnification
Concept of “scale-free” measurement
Can apply to time series analysis
also
Relationships to chaos and
self-organization
Basin of a chaotic attractor has a fractal shapeSlide32
Fractals
Non-fractalFractalSlide33
Fractals
Mandelbrot set
Julia setSlide34
Fractals
Nature: Branching structures
Nature: LandscapesSlide35
FractalsSCALE:
The magnified piece of an object is an exact copy of the whole object.Slide36
FractalsPrinciple extends to nonlinear time series analysisSlide37
Fractals
N (r) balls of radius r needed to cover the objectMake radius smaller, recount
Slope = fractal dimensionSlide38
Fractals
Power law distribution of object sizesSlide39
Near 0
fixed point
1.0
line or an open circle
1.0 – 2.0
chaotic system that has self-organized into a low-dimensional configuration (self-organized criticality). Pink noise
2.0
Points are evenly distributed around a two-dimensional surface. Brownian motion in 2-D.
Beware: Some rather famous chaotic attractors have dimensions that are very close to 2.0.
2.0 – 3.0
Motion is punctuated with large bursts. Or, a physical landscape that relatively flat (~ 2.0), or relatively rugged (~3.0).
Many chaotic attractors fall into this range.
3.0
Brownian motion in 3-D.
>3.0
Chaos is likely in a time series.
Beware: (a) The correlation dimension is not an effective test for chaos. (b) Not all chaos occurs in a chaotic attractor.
Very large
Probably white noiseSlide40
Optimum variability principle
Sick or deficient systems display lower complexity than healthy onesSlide41
Self-Organization
Can a global structure emerge from local interactions among agents (Subsystems)?
Systems in far-from-equilibrium conditions (chaos, high entropy) tend to self-organize
Create their own structure without any outside help. Entropy is reduced.Bottom-up process begins with bilateral interactions among agents
(Holland)Slide42
Self-Organization
Create structure by forming feedback loops among subsystems
Several modelsInformation flows among subsystemsDriver-slave relationships (Haken
): Dynamics of one subsystem affects output of another. (usually one-way).Sudden change in system organization is characterized as a
phase shift.Slide43
Self-Organization
Edge of chaos effect (Waldrop):Systems are critically poised to self-organize, dissolve, and re-organize in adaptation to their environments
.Some controversy over the construction of simulationsNow a moot point in light of real-world examples
Optimum variability principleHence the essence of the Complex Adaptive System
Ashby’s Law of Requisite Variety: The complexity of the control system needs to be at least great as the complexity of possible system states it needs to control
Optimum variability principle againSlide44
Self-Organization
Kauffman: NK[C] modelDifferentiation scenarioNK distribution
Niche hopping scenarioNK[C] complexity of interacting agents on a hilltop.Goldmine of strategic concepts.
There will be
many
other agents in a low-K environment.
Expect competitive interactions.
Opportunities for cooperation?Slide45
Self-OrganizationAdaptive
walkHop with a pogo stick
Parallel search parties explore
optionsSlide46
Try a random idea sometimes
Look for patternsSlide47
Self-OrganizationSandpiles
and avalanche (Bak)
Distribution of large and small piles after the avalanche is a power law distribution.Slide48
Out of control?
The hive mentalityCollective intelligenceWork systems self-organize from the bottom
upPhysical boundaries shape how things self-organizeBoundaries that used to exist between organizations have broken down
e-comGiving rise to networks.Slide49
Agent-Based Modeling
The number of interactions among agents is too complex to calculate individually.Agent-based programs devised to evaluate outcomes of thousands of possible interactions.
This is how “complexity theory” got its name.Interactions among agents (people, grains of sand) are thought to underlie s/o processes.Slide50
Synchronization
Special Case of S/ORhythmic physical movementsDiscrete events close-enough in timeAutonomic arousal or EEG activity while people are doing something else
Emotional contagionSync
PhenomenaSync of clocksHyper-synchronization of neuron firings during
epileptic seizureElectrodermal response of two people in conversationCoordination of efforts in a work team
Shift in weight from one foot to another during conversation.Slide51
SynchronizationGeneral principles
Two oscillatorsFeedback loop between themControl parameter that speeds up
Speed up enough phase lockSlide52
SynchronizationRhythmic Movements
Are the agents in-phase or out of phase with each other?
