SCTPLS Annual Conference, Raleigh, NC Nonlinear Dynamics Workshop, 2018 APPLICATIONS GALORE Applications Galore 1. Friction-free introduction to NDS concepts and how they connect. (Stephen Guastello) 2. DAVID KATERNDAHL Medical practice 3. DAVID SCHULDBERG Clinical psychology, psychotherapy 4. J. BARKLEY ROSSER Jr. Economics
By the end of the day itll all look easy So lets get started Friction-free intro THE PARADIGM SHIFT ELEMENTARY NONLINEAR DYNAMICAL SYSTEMS
Attractors and stability Bifurcations and instability Chaos Fractals COMPLEXITY When agents and variables interact
Self-organization Synchronization Emergence Catastrophe theory Phase shifts Entropy Nonlinear methods 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 R2
The qualitative aspects of the phenomena Paradigm Shift Conventional Paradigm Nonlinear Dynamics Linear relationships
Nonlinear relationships Static situations Changes over time Different kinds of changes display different dynamics Paradigm Shift Conventional Paradigm
Seemingly random process Outcomes are proportional in inputs Simple cause and effect Nonlinear Dynamics Determined by simple functions Little things can have big consequences, vice-versa Control variables that behave differently
Emergent phenomena Paradigm Shift Conventional response Nonlinear response Ignore, stifle random blips Maintain equilibrium, stability, and control Look for disruptions
external to the system It was no blip Navigate a repertoire of nonlinear change processes Equilibrium replaced by attractors Look for intrinsic dynamics Conventional Paradigm
Nonlinear Dynamics 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). 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 selforganizing processes Attractors Fixed points An incoming point can proceed in directly Or spiral in
Attractors Attractor basin Area around an attractor where the attracting force can operate. Some attractors are stronger than others Chaos in multi-attractor systems This point almost
escaped the grip: Attractors Repellor -- An inverseattractor Incoming points (objects) are deflected outward in any direction. System of 3 repellors Attractors Saddle
Has properties of an attractor and a repellor. You can visit, but you cant stay too long. Perturbed pendulum Attractors Fixed point attractor + saddle Unfolding of a romantic relationship over time Industry introduces new products
Attractors Limit cycles Oscillators, sine waves Dampened, control parameter is involved There are many in real life Economics Biology Signal processing
Stability, instability Principle of structural stability All points are behaving according to the same mathematical rule. Attractors, usually Repellors and saddles, no. Bifurcations, no.
Chaos YES if its a chaotic attractor NO if not. Not all chaos is a chaotic attractor 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.
Bifurcations A split in a dynamical field where different dynamics exist in each local region. Represent a pattern of instability Types Simplest is a critical point tipping point Points follow different trajectories
Hopf: A fixed point turns into a limit cycle Entire dynamic fields come and go depending a control parameter Two dynamic fields annihilate each other producing a new dynamic field. Bifurcations Pitchfork with 1 critical point Seen in cusp catastrophe models
Bifurcations Logistic map Verhulst equation Iteration: x2 = cx1(1 x1) Critical points Logistic map Chaos Seemingly random events that are predictable with simple but special equations. First discovered by Henri Poincar while trying to solve
the 3-body problem. Strange attractor discovered by E. Lorenz, meteorologist. Chaos Properties of chaos Expanding and contracting trajectories Sensitivity to initial conditions Two points start arbitrarily close together, but become far apart as iteration progresses. Non-repeating sequences
Bounded Latter 2 are matters of degree Chaos A chaotic time series 1.0 0.9 0.8 0.7 0.6 X
0.5 0.4 0.3 0.2 0.1 0.0 ITERATIONS Chaos Lorenz attractor Order parameters: X, Y, Z Control parameters: A, B,
C Pathways to Chaos Attractor fields with 2 or more attractors Coupled oscillators Newhouse et al: 3 is sufficient Not all combinations produce chaos
Pathways to Chaos Logistic map when the control parameter > 3.56 Other bifurcations structures too Chaos At least 60 chaotic systems are currently known Assess generically for level of complexity
Lyapunov Exponent Fractal Dimension Entropy Lyapunov Exponent Properties Interpretation Based on successive differences in y values
Large negative = fixed point Spectrum of values Small negative = dampened oscillator Indicates level of turbulence Generalizes as ||y||=elt If largest l > 0, system
is chaotic Converts to a fractal dimension: DL = el (Kaplan-Yorke) 0 = oscillator Small positive = aperiodic, selforganized criticality Large positive = chaos Turbulent air flow
Fractals Geometric items in fractional dimensions Produced 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 shape
Fractals Non-fractal Fractal Fractals Mandelbrot set Julia set Fractals Nature: Branching structures
Nature: Landscapes Fractals SCALE: The magnified piece of an object is an exact copy of the whole object. Fractals Principle extends to nonlinear time series analysis Fractals N (r) balls of radius r
needed to cover the object Make radius smaller, recount Slope = fractal dimension Fractals Power law distribution of object sizes Near 0
fixed point 1.0 line or an open circle 1.0 2.0 chaotic system that has self-organized into a lowdimensional configuration (self-organized criticality). Pink noise
2.0 Points are evenly distributed around a twodimensional 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 noise Optimum variability principle Sick or deficient systems display lower complexity than healthy ones 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) Self-Organization Create structure by forming feedback loops among subsystems
Several models Information flows among subsystems Driver-slave relationships (Haken): Dynamics of one subsystem affects output of another. (usually oneway). Sudden change in system organization is characterized as a phase shift.
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 simulations Now a moot point in light of real-world examples Optimum variability principle Hence the essence of the Complex Adaptive System Ashbys 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 again Self-Organization Kauffman: NK[C] model Differentiation scenario NK distribution Niche hopping scenario NK[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? Self-Organization Adaptive walk Hop with a pogo stick Parallel search parties explore options
Try a random idea sometimes Look for patterns Self-Organization Sandpiles and avalanche (Bak) Distribution of large and small piles after the
avalanche is a power law distribution. Out of control? The hive mentality Collective intelligence Work systems self-organize from the bottom up Physical boundaries shape how things self-organize Boundaries that used to exist between organizations
have broken down e-com Giving rise to networks. 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. Synchronization Special Case of S/O Rhythmic physical movements Discrete events closeenough in time Autonomic arousal or
EEG activity while people are doing something else Emotional contagion Sync Phenomena Sync of clocks Hyper-synchronization of neuron firings during epileptic seizure Electrodermal response of two people in conversation Coordination of efforts in a
work team Shift in weight from one foot to another during conversation. Synchronization General principles Two oscillators Feedback loop between them Control parameter that speeds up
Speed up enough phase lock Synchronization Rhythmic Movements Discrete Events Are the agents in-phase or out of phase with each other?
Did events occur within a specified time window? Autonomic Arousal Nonlinear time series analysis Recurrence plots 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. Emergence Agents interact Eventually patterns emerge Consolidate in the form of institutions Patterns affect the
behavior of new individuals entering the system Karl Weicks studies of experimental cultures. Emergence Lite Hierarchical structures form with little downward motion Emergence Hardcore
Any time A affects dynamics of B, not reciprocal (Haken) Catastrophe theory All discontinuous changes of events can be explained by one of 7 elementary topological forms.
Singularity theorem: Given a maximum of 4 control parameters there is only one behavior response surface Forms differ in their levels of complexity. Cusp is the most commonly used. Ties up some basic
dynamics, selforganization, phase shifts, entropy levels, some types of emergence processes Catastrophe theory Models with 1 order parameter CUSPOID series The The
The The 1 2 3 4 fold cusp swallowtail butterfly
control control control control parameter parameters parameters parameters Catastrophe theory
Models 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 parameters Catastrophe theory Classification 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 Catastrophe theory The Cusp 2 attractors, 1 repellor, 1 saddle, 2 control parameters
BIFURCATION explains large v small differences. ASYMMETRY explains proximity to the threshold of change. df(y)/dy = y3 by a 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. Cusp Applications Often shown with its bifurcations set in many of
the early applications. Illustrates gradients instead of control parameters Hysteresis Behavior change back and forth, or up and down the manifold. Presence strongly suggests a cusp dynamic. Cusp Applications Stock market crashes
Euler buckling, workload Object choice Phase Shift Two-state phase shifts are cusp catastrophes Other Catastrophe Models Fold
Swallowtail Other Catastrophe Models Butterfly Wave Crest Other Catastrophe Models Hair Mushroom
Entropy Heat loss Motion of molecules (Shannon) information & entropy Hs = Si [pi log2(1/pi)]
Information is what you need to predict a system state Entropy is the information you dont have for complete prediction Information + Entropy = HMAX Hs HMAX when odds of each state are equal Categorical states Not sensitive to temporal ordering Cross-entropy and other spin-off metrics Entropy Does heat loss heat-death of a system? No, self-organizes, creates hierarchies to conserve energy
How systems defy 2nd law of thermodynamics Topological 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). Essence of Effective Models Questions? Thats all folks!
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