# 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
• $(x_1,x_2,\dots,x_n)^T = \vec f(x_1,x_2,\dots,\x_n,t).$

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: