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# Analysis of Chaotic Dynamics - 2. Stability

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Stability analysis normally deals with the effects of perturbations on the state of a system. For stable systems one expects that small perturbations of the state of the system have only limited effects. Regarding technical systems it is important to know whether there exists a practically useful domain of attraction of the steady state solution or not. In technical systems, one not only has perturbations acting on the state of the system but also perturbations of the system itself. To numerically study the stability of a nonlinear dynamic system we use Lyapunov exponents. By means of Lyapunov exponents we can obtain stability diagrams in parameter space. We have already observed that variation of parameters may often essentially change behaviour. Lyapunov exponents quantitatively characterize the average exponential divergence or convergence of neighbouring trajectories with respect to a reference trajectory. To measure the average speed with which a given displacement is amplified, we introduce a vector a which measures the initial distance between two neighbouring trajectories. As time increases, the distance may change and we would like to know how the vector a evolves with time. Therefore we take the norm of "a of t" which means the length of a. The mean exponential rate of divergence of the two initially close trajectories is then defined by the limit of 1 over t times the natural logarithm of the quotient a of t over a. Hence, we consider the stretching of an infinitesimally small distance a after an infinitely long time. Sigma is called Lyapunov exponent. The Russian mathematican Oselédec could prove the existence of sigma and that it is finite. This simple notion of the divergence of a normalized vector can be generalized to higher dimensions. The number of exponents coincides with the dimension of the state space. For a continuous dynamic system the behaviour along the flow is in the average constant, consequently, the exponent sigma along the flow is equal to zero. In addition to the case of the zero exponent along the flow, negative exponents will signal periodic orbits. At least one positive exponent indicates divergence of initially close trajectories and therefore a chaotic orbit. In the domain of periodic solutions zero exponents mean the loss of stability of one orbit and gain of stability of another orbit, i.e. zero exponents characterize bifurcations. By means of Lyapunov exponents we can generate stability diagrams, indicating the system's behaviour e.g. for damping coefficient d against forcing amplitude a. We generated a sequence of such diagrams by starting at different initial conditions. The initial
velocity was kept constant at x-dot = 0, and only the initial displacement x was varied, starting at x = -2. Blue areas indicate negative, yellow and red positive exponents, and green areas zero exponents, which indicate bifurcations. In these graphs we stepwise increase the displacement of x by delta x = 0.5. The varying structure within the stability diagram indicates different long-term behaviour depending on initial states for identical values of the system's parameters. Superimposing all diagrams resulting from our calculations by using the Lyapunov exponents approach yields this picture. Hatched regions indicate multiple steady states. That is, the same oscillator with fixed values of parameters may have different longterm behaviour. This picture represents a small portion of Ueda's diagram already used before. For a large range of the parameters a and d the complete diagram shows a variety of different behaviours. Ueda also observed different kinds of behaviour for the same parameter values. Again these regions are indicated by the vertically hatched subdomains. Let's now restrict ourselves to the most interesting area for a from 5 to 15 and d from 0 to 0.2. The results obtained via Lyapunov exponents confirm Ueda's observations. The phenomenon of multiple or coexisting final behaviours in the same system is an important common feature of nonlinear dynamic systems.
Besides different periodic oscillations even chaotic motions may coexist; that is, different initial states may lead to completely different steady state solutions. As an example we pick out of the parameter space a = 12, d = 0.1, all other parameters being fixed. Here we have coexistence of a periodic oscillation and a chaotic motion. The time history of the chaotic motion is already well known to us.
