Linear vs. Nonlinear Jaeseung Jeong, Ph.D Department of Bio and Brain Engineering, KAIST

Chaos 1. the formless shape of matter that is alleged to have existed before the Universe was given order. 2. complete confusion or disorder. 3. Physics; a state of disorder and irregularity that is an intermediate stage between highly ordered motion and entirely random motion. Nonlinear dynamics and Chaos : the tiniest change in the initial conditions produces a very different outcome, even when the governing equations are known exactly - neither predictable nor repeatable

800 BC, Greek “χaos” – complete absence of order Aspect of chaos (1) Isaac Newton (1642 – 1727) “get headaches contemplation the three-body gravitational problem such as Sun, Moon, and Earth Nonlinear dynamics and Chaos

Hadamard and Duhem They were interested in the movement of a ball on a negatively curved surface and on the failure to predict its trajectory due to the lack of knowledge of its initial condition.

(2) King Oscar II (1829 – 1907) offered a prize of 2500 crowns to anyone solve the n-body problem  stability of the Solar System Nonlinear dynamics and Chaos

N-body problem The classical n-body problem is that given the initial positions and velocities of a certain number (n) of objects that attract one another by gravity, one has to determine their configuration at any time in the future.. Nonlinear dynamics and Chaos

(3) Jules Henri Poincarè (1854 – 1912) Won the Oscar II’s contest,  not for solving the problem, but for showing that even the three-body problem was impossible to solve. (over 200 pages ) Nonlinear dynamics and Chaos “…it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon” - in a 1903, essay "Science and Method"

N-body problem This problem arose due to a deterministic way of thought, in which people thought they could predict into the future provided they are given sufficient information. However, this turned out to be false, as demonstrated by Chaos Theory. Nonlinear dynamics and Chaos

Systems behaving in this manner are now called “chaotic.” They are essentially nonlinear, indicating that initial errors in measurements do not remain constant, rather they grow and decay nonlinearly (usually exponentially) with time. Since prediction becomes impossible, these systems can appear to be irregular, but this randomness is only apparent because the origin of their irregularities is different: they are intrinsic, rather than due to external influences.

What is chaos? The meteorologist E. Lorenz He modeled atmospheric convection in terms of three differential equations and described their extreme sensitivity to the starting values used for their calculations. The meteorologist R May He showed that even simple systems (in this case interacting populations) could display very “complicated and disordered” behavior. D. Ruelle and F. Takens They related the still mysterious turbulence of fluids to chaos and were the first to use the name ‘strange attractors.’

Nonlinear dynamics and Chaos Lorenz attractor

Nonlinear dynamics and Chaos

The Logistic equation X n+1 =AX n (1-X n )

The Logistic equations

R

Laminar(regular) / Turbulent(chaotic) Turbulent of gas flows Nonlinear dynamics and Chaos

High flow rate : Laminar  Turbulent Department of BioSystems Nonlinear dynamics and Chaos

What is Chaos? M Feigenbaum He revealed patterns in chaotic behavior by showing how the quadratic map switches from one state to another via periodic doubling. TY Li and J Yorke They introduced the term ‘chaos’ during their analysis of the same map. A. Kolmogorov and YG Sinai They characterized the properties of chaos and its relations with probabilistic laws and information theory.

Taffy – pulling machine Nonlinear dynamics and Chaos

The strength of science It lies in its ability to trace causal relations and so to predict future events. Newtonian Physics Once the laws of gravity were known, it became possible to anticipate accurately eclipses thousand years in advance. Determinism is predictability The fate of a deterministic system is predictable This equivalence arose from a mathematical truth: Deterministic systems are specified by differential equations that make no reference to chance and follow a unique path.

Chaos systems Newtonian deterministic systems (Deterministic, Predictable) Probabilistic systems (Non-deterministic, Unpredictable) Chaotic systems (Deterministic, Unpredictable)

Fractal As a non-fractal object is magnified, no new features are revealed. As a fractal object is magnified, ever finer features are revealed. A fractal object has features over a broad range of sizes. Self-similarity The magnified piece of an object is an exact copy of the whole object.

Fractal measure : Scaling law or power law Scaling: The value measured of a property, such as length, surface area, or volume, depends on the resolution used to make the measurement is called the scaling relationship. The simplest form of the scaling relationship is that the measured value of a property Q(r) depends on the resolution used to make the measurement with the equation: Q(r)=Br b (B and b are constants). This form is called a power law. Log Q(r) Log r

Fractal – Self-similarity

Fractal Art

Fractal music

Dynamical system and State space A dynamical system is a model that determines the evolution of a system given only the initial state, which implies that these systems posses memory. The state space is a mathematical and abstract construct, with orthogonal coordinate directions representing each of the variables needed to specify the instantaneous stae of the system such as velocity and position Plotting the numerical values of all the variables at a given time provides a description of the state of the system at that time. Its dynamics or evolution is indicated by tracing a path, or trajectory, in that same space. A remarkable feature of the phase space is its ability to represent a complex behavior in a geometric and therefore comprehensible form (Faure and Korn, 2001).

Phase space and attractor

For any phenomena, they can all be modeled as a system governed by a consistent set of laws that determine the evolution over time, i.e. the dynamics of the systems.

Linear vs. Nonlinear Conservative vs. Dissipative Deterministic vs. Stochastic A dynamical system is linear if all the equations describing its dynamics are linear; otherwise it is nonlinear. In a linear system, there is a linear relation between causes and effects (small causes have small effects); in a nonlinear system this is not necessarily so: small causes may have large effects. A dynamical system is conservative if the important quantities of the system (energy, heat, voltage) are preserved over time; if they are not (for instance if energy is exchanged with the surroundings) the system is dissipative. Finally a dynamical system is deterministic if the equations of motion do not contain any noise terms and stochastic otherwise.

Attractors A crucial property of dissipative deterministic dynamical systems is that, if we observe the system for a sufficiently long time, the trajectory will converge to a subspace of the total state space. This subspace is a geometrical object which is called the attractor of the system. Four different types of Attractors: Point attractor: such a system will converge to a steady state after which no further changes occur. Limit cycle attractors are closed loops in the state space of the system: period dynamics. Torus attractors have a more complex ‘donut like’ shape, and correspond to quasi periodic dynamics: a superposition of different periodic dynamics with incommensurable frequencies (Faure and Korn, 2001; Stam 2005).

Stam, 2005

Chaotic attractors The chaotic (or strange) attractor is a very complex object with a so-called fractal geometry. The dynamics corresponding to a strange attractor is deterministic chaos. Chaotic dynamics can only be predicted for short time periods. A chaotic system, although its dynamics is confined to the attractor, never repeats the same state. What should have become clear from this description is that attractors are very important objects since they give us an image or a ‘picture’ of the systems dynamics; the more complex the attractor, the more complex the corresponding dynamics.

Three-dimensional Lorenz attractors

Characterization of the attractors I If we take an attractor and arbitrary planes which cuts the attractor into two pieces (Poincaré sections), the orbits which comprise the attractor cross the plane many times. If we plot the intersections of the orbits and the Poincaré sections, we can know the structure of the attractor.

Characterization of the attractors II The dimension of a geometric object is a measure of its spatial extensiveness. The dimension of an attractor can be thought of as a measure of the degrees of freedom or the ‘complexity’ of the dynamics. A point attractor has dimension zero, a limit cycle dimension one, a torus has an integer dimension corresponding to the number of superimposed periodic oscillations, and a strange attractor has a fractal dimension. A fractal dimension is a non integer number, for instance 2.16, which reflects the complex, fractal geometry of the strange attractor.

Fractal dimension of the Attractor

Characterization of the attractors III Lyapunov exponents can be considered ‘dynamic’ measures of attractor complexity. Lyapunov exponents indicate the exponential divergence (positive exponents) or convergence (negative exponents) of nearby trajectories on the attractor. A system has as many Lyapunov exponents as there are directions in state space.

Characterization of the attractors IV A chaotic system can be considered as a source of information: it makes prediction uncertain due to the sensitive dependence on initial conditions. Any imprecision in our knowledge of the state is magnified as time goes by. A measurement made at a later time provides additional information about the initial condition. Entropy is a thermodynamic quantity describing the amount of disorders in a system.

Control parameters and multistability Control parameters are those system properties that can influence the dynamics of the system and that are either held constant or assumed constant during the time the system is observed. Parameters should not be confused with variables, since variables are not held constant but are allowed to change. Multistability: For a fixed set of control parameters, a dynamical system may have more than one attractor. Each attractor occupies its own region in the state space of the system. Surrounding each attractor there is a region of state space called the basin of attraction of that attractor. If the initial state of the system falls within the basin of a certain attractor, the dynamics of the system will evolve to that attractor and stay there. Thus in a system with multi stability the basins will determine which attractor the system will end on.

The escape time plot gives the basin of attraction.

Bifurcations In a multistable system, the total of coexisting attractors and their basins can be said to form an ‘attractor landscape’ which is characteristic for a set of values of the control parameters. If the control parameters are changed this may result in a smooth deformation of the attractor landscape. However, for critical values of the control parameters the shape of the attractor landscape may change suddenly and dramatically. At such transitions, called bifurcations, old attractors may disappear and new attractors may appear (Faure and Korn, 2001; Stam 2005).

Bifurcations

This EEG time series shows the transition between interictal and ictal brain dynamics. The attractor corresponding to the inter ictal state is high dimensional and reflects a low level of synchronization in the underlying neuronal networks, whereas the attractor reconstructed from the ictal part on the right shows a clearly recognizable structure. (Stam, 2003)

Route to Chaos Period doubling As the parameter increases, the period doubles: period- doubling cascade, culminating into a behavior that becomes finally chaotic, i.e. apparently indistinguishable visually from a random process Intermittency A periodic signal is interrupted by random bursts occurring unpredictably but with increasing frequency as a parameter is modified. Quasiperiodicity A torus becomes a strange attractor.

R

Intermittency

The importance of dynamical stationarity Time series generated from nonlinear dynamical systems exhibit nonstationary (i.e. time-dependent) based on statistical measures (weak statistics) including the mean and variance, despite that the parameters in the dynamical process all remain constant. It indicates that the statistical stationarity of the time series does not imply its dynamical stationarity. Given that the EEG is possibly generated by the dynamical, cognitive process of the brain, the dynamical nonstationarity of the EEG can reflect on the state transition of the brain.

AD/HD: Attention-Deficit/Hyperactivity disorder Correspondence with cognitive science (for a pretreated data): Cognitive tasks (rest, image recognition, games…) = Brain Dynamical State Change in cognitive States (Attention, Brain Functions …) = Nonstat. Detection Main Hypothesis: Since ADHD could have shorter characteristic time for attention, we could expect same order behavior inside a Cognitive State, which could be found analyzing the time criterion in loss of Dynamical Nonstationarity. Definition of the Dynamical stationarity: For two consecutive windows of a non stationary dynamical system time series, there should be change in dynamic from passage of one windows to another one. Dynamical nonstationarity