Furman University Electronic Journal of Undergraduate Mathematics Volume QUANTIFYING CHAOS IN DYNAMICAL SYSTEMS WITH LYAPUNOV EXPONENTS MICHAEL VAN OPSTALL Abstract In this pap er we analyze the dynamics of a four dimensional me chanical system which exhibits sensitive dep endence on initial conditions The aim of the pap er is to intro duce the basic ideas of chaos theory while assuming only a course in ordinary dierential equations as a prerequisite Introduction Dynamical systems in short are systems which exhibit change As such the eld of dynamical systems is varied and rich Many dynamical systems can b e mo deled by systems of dierential equations or discrete dierence equations Such systems are called deterministic Examples of such systems include those of classical mechanics Sensitive dependence on initial conditions is a phenomenon where slight distance b etween the initial conditions of a system grows exp onentially Determin istic dynamical systems that exhibit a sensitive dep endence on initial conditions are known as chaoticManyphysical systems are chaotic from the driven simple p endulum to the more complex system mo deled in this pap er Dynamical systems are classied as discrete or continuous A discrete dynamical system given by one or more dierence equations is one in which a function f is iterated on an initial condition x The set of all p oints generated by iterating f b eginning with x is known as the orbit of x under f Acontinuous system is generally given by one or more dierential equations Continuous orbits are known as trajectories There are several diculties in working with chaotic systems Systems of dif ferential equations that b elievechaotically are always nonlinear This nonlinearity makes an analytic solution to these equations dicult In addition to the nonlinear ity a continuous system which exhibits sensitive dep endence on initial conditions must have dimension of at least three that is it must have three indep endent variables The system discussed in the present pap er has degree four and hence cannot b e easily visualized Despite these diculties the fundamental concepts of the science are accessible to anyone who has taken a course in ordinary dierential equations OneDimensional Discrete Systems Despite their simple nature systems in a single variable can b e used to mo del many things One go o d example is the logistic map x x l x which is used as a simple mo del for p opulation n n n Received by the editors Septemb er Mathematics Subject Classication D Key words and phrases Chaos Dierential Equations Dynamical Systems This pap er was written while the author was an undergraduate at Hop e College MICHAEL VAN OPSTALL Figure The logistic map with with an attracting xed p oint left and another with it a rep elling xed p oint growth Some features of dynamical systems are easiest to demonstrate in single dimensional systems so a few are describ ed here The orbit of the function is computed according to the relation x f x n n The logistic map describ ed ab ove is an example of a onedimensional discrete sys tem Apoint where the functions value is unaected by further iteration ie x x is called a xedpoint A xed p ointwhich is approached by orbits n n is known as an attractor and one from which orbits diverge is a rep eller Figure represents a logistic map with an attracting xed p oint and a chaotic logistic map with a rep elling xed p oint One way to quantify chaotic b ehavior in a system is to measure the divergence n between orbits of twopoints with small initial separation Assume f is the nth iteration of a function f Then for two dierent initial conditions x and x the n n separation b etween these orbits is given b y jf x f xj as a function of the numb er of iterations If we assume that the separation of the tra jectories grows or shrinks exp onentially wehave n n n jf x f xj e and is called the Lyapunov exp onent If wetake the initial separation between tra jectories to b e small we obtain n n n f x f x df log log n n dx n n Noting that x f x f x wenddf dx using the chain rule n n n n Y df n n f f f f f x f x m dx m QUANTIFYING CHAOS IN DYNAMICAL SYSTEMS WITH LYAPUNOV EXPONENTS From this wearrive at our nal formula for the Lyapunov exp onent of a one dimensional discrete system n n Y X lim log f x lim log jf x j m m n n n n m m This exp onent represents the average exp onential rate of divergence of nearby orbits A zero exp onent implies linear divergence A p ositive exp onent indicates sensitive dep endence on initial conditions as p oints initially close together will diverge exp onentially along neighb oring tra jectories Negative exp onents are found in systems where tra jectories converge so the initial separation