For the same parameter values but starting at a different initial state we obtain the time history of the harmonic oscillation. In the three-dimensional ring model the coexistence of the periodic motion and the chaotic motion is obvious. The Poincaré map will finally be used to visualize the different final behaviours in the surface of section. Starting with this initial state, the motion settles down to a strange attractor of a familiar shape. For another initial condition the final behaviour is represented by a single point. Different initial states obviously become entrained to different final behaviours. In cases where different final motions are possible, the determination of the domain of attraction of the final motions must be done for the given state space. Simply using simulations and evaluating many starting conditions can be very time consuming. A more pragmatic approach is the cell mapping method, first described by C.S. Hsu from Berkeley, U.S.A. It determines the evolution of a system based upon a large collection of very small cells. For our example we subdivide the domain of interest of the x, x-dot plane or surface of section into small subdomains. For convenience we use rectangular cells, but in general the shape of the cells can be arbitrary. Within each cell we use a certain number of initial sample points. Starting from these points, in the course of time the system evolves along trajectories. The end point of each trajectory after one forcing period is determined by time integration. The relative probability of a cell mapped onto other cells is calculated by determining what fraction of the points ends within a target cell. This information can be used in the context of Markov chain theory to determine steady states in the surface of section. For the parameter value a = 12 the chaotic solution is represented by this strange attractor. The coexisting one-periodic orbit is represented by a single point. The steady state solutions from the cell mapping cover the same area of the surface as the Poincaré map. Let us now supplement the attractors with their domains of attraction: From the green area,
the system ends up in the strange attractor with 100 % probability. Starting in the yellow area with 50 - 100 %,
and with 0 to 50 % probability from the violet area. The probability distribution for the periodic solution is obtained in the same way.
Starting in "green" we end up with 100 %, starting in "yellow" we end
up with 50 to 100 % and starting in "violet" we eventually end up with 0 to 50 % in the periodic solution.
Finally we will study the time behaviour for the two coexisting domains of attraction. To settle down to the strange attractor from the most distant area of the domain of attraction it takes about 14 seconds.
From the outermost cells of the domain of attraction of the periodic solution it takes on the average 16 seconds to end up at the steady state. By using the cell mapping method, similar portraits for other system's parameters and other phase angles can easily be obtained.
TVD-Verfahren
Einfügungsdämpfung
Folge <Mathematik>
Sterbeziffer
Hausdorff-Dimension
Gruppenoperation
Zahlenbereich
Anfangswertproblem
Trajektorie <Mathematik>
Massestrom
Sigma-Algebra
Analysis
Raum-Zeit
Computeranimation
Stabilitätstheorie <Logik>
Dynamisches System
Logarithmus
Mittelwert
Existenzsatz
Uniforme Struktur
Inverser Limes
Fließgleichgewicht
Abstand
Störungstheorie
Analysis
Divergenz <Vektoranalysis>
Parametersystem
Dicke
Ljapunov-Exponent
Exponent
Quotient
Kraft
Zeitbereich
Orbit <Mathematik>
Endlich erzeugte Gruppe
Störungstheorie
Physikalisches System
Vektorraum
Frequenz
Frequenz
Arithmetisches Mittel
Diagramm
Koeffizient
Mathematiker
Normalvektor
Aggregatzustand
Geschwindigkeit
Resultante
Parametersystem
Subtraktion
Ortsoperator
Exponent
Green-Funktion
Ungerichteter Graph
Physikalisches System
Rechnen
Chaotisches System
Raum-Zeit
Computeranimation
Dynamisches System
Algebraische Struktur
Multiplikation
Spannweite <Stochastik>
Diagramm
Flächeninhalt
Nichtunterscheidbarkeit
Fließgleichgewicht
Pendelschwingung
Aggregatzustand
Varietät <Mathematik>
Ebene
Punkt
Zahlenbereich
Anfangswertproblem
Trajektorie <Mathematik>
Chaotisches System
Physikalische Theorie
Raum-Zeit
Computeranimation
Unterring
Harmonischer Oszillator
Flächentheorie
Stichprobenumfang
Fließgleichgewicht
Modelltheorie
Parametersystem
Bruchrechnung
Determinante
Kraft
Zeitbereich
Orbit <Mathematik>
Attraktor
Physikalisches System
Kette <Mathematik>
Frequenz
Integral
Flächeninhalt
Konditionszahl
Evolute
Dimension 3
Poincaré-Abbildung
Garbentheorie
Repellor
Aggregatzustand
Diskrete Wahrscheinlichkeitsverteilung
Flächeninhalt
Green-Funktion
Computeranimation
Flächeninhalt
Kraft
Zeitbereich
Zwei
Attraktor
Computeranimation
Parametersystem
Phasenverschiebung
Zeitbereich
Kraft
Zwei
Ähnlichkeitsgeometrie
Physikalisches System
Computeranimation