Glossary
|A | B
| C | D | E | F
| G | H | I | J
| K | L |
| M | N | O | P
| Q | R | S | T
| U | V | W | X
| Y | Z
A
Autonomous: An
Autonomous differential equation
does not explicitly use the independent variable (often
denoted by t) in its formulation, i.e. it takes the form x' = f(x).
Reaction rate constants are usually temperature
dependent; the rate of a reaction usually increases as the temperature
rises. The temperature dependence often follows Arrhenius'
equation: k(T) = A exp(-Ea/RT) where T is the absolute temperature, R the
universal gas constant, Ea is the activation energy (specific to each reaction),
and A is the "pre-exponential" or "frequency" or "entropy"
factor.
B
Boundary Conditions:
Auxiliary conditions for a differential equation
that solutions must satify at two or more different values
of the independent variable. For example, the solution u to the
differential equation might be required to satisfy
u(0)=0 and u(1)=0.
Boundary Value Problem:
Any system of differential equations coupled with two or
more boundary conditions is a boundary value problem
(BVP).
C
Carrying Capacity:
The maximal steady state solution for a differential equation that
describes a population. For example, in the logistic model
p' = rp(1-p/K) for a population, the parameter K is the
carrying capacity, because p(t) = K is a maximal steady solution.
Closed-form Solution:
This is simply a formula for a solution to
the given differential equation. For example, the differential equation
x' = ax has a closed form solution of x(t) = x0 eat.
Almost all differential equations have solution but
most do not have closed form solutions.
Critical Point:
See Equilibrium Point.
D
Many reactions follow elementary differential rate
laws such as v = k f([A], [B], ...) where f([A], [B], ...) is
a function of the concentrations of reactants and products. That is, the
rate varies as the concentrations change. A proportionality constant, k,
is called the rate constant of the reaction.
Damping: In
physical systems such as pendulums and bungee cords, the slowing effect
of a force such as friction or air resistence is referred to as damping.
The effect of damping is
generally proportional to velocity and thus takes the form a x'
in the differential equation.
Demand Curve:
A demand curve shows the quantity of some commodity
that buyers would purchase at various prices that might be
charged per unit.
E
It is convenient to refer to the extent
of reaction. As the reactants are consumed and the products
are produced, their concentrations change. If the initial concentrations
of A, B, P and Q are [A], [B], [P] and [Q], resp., then the extent of reaction
is defined: x = -([A]-[A]0)/na
= -([B] - [B]0)/nb = ([P]-[P]0)/np
= ([Q]-[Q]0)/nq. Alternately, each species concentration
is a function of the extent of reaction: [A] = [A]0 - nax,
etc.
Eigenvalue:
A number 
is an eigenvalue
of the square matrix A if there exists a nonzero vector x
such that 
. Eigenvalues
are used to determine the stability of critical points of systems of first-order,
autonomous differential equations.
Eigenvector:
A nonzero vector x is an eigenvector
of the square matrix A if there exists a nonzero number
such that . Eigenvectors
are used to determine stable and unstable directions of critical points
of systems of first-order, autonomous differential equations.
Equilibrium Point: A
point which is invariant under the flow for a given dynamical system. It
is the graph of a constant solution. One finds equilibrium points by finding
where the right hand sides of the given system of differential equations
are all zero, i.e. where there is no change in the dependent variables.
F
First Order Equation: A
first order differential equation is one that only involves the first derivative.
In general the order of a differential
equation refers to the highest derivative appearing in the equation.
G
H
I
Initial Conditions: An
initial condition for a first order differential equation
is an auxiliary condition specifying the value of the solution at
some value of the independent variable. These typically
take the form x(t0)=x0.
One objective of chemical kinetics is to solve
the differential rate law d[G]/dt = k f([A], [B], ...), and thereby express
each species concentration as a function of time: [G](t). Since solution
requires integration, we call it the integrated
rate law.
Integrator:
The numerical method used to compute an approximate solution to a differential
equation or system of differential equations. The best known integrators
are Euler's method and the Runge-Kutta methods, although there are many
other approaches that are superior to these.
Isosector:
J
Jacobian:
The Jacobian matrix of an
autonomous
system x'=f(x) of differential equations at a point
x0 is the matrix
Df(x0)
of partial derivatives of the right-hand side evaluated at that point.
K
L
Linear Term: A
linear term of a differential equation is a term of the form a(t) x^(n)
where a is a function of only the independent
variable, and the (n) denotes the nth
derivative. See also nonlinear term.
M
The marginal
effect of a change in a parameter on a system is an economic
description of the derivative of the system with respect to the
parameter in question.
A reaction mechanism
is a set of steps at the molecular level. Each step involves combinations
or re-arrangements of individual molecular species. The steps in combination
describe the path or route that reactant molecules follow to reach the
product molecules. The result of all steps is to produce the overall balanced
stoichiometric chemical equation for reactants producing products.
N
Nondimensional equations:
Frequently it is useful to remove the dependence of a variable
on a particular set of units and constants. To do so, one defines a new
variable which is scaled by the constant one wishes to remove, and writes
the equations in terms of the new variable. The resulting equations are
called nondimensional(ized) equations.
Nonlinear Term: A
noninear term of a differential equation is a term involving powers of
x^n or special functions such as sin(x) and exp(x). The (n)
denotes the nth derivative. See also linear term.
Nullcline: In
a two-dimensional system of differential equations the nullclines
are the curves where the vector field is either horizontal or vertical.
The horizontal nullcline is found by
setting y' = 0 since this says that there is no vertical component
of the vector field along this curve. Similarly, to find the vertical nullcline
we set x' = 0.
O
Optimal Control Theory:
Optimal Control Theory is used to solve for the optimal levels of
variables that are under the decision-maker's control
(control variables) over an interval
of time.
The optimal time paths for the control variables imply, via a
set of differential equations, time paths for variables describing the
system (state variables).
When the rate law has the special form of a product
(or quotient) of powers, f([A], [B], ...) = [A]a [B]b
[P]p [Q]q then a is the order
of the reaction with respect to A, b is the order w.r.t.
B, etc. Note that order may be positive, negative, integer, or non-integer.
Further, the sum a + b + p + q is the overall order of the reaction
rate law.
NOTE: there is no necessary relation between
orders and stoichiometric coefficients. That is, a might differ
from na.
P
Parameters: Quantities
in a differential equation that are constant. For example, in the differential
equation x' = ax, a is a parameter. However, in the system
of differential equations x' = ax, a' = 1 a is not a parameter
since it is described by a differential equation.
Phase Plane: A
solution of a system of differential equations x'
= f(x,y), y' = g(x,y) is a pair of functions (x(t), y(t)). This
solution is usually plotted in the (x,y)-plane
which is referred to as the phase plane. This
is a 2-dimensional version of the phase space.
Phase Portraits: A
phase portrait is a plot of the phase
plane showing multiple solutions to a given differential
equation.
Phase Space: A
solution of a system of differential equations
is a vector of functions 
.
These solutions live in R^n which is referred
to as the phase space. When n=2
the phase space is often called the phase plane.
Q
R
The rate of a chemical
reaction is defined in such a way that it is independent of
which reactant or product is monitored. We define the rate, v, of a reaction
to be v = (1/ng) d[G]/dt where ng is the signed (positive
for products, negative for reactants) stoichiometric coefficient of species
G in the reaction. Namely, v = (-1/na) d[A]/dt = (1/np)
d[P]/dt, etc.
S
Saddle Point:
For a planar system of first-order, autonomous differential
equations a critical point where the eigenvalues
of the Jacobian matrix evaluated at the critical
point are real and of opposite sign. Thus there exists a stable curve containing
the critical point such that solutions on this curve approach the critical
point as the independent variable goes to positive infinity, and an unstable
curve containing the critical point such that solutions on this curve approach
the critical point as the independent variable goes to negative infinity.
Separatrix:
Solution: Any
function which, when substituted into a differential equation, makes it
an identity. In other words, a function is a solution to a differential
equation if it makes the equation true. The function may be defined implicitly.
Spring Coefficient:
Stable Node:
For a planar system of first-order, autonomous differential
equations a critical point where the eigenvalues
of the Jacobian matrix evaluated at the critical
point are real and negative. Thus solution curves near the equilibrium
point limit on that point as the independent variable goes to positive
infinity.
Stablity:
The stability of an equilibrium
describes the behavior of solutions having nearby initial conditions. An
equilibrium is stable if all nearby solutions approach the equilibrium
as t goes to infinity. Otherwise it is said to be unstable.
Steady State:
A steady state solution to a differential equation is an
equilibrium solution.
Stiff: Loosely
speaking, a stiff differential equation
is one for which there are regions of phase space where the velocity or
magnitude of the vector field changes rapidly. Stiff
differential equations often require special integrators,
such as Gear's Method, that are sensitive to these rapid changes of magnitude.
Stoichiometry:
determines the molar ratios of reactants and products in an overall chemical
reaction. We express the stoichiometry as a balanced chemical equation.
For kinetics it is convient to write this as products minus reactants:
npP + nqQ - naA - nbB (instead
of the conventional equation naA + nbB ---> npP
+ nqQ). This indicates that na and nb
moles of reactants A and B, resp., produce np and nq
moles of products P and Q.
System of differential
equations: A system of differential
equations is usually n first order
differential equations and is written
when the equations are autonomous. The phase
space for a system of differential equations
is n dimensional.
T
Trajectory:
The graph of a family of solutionsthat pass through
a given point (or initial condition) in phase space.
Transversality Condition:
A Transversality condition provides a means of determining
an upper bound on the duration of a solution in time. I.e.
it allows us to find when the solution should stop.
It is called a transversality condition because it appears
graphically when a solution curve transversely crosses a nullcline or other
curve describing a terminal condition.
U
Unstable Node:
For a planar system of first-order, autonomous differential
equations a critical point where the eigenvalues
of the Jacobian matrix evaluated at the critical
point are real and positive. Thus solution curves near the equilibrium
point limit on that point as the independent variable goes to negative
infinity.
V
W
X
Y
Z
Zeros of a Solution: