- Asymptotic stability A
fixed point is asymptotically stable if
it is stable and nearby initial conditions tend to the fixed point in
positive time. For limit cycles ,
it is called orbital asymptotic
stability and then there is an associated phase shift. A fixed point is
locally stable if the eigenvalues of the
linearized system have
negative real parts. A limit cycle is orbitally asymptotically stable
if the Floquet multipliers of the
linearized system lie inside the
unit circle with the exception of a multiplier with value 1.
- Attractor An attractor is a trajectory of a dynamical system
such that initial conditions nearby it will tend toward it in forward
time. Often called a stable attractor but this is redundant.
- Averaging A method in which one can average over the period of
some system when one of the variables evolve slowly compared to length
of the period.
- Bifurcation point This is a point in parameter space where we
can expect to see a change in the qualitative behavior of the system,
such as a loss of stability of a solution or the emergence of a new
solution with different properties.
- Bifurcation diagram This is a depiction of the solution to a
dynamical system as one or more parameters vary. Typically, the
horizontal axis has the parameter and the vertical axis has some
aspect of the solution, such as, the norm of the solution, the maximum
and/or minimum values of one of the state variables, the frequency of
a solution, or the average of one of the state variables.
- Bistability The presence of two
coexistent attractors in a
dynamical system. For example, two stable fixed points or a stable
fixed point and a stable limit cycle. Birhymicity has been used
to define a system with two stable limit cycles.
- Chaos A long time behavior of a dynical system characterized by
a great deal of irregularity. Typically, two nearby trajectories will
diverge exponentially.
- Continuation branch A curve of fixed points, limit cycles, etc
as a function of some parameter.
- Domain of attraction Also called basin of attraction is
the region in state space of all initial conditions that tend to a
particular solution such as a limit cycle, fixed point, or other
attractor.
- Eigenvalues Complex numbers, that satisfy where A is an matrix and x is some vector. In this
case, x is called an eigenvector.
- Fixed Point A special trajectory of the dynamical system which
does not change in time. It is also called an equilibrium,
steady-state, or singular point of the system.
- Floquet Multipliers Complex numbers, that satisfy where is an periodic
matrix and is a periodic vector.
- Global stability When referring to an attractor, this means that
the domain of attraction is the entire phase space. When referring to
a dynamical system, it means that all initial conditions tend to one
of the attractors.
- Hard loss of stability This term is often used when there is a
subcritical bifurcation which turns around and stabilizes. Most
typically, it is used in reference to a Hopf bifurcation. As some
parameter is changed, the switch in behavior is sudden rather than
gradual. See also Hysteresis.
- Heteroclinic orbit A trajectory which tends to fixed points
in both positive and negative time. The fixed points do not have to be
the same.
- Hopf bifurcation This is the appearance of a new branch of
periodic solutions from a branch of fixed points. The classic earmark
of this is that a pair of complex conjugate
eigenvalues of the
linearized system change from negative to positive real parts.
- Homoclinic A trajectory that tends to the same fixed point in
positive and negative time.
- Hysteresis As a parameter is increased, the behavior makes a
sudden jump at a particular value of the parameter. But as the
parameter is then decreased, the jump back to the original behavior
does not occur until a much lower value. In the region between the
two jumps, the system, is bistable.
- Invariant set
Any orbit or region in phase space which
does not change under the differential equations. For example, a
fixed point or a periodic orbit are both invariant sets.
- Limit cycle An isolated periodic orbit (as compared to the
solutions to for which there are infinately many periodic
solutions passing through any region in phase space.)
- Limit point A saddle-node bifurcation for fixed points. Two
fixed points coalesce and disappear as a parameter varies. Also called
a turning point.
- Linearization If the dynamical system is then the
linearization is the dynamical system, where A is the
derivative of F evaluated at a solution.
- Local stability A fixed point is locally stable or stable
if nearby initial conditions remain nearby in positive time.
- Liapunov function Any function,
V(X) of
n variables which vanishes at a fixed point, say, X=0
, is positive in some neighborhood of the fixed point, and
satisfies dV/dt < 0 along trajectories of the dynamical
system.
- Nullclines Curves drawn in the phase portrait along which one
of the state variables does not change in time. Used frequently in
2-dimensional systems.
- Period doubling bifurcation A bifurcation in which the system
switches to a new behavior with twice the period of the original
system. The hallmark of this is a Floquet multiplier of -1.
- Period doubling cascade A sequence of period doubling
bifurcations ultimately ending in chaotic behavior.
- Periodic orbit A trajectory which after time , the period,
comes back to its initial point.
- Phase plane A two-dimensional phase portrait of a
two-dimensional dynical system.
- Phase portrait A plot of two or more dynamical variables
against each other.
- Phase space The set of all possible initial conditions for a
dynamical system. The dimension of the phase space is the
number of intial conditions required to uniquely specify a trajectory;
it is the number of variables in the dynamical system.
- Orbit Trajectory of differential equation.
- Saddle-loop A bifurcation in which a limit
cycle
collides with
a saddle-point at a homoclinic orbit. As the bifurcation is
approached, the period of the limit cycle tends to infinity as
.
- Saddle-node loop A bifurcation in which a
limit cycle collides
with a saddle-node point.
As the bifurcation is approached, the
period of the limit cycle tends to infinity as the
.
- Saddle-node point A limit point.
- Saddle point A fixed point that has at least one positive
eigenvalue and one negative eigenvalue in its
linearization. More
generally, a fixed point for which there are trajectories that tend to
the fixed point in both positive and negative time.
- Sink A locally asymptotically stable fixed point.
- Source A fixed point that is unstable and not a saddle-point.
All trajectories move away in positive time.
- Stable manifold The set of all points in phase space
which are attracted to a fixed point or other
invariant orbit in
positive time .
- Subcritical bifurcation The branch of solutions occurs for
parameter values on the opposite side of the loss of
stability. Often indicates that the new branch is unstable.
- Supercritical bifurcation The branch of solutions occurs for
parameter values on the same side as the loss of
stability. Often indicates that the new branch is stable.
- Trajectory The solution to a dynamical system in forward and
backward time passing through a specified initial condition.
- Unstable manifold The set of all points in phase space
which are attracted to a fixed point or other
invariant orbit
in
negative time .