Introduction to Dynamic Systems:Introduction to Dynamic Systems J. Devin McAuley Center for Neuroscience, Mind & Behavior Department of Psychology Bowling Green State University Email: mcauley@bgnet.bgsu.edu SMPC Workshop on Music and Dynamics, The Neurosciences Institute, San Diego, August 5th 2005


Millenium Bridge (London):J. Devin McAuley 2 Millenium Bridge (London)


What are Dynamic Systems?:J. Devin McAuley 3 What are Dynamic Systems? Dynamic systems specify change over time. This is useful because … Most everything in our world changes over time. One important distinction to make: Deterministic versus non-deterministic processes


Characterizing Change Over Time:J. Devin McAuley 4 Characterizing Change Over Time Four mathematical ideas used to characterize change over time: Steady States (Fixed points) Oscillations (Limit cycles) Chaos Noise


Music and Dynamics:J. Devin McAuley 5 Music and Dynamics In this workshop, our interest is in music and musical behavior. From this perspective, the study of dynamic systems provides tools for describing and understanding the dynamic characteristics of musical behavior.


Slide 6:J. Devin McAuley 6 The goal of this tutorial is to introduce basic concepts and tools used in the study of dynamic systems


Slide 7:J. Devin McAuley 7 Because this is a workshop, the goal is to have an interactive tutorial, not a lecture. Be a dynamic workshop participant and ask questions along the way! No background in dynamics is assumed, so no question is too simple! Some questions may be too hard though You can ask Ed those! 


Types of Dynamic Systems:J. Devin McAuley 8 Types of Dynamic Systems Discrete vs. Continuous Linear vs. Non-linear


Discrete Systems:J. Devin McAuley 9 Discrete Systems Typically expressed as difference equations. State variable, X, changes in discrete steps: n = 0, 1, 2, 3, …. X0 is the initial state (condition).


Discrete Systems:J. Devin McAuley 10 Discrete Systems Iteration – enumeration of successive states starting with initial state. X0, X1, X2, … Xn Fixed point: Xn+1 = Xn or (Xn+1 – Xn = 0) Two methods of investigation Numerical Graphical


Numerical Methods:J. Devin McAuley 11 Numerical Methods


Graphical Methods:J. Devin McAuley 12 Graphical Methods


Graphical Methods Continued …:J. Devin McAuley 13 Graphical Methods Continued …


Continuous Systems:J. Devin McAuley 14 Continuous Systems Typically expressed as differential equations. State variable, x, is continuous in time. dx/dt describes instantaneous rate of change. Fixed point occurs when dx/dt = 0!


Continuous Systems Continued …:J. Devin McAuley 15 Continuous Systems Continued … Of interest is how state variable, x, changes as a function of time, t. However, here’s the problem. It is not clear how we determine x(t).


Continuous Systems Continued …:J. Devin McAuley 16 Continuous Systems Continued … Two methods of investigation Numerical Integration Discrete approximation Graphical (Geometric analysis)


Numerical Methods: Integration:J. Devin McAuley 17 Numerical Methods: Integration But, as we will see, this method only works well for linear systems.


What Do Derivatives Mean?:J. Devin McAuley 18 What Do Derivatives Mean?


Graphical Methods Continued …:J. Devin McAuley 19 Graphical Methods Continued …


Graphical Methods Continued …:J. Devin McAuley 20 Graphical Methods Continued …


Summary:J. Devin McAuley 21 Summary Discrete Systems Expressed as difference equations Change occurs in discrete steps Steady state (fixed point) occurs when Xn+1 = Xn Both numerical and graphical methods for analysis. Continuous Systems Expressed as differential equations Change is continuous in time Steady state (fixed point) occurs when dx/dt = 0. Both numerical and graphical methods for analysis.


A Few Comments on Stability:J. Devin McAuley 22 A Few Comments on Stability Discrete systems Fixed points: states, X*, where Xn+1 = Xn. What happens for values of X different from X*? Continuous systems Fixed points: states, x*, where dx/dt = 0. What happens for values of x different from x*? Both questions are about stability.


Stability Continued …:J. Devin McAuley 23 Stability Continued … Fixed points can be stable (an “attractor”) or unstable (a “repeller”) Fixed points are considered stable or unstable based on what happens in regions around them. The key question is whether positions near the fixed point attracted to the fixed point or are repelled from it?


Stability Continued …:J. Devin McAuley 24 Stability Continued … Consider a cont. system with fixed point, x*: If values of x0 near x*, eventually get to x*, then the fixed point is locally stable. If all values of x0 eventually get to x*, then the fixed point is globally stable. The same is true for discrete systems & X*.