b etween twopoints will decrease in time Noting that the formula for a singledimensional Lyapunov exp onent is simply an average of the logarithm of the size of the derivative a formula for continuous systems can b e obtained For a continuous system the mean b ecomes the exp ected value of log jf xj and wehave the following formula for the Lyapunov exp onent for a single dimensional continuous system Z f xlogjf xjdx Higher Dimensional Systems For many real systems a singledimensional mo del is inadequate Unfortunately along with a b etter multidimensional mo del we gain more problems in calculating the Lyapunov exp onent of a system The equation derived for single variable discrete systems do es not directly applyand nonlinear dierential equations p ose problems as they are dicult or imp ossible to solve Wemust often resort to numerical metho ds to solve these problems Phase Space The phase space of a system is the ndimensional space in which the p oints of an ndimensional system reside A graph of tra jectories in the phase space is known as a phase diagramFor two dimensional systems the phase space lies in the plane known as the phase plane and is easily visualized For higher dimensional systems however the phase space is often pro jected into two dimensions for easy viewing In our system describ ed b elow the four variables dening the phase space were paired to pro duce two phase diagrams Atwo dimensional phase diagram often plots the velo city of a b o dy against its p osition Attractors In the onedimensional case p oints to which orbits converged were known as attracting xed p oints The xed p oint is a sp ecial case of an attractor In higher dimensional spaces tra jectories with small initial separation are sometimes pulled together into a single tra jectory an attractor In these higher dimensional systems these attractors can b e curves or surfaces An attractor in a chaotic system is known as a strange attractor Lyapunov Exp onents In an ndimensional dynamical system wehave n Lyapunov exp onents Each represents the divergence of k volume k length k k area etc The sign of the Lyapunov exp onents indicates the b ehavior of nearby tra jectories A negative exp onent indicates that neighb oring tra jectories converge to the same tra jectory A p ositive exp onent indicates that neighb oring tra jectories diverge When tra jectories diverge exp onentially a slight error in mea surement of the initial p oint could b e catastrophic as the error grows exp onentially as well If in equation is taken to b e the slight error in measuring a systems MICHAEL VAN OPSTALL Figure Left Convergence of tra jectories Center two concentric circular tra jectories Right divergence of tra jectories state eventuallythiserrorgrows in accordance with the Lyapunov exp onent Fig ure represents the three typ es of tra jectory b ehavior Any measurementtaken has some error The Lyapunov exp onent aords us a measure of how quickly this error grows If the Lyapunov exp onent is negative error actually decreases Consider the damp ed p endulum a slight error in measure ment do es not lead to a large overall error since the p endulum eventually comes to rest We are primarily interested in systems where one or more of the Lyapunov exp onents is p ositive In accordance with our informal denition of chaos b ehavior of a system exhibiting sensitive dep endence on initial conditions we can dene a chaotic system as one with at least one p ositiveLyapunov exp onent Predictability in a system is lost here and measurement error grows exp onentially Experimental Setup Physical Setup The physical dynamical system we studied consisted of a p endulum of length and mass m attached to a blo ckofmassM oscillating on the end of a spring with spring constant k This apparatus w as forced with forcing p function f tA cos t A sin twhich is the motion of a camshaft of length displaced A units from the axis of rotation driven with frequency Sensors connected to a Realtime VAX recorded the p osition of the cart and angular displacementofthependulumx and resp ectively To obtain data for the carts velo city and the p endulums angular velo cityv and resp ectively the data was generated by taking numerical derivatives actually the slop e b etween neighb oring p oints A diagram of our system app ears in Figure Phase Diagrams After recording data for dierent frequencies of forcing twodimensional phase plots were pro duced for v vs x and vs The system was chaotic at high driving frequencies The phase diagrams are given in Figure Figure illustrates chaotic and p erio dic time series The problem of exp erimental noise is quite evident in these gures Spurious data p oints can cause problems when calculating the