Discrete Events
Did events occur within a specified time window? Slide53
Autonomic Arousal
Nonlinear time series analysis
Recurrence plotsSlide54
Emergence
The whole is greater than the sum of its parts.Basis of a scientific sociology (Durkheim)
Are there phenomena, e.g. social institutions, that cannot be reduced to the psychology of individuals?Gestalt psychology arrived a decade later.Slide55
Emergence
Agents interactEventually patterns emergeConsolidate in
the form of institutionsPatterns affect the behavior of new individuals entering the systemKarl Weick’s studies of experimental cultures.
Emergence Lite –Hierarchical structures form with little downward motion
Emergence Hardcore –Strong downward influence
Types
Phase shifts
1/f distributions
Boundary conditionsSlide56
Emergence
Classic Hierarchy Concept
Hierarchies Generally
Any time A affects dynamics of B, not reciprocal
(Haken)Slide57
Catastrophe theory
All discontinuous changes of events can be explained by one of 7 elementary topological forms.Forms differ in their levels of complexity. Cusp is the most commonly used.
Singularity theorem:Given a maximum of 4 control parameters there is only one behavior response
surfaceTies up some basic dynamics, self-organization, phase shifts, entropy levels, some types of emergence processesSlide58
Catastrophe theoryModels with 1 order
parameter
The foldThe cuspThe swallowtailThe butterfly
CUSPOID series
1 control parameter2 control parameters3 control parameters
4 control parametersSlide59
Catastrophe theoryModels with 2 order parameters
The wave crest
(hyperbolic umbilic)
The hair (elliptic umbilic)
The mushroom (parabolic umbilic)
UMBILIC series
3 control
parameters
3 control parameters with an interaction between order parameters
4 control parameters with an interaction between order parametersSlide60
Catastrophe theoryClassification theorem:
Given a fixed number of control parameters
(up to 5) there is only one surface associated with them (choose 1 or 2 order parameters).
If we know the number of stable states of the behavioral surface, we know how many control parameters are operating.
The number of control variables and the behavioral spectrum are unbreakable packages.If we know how many behavioral states there are, we know the number of control parameters in the process and what they do.Slide61
Catastrophe theory
The Cusp2 attractors, 1 repellor, 1 saddle, 2 control parametersBIFURCATION explains large v small differences.
ASYMMETRY explains proximity to the threshold of change.df(y)/
dy = y3 – by – aSlide62
Catastrophe theory
Each model has 3 characteristic equations:The response surface is perhaps the most important
e.g. df(y)/
dy = y3 – by –
a for the cusp.The bifurcation
set contains
critical points where behavior changes.
It is the
derivative
of the response
surface
The potential
function
captures the dynamics when the system is standing still.
It is the
integral
of the response
surface
Useful for defining statistical distributions associated with each catastrophe model.Slide63
Cusp Applications
Often shown with its bifurcations set in many of the early
applications.Illustrates gradients instead of control parametersHysteresis – Behavior change back and forth, or up and down the manifold.
Presence strongly suggests a cusp dynamic.Slide64
Cusp Applications
Stock market crashes
Euler buckling, workload
Object choiceSlide65
Phase Shift
Two-state phase shifts are cusp catastrophesSlide66
Other Catastrophe Models
Fold
SwallowtailSlide67
Other Catastrophe ModelsButterfly
Wave CrestSlide68
Other Catastrophe ModelsHair
MushroomSlide69
Entropy
Heat lossMotion of molecules(Shannon) information & entropy
Hs =
Si [
pi
log
2
(1/
p
i
)]
Information is what you need to predict a system state
Entropy is the information you don’t have for complete prediction
Information + Entropy = H
MAX
Hs
H
MAX
when odds of each state are equal
Categorical states
Not sensitive to temporal ordering
Cross-entropy and other spin-off metricsSlide70
Entropy
Does heat loss heat-death of a system?
No, self-organizes, creates hierarchies to conserve energyHow systems defy 2
nd law of thermodynamicsTopological
– motion of the system generates information
(Prigogine)
Lyapunov exponent
Kolmogorov-Sinai (
Shannon-based for 2+ dimensional categories)
Shannon still used, but Info = Entropy
Other types
intended for continuous measurements (
ApEn
,
SampleEn
).Slide71
Essence of Effective ModelsSlide72
Questions?
That’s all folks!