Basin of Attraction:J. Devin McAuley 25 Basin of Attraction x0


Next, we’re going to distinguish linear from non-linear systems:Next, we’re going to distinguish linear from non-linear systems


Linear Systems:J. Devin McAuley 27 Linear Systems In a linear dynamic system, the evolution of the state variable over time is described by a straight line. May be discrete: Or continuous:


Non-linear Systems:J. Devin McAuley 28 Non-linear Systems In a non-linear dynamic system, the evolution of the state variable over time is not described by a straight line. May be discrete: Or continuous:


Non-linear Systems Continued …:J. Devin McAuley 29 Non-linear Systems Continued … Biologically systems are generally described by non-linear differential equations. However, non-linear differential equations can be difficult (or impossible) to integrate! What should I do? Use numerical methods for discrete approximation Use graphical methods (geometric analysis)


Let’s now look at two examples:1. A discrete linear system2. A continuous non-linear system:Let’s now look at two examples:1. A discrete linear system2. A continuous non-linear system


Case Study 1: Growth & Decay:J. Devin McAuley 31 Case Study 1: Growth & Decay Let’s revisit the linear difference equation, and investigate the dynamics for different values of the parameter, a.


Cobweb Method: a > 1:J. Devin McAuley 32 Cobweb Method: a > 1


Cobweb Method: a = 1:J. Devin McAuley 33 Cobweb Method: a = 1


Cobweb Method: 0 < a < 1:J. Devin McAuley 34 Cobweb Method: 0 < a < 1


Summary: Six Possible Behaviors:J. Devin McAuley 35 Summary: Six Possible Behaviors Decay 0 1 Steady-state behavior a = 1 Alternating growth a < -1 Alternating decay -1 < a < 0 Periodic cycle a = -1


Quiz! Which of these are stable?:J. Devin McAuley 36 Quiz! Which of these are stable? Decay 0 1 Steady-state behavior a = 1 Alternating growth a < -1 Alternating decay -1 < a < 0 Periodic cycle a = -1


Stability Revisited …:J. Devin McAuley 37 Stability Revisited … So far, we’ve used the concept of stability only in relation to steady states (fixed points). However, in the next example, we shall see that the concept of stability also applies to periodic behavior. Stable periodic behavior, of the sort we will examine, is called limit cycle behavior.


Case Study 2: Periodic Behavior:J. Devin McAuley 38 Case Study 2: Periodic Behavior One simple way to describe periodic behavior is by a uniform trip around the perimeter of a circle. Specifying a circle requires two dimensions.


Polar and Rectangular Coordinates:J. Devin McAuley 39 A position on the circle can be described in either rectangular (x,y) coordinates or polar coordinates (, r). Polar coordinates are very convenient for describing oscillations. Polar and Rectangular Coordinates


Polar and Rectangular Coordinates:J. Devin McAuley 40 Polar and Rectangular Coordinates r (x, y)  180 0, 360 270 90


A Simple Model:J. Devin McAuley 41 A Simple Model Consider the following 2D non-linear differential equation:


Slide 42:J. Devin McAuley 42 Let’s start out by looking at each equation separately


Slide 43:J. Devin McAuley 43 Where are the fixed points?


Slide 44:J. Devin McAuley 44 Where are the fixed points?


Limit Cycle Trajectory for dr/dt = 0::J. Devin McAuley 45 Limit Cycle Trajectory for dr/dt = 0: r = 1  180 0, 360 270 90


What happens when r > 1?:J. Devin McAuley 46 What happens when r > 1?


Illustration of r0 > 1:J. Devin McAuley 47 Illustration of r0 > 1 r = 1  180 0, 360 270 90 r0


What about when 0 < r < 1?:J. Devin McAuley 48 What about when 0 < r < 1?


Illustration of 0 < r0 < 1:J. Devin McAuley 49 Illustration of 0 < r0 < 1 r = 1  180 0, 360 270 90 r0


Quiz! What happens when r0 = 0?:J. Devin McAuley 50 Quiz! What happens when r0 = 0? r = 1  180 0, 360 270 90 r0


Is the fixed point at r0 = 0 stable?:J. Devin McAuley 51 Is the fixed point at r0 = 0 stable? r = 1  180 0, 360 270 90 r0 No!


Does a change in r affect ?:J. Devin McAuley 52 Does a change in r affect ? r = 1  180 0, 360 270 90 No!


Summary:J. Devin McAuley 53 Summary Discrete versus Continuous Systems Linear versus Non-linear systems Two General Methods of Investigation Numerical Graphical Key concepts State variable Fixed points (steady states) Local versus global stability Circle description of periodic behavior/polar coordinate notation Amplitude r Phase  Stable limit cycle oscillation


Other Important Ideas …:J. Devin McAuley 54 Other Important Ideas … Sensitivity to initial conditions (value of x0) Bifurcations and period doubling Chaos versus quasi-periodicity A good example of a model that demonstrates these concepts is the logistic map:


Suggested Readings:J. Devin McAuley 55 Suggested Readings Glass, L. & Mackey, M. (1988). From Clocks to Chaos: The Rhythms of Life, New Jersey: Princeton University Press. Kaplan, D. & Glass, L. (1995). Understanding Non-Linear Dynamics. New York: Springer-Verlag. Port, R. & Van Gelder, T. (1995). Mind as Motion: Explorations in the Dynamics of Cognition. Cambridge, MA: MIT Press. Strogatz, S. (2003). Sync: The Emerging Science of Spontaneous Order. New York: Hyperion Books. Thelen, E. & Smith, L. (1994). A Dynamic Systems Approach to the Development of Cognition and Action. Cambridge, MA: MIT Press.