PH3160 Non-Linear Phenomena and Chaos.

  Glossary of Terms used in ph3160 ... compiled by Andrea Johnson.  

Affine:
         A linear function which has a constant derivative.

Attractor:
         Stable fixed point.  All the flow is towards an attractor.

Basin of attraction:
         Set of initial conditions giving rise to trajectories that
         approach a given attractor as t goes to
         infinity, is the basin of attraction for that attractor.

Bifurcation:
         Splitting into two branches.  An example of this is the roots 
         of the equation f(x,c) = x^2 + c  as c changes from positive values 
         where there are no real roots, through 0, one root, to negative 
         values where there are two real roots.

Bistability:
       If the origin is locally stable and not globally stable 
       (see global/local motion) then the stable large amplitude branches 
       are produced this gives jumps which are catastrophic changes 
       brought about by small changes in the control parameter.


Cantor set:
         If you begin with a line and remove the middle third then
         remove the middle third of the remaining segments and so on.  
         The Cantor set is the dust of particles that remains.  
         They are infinitely many but their total length is zero.

Cellular automata:
         Dynamical systems that are completely discrete in time and space.

Centre:
       The case where trajectories are in the form of closed curves and
       correspond to periodic solutions.  
       An example of this is the behaviour of a simple pendulum with the 
       centre at the origin of the phase plane portrait.

Control space:
        A diagram of equilibrium points against a control parameter. 
        For example, the control space
        diagram for the map f(x,c) = x^2 + c would be a parabola in x-c
        space reflected about the c axis.


Direction Field:
         A plot indicating the general direction trajectories would take
         on a phase diagram.  

Duffing equation:
           eg.  x " + a x' + b x  + c x^3= 0. 
           This is the model for a simple oscillator as far as the cubic term 
           and subject to friction.
           There are several varieties in which the cubic non linearity is 
           common to each.

Elliptic point:
         Closed trajectories around points of equilibrium are known as 
         neutral centres or elliptic points.

Equilibrium point:
         The point at which all thermodynamic variables such as 
         temperature, volume and pressure are constant throughout time 
         and space.  
         It is the point at which dy/dx goes to 0/0.

Ergodic theory.
         The system is said to be a Ergodic when the time averages (the
         average of time intervals) are the same as the control space averages, 
         where the control space average is weighted by the probability that 
         the trajectory visits a particular portion of the control space.

Feigenbaum number:
          4.66201609...  It can be applied to a mapping that goes through a
          sequence of period-doubling bifurcations on the way  to chaos.  
          Common ratio in parameter changes over certain periods.

Fixed point:
           If the evolution of a system is unchanging it has reached a 
           fixed point.

Flows:
         Models with time and space values continuous, corresponding to a 
         finite number of ordinary difference equations.


Fractal dimension:
            Although shapes such as Koch's snowflake curve can be 
            represented in a 2 dimensional plane they have a non-integer 
            dimension.  This can be explained by realising that the curve
            'fills space' more efficiently than a 1 dimensional line but 
            is equally not a 2 D plane.  This fractal dimension can be 
            used on real physical examples such as coast lines.

Global/Local motion:
          Global motion is stable to small perturbations but not to big 
          ones around a fixed point,            
          whereas local motion is  unstable to small perturbations and 
          stable to big ones.

Hamiltonian systems:
         Hamiltonian systems are not dissipative, ie. their complex
         behaviour does not diminish towards simple behaviour.

Hausdorff dimension:
         To define the d-dimensional volume of a sharp point for an 
         arbitrary, non-integer d the Hausdorff dimension of the space is 
         the value for which the d-dimension volume changes from infinity 
         to zero.

Heteroclinic orbit:
           An orbit which connects two distant saddle points.

Homoclinic orbit:
           Trajectories that start and end on the same saddle points.

Homoclinic point
       Topological dodge - cutting and unwrapping the phase plane to
       make it cylindrical so that for orbits on the cylindrical phase 
       space, 2 heteroclinic points go towards one homoclinic point.

Hopf bifurcation:
        Hopf bifurcation occurs in 2 D systems when a stable fixed point
        becomes unstable to form a limit cycle or a stable limit cycle
        becomes unstable to a fixed point as a control parameter is varied.

Hyperbolic point:
         A point in phase space which is bisected by a pair of separatrixes 
         whereby it is stable along one and unstable along the other.

Intermittency:
         Where the behaviour of a system switches back and forth between
         apparently regular behaviour and chaotic behaviour.

Invariant manifolds:
         A set of points performing trajectories heading directly
         towards or away from a saddle point.
         There are two types, stable manifolds and unstable manifolds.

Invariant torus:
         A torus where any trajectories on its surface do not change,
         despite any change in control space.

Invertible map:
            A map is invertible if x_(n + 1) = M (x_(n)) can be solved
            uniquely given x_(n).  Then the inverse mapping can be 
            defined M^(-1) such that x_(n) = M^(-1) (x_(n +1)).

Jacobian matrix:
        Defined as a matrix whose element in the ith row and jth column 
        is  df_i/ dx_j.

KAM theory:
         Kolmogorov-Arnold-Moser theory explains the mechanism whereby
         chaos is created in the standard mapping.

Limit cycle:
         In 2 D flows, the possibility of another type of attractor
         arises.  It is an isolated closed trajectory called a limit cycle.

Linear operator:
         For a linear operator L, linear superposition holds, ie.  L 
         acting on af(x)  and bg(x), L(af(x) + bg(x) ) =  aL(f(x)) + bL(g(x)). 
         Where  a and b are constants and f(x) and g(x) are suitably regular 
         functions.


Lyapunov exponents:
         The gradient of a function at its fixed point, x*, 
         f'(x) = [df/dx]_x=x*. 
         The value of the exponent determines the stability of the point

Map/mapping:
         A set of equations that give the state of a system at time t + 1 when
         the state at time t is known.  
         All the states of the system are continuous ie. arbitrary real
         numbers, but time steps remains discrete.

Nodes:
         Fixed points which can be stable (eigenvalues real and negative) 
         or unstable (real and positive).

Nonlinear operator:
         A non-linear operator does not satisfy the equation shown under 
         linear operator.

Nullclines:
         Curves for which x' = 0 and y' = 0 where the direction
         vector of flow is parallel to the y-axis and x-axis respectively.

Phase space:
       A plot of a system's motion as described by the coordinates and momenta 
       of a point.  The coordinates can be a pair of coupled first order 
       differential equations describing a pendulum for example.

Pitchfork bifurcation:
         Common in physical problems having symmetry, normal form: 
         x' = rx - x^3. It is subcritical if x(t)
         goes towards 0 and an unstable solution suddenly becomes stable
         and throws off two unstable branches.  
         It is supercritical if x(t) goes towards infinity and a stable 
         solution suddenly becomes unstable and throws off two stable branches.

Poincare  section:
         A line segment which is chosen to cut through the limit cycle
         to reduce the original 2 dimensional problem to examining it in 
         1 dimension. 
         Or similarly a 2-D section sampling motion in 3 D. 

Potential:
            For a function f(x) the potential V(x) is defined as 
            f(x) = -dV/dx.

Quasiperiodicity:
            An attractor which is a closed trajectory on a toroidal
            surface in 2 dimensional phase space.

Repellor:
            An unstable fixed point.  All the flow is away from a
            repellor.  Opposite to an attractor.

Resonance:
         A phenomenon occurring when the driving frequency of a system
         is close to the natural frequency which leads to a maximum 
         amplitude oscillations.

Relaxation oscillations:
        Oscillations which are non-sinusoidal in behaviour and which
        switch rapidly from one extreme of time scales to another.

Saddle-node bifurcations:
        Basic mechanism by which fixed points are created or destroyed. 
        A stable fixed point and an unstable fixed point suddenly appear 
        as some parameter r passes through a critical value (the simplest 
        example is, for the one dimensional case,
        x' = r - x^2, where r is less than or equal to nought).

Saddle point:
            Point of unstable equilibrium also known as hyperbolic point.

Self-parity:
         A property of fractals whereby a section of phase space can be 
         enlarged for infinity and it will continue to show detail 
         (there is no physical analogue) and self-similar patterns recur.

Separatrix:
            Consists of a pair of heteroclinic orbits, it separates
            different basins of attraction.

Stability:
         When there is no flow.

Stable manifold:
         Set of points performing trajectories heading directly towards 
         a fixed point or periodic orbit.
         Opposite to an unstable manifold, both are types of invariant
         manifold.

Strange attractors:
            An attractor in phase space with fractal dimension.

Transcritical bifurcation:
        Where a fixed point exists for all parameter values but the
        stability of this point may change as the parameter varies.  
        Example x' = rx - x^2 as r goes from negative to positive.

Unstable manifold:
         Set of points performing trajectories heading away from a fixed
         point or periodic orbit.
         Opposite to stable manifold.   A type of invariant manifold.

Van der Pol equation:
         x" + x + e*(x2 - 1)x' = 0, where e is greater or equal to
         zero, i.e. linear restoring force and non linear damping.





MW - 18/03/97