A geometric formulation of the theory of differential equations says that a differential equation is a vector field on a manifold . To understand this definition we present an informal description of a manifold and a vector field.
A manifold is any smooth geometric space (line, surface,
solid). The smoothness condition ensures that the manifold cannot have any sharp
edges. An example of a one-dimensional manifold is an
infinite straight line. A
different one-dimensional manifold is a circle. Examples of two-dimensional
manifolds are the surface of an infinite cylinder,
the surface of a sphere, the
surface of a torus, and the unbounded real plane (Fig. 1).
Three-dimensional manifolds are harder to visualize. The simplest example of a
three-dimensional manifold is unbounded three-space, 
. The surface
of a cone is an example of a two-dimensional surface that is not a manifold. At
the apex of the cone is a sharp point, which violates the smoothness
condition for a manifold. Manifolds are useful geometric objects because the
smoothness condition ensures that a local coordinate system can be erected at
each and every point on the manifold.
Figure 1: (a) The surfaces of the three-dimensional
objects are two-dimensional manifolds. (b) Examples of objects that are not
manifolds.
A vector field is a rule that smoothly assigns a vector (a directed line segment) to each point of a manifold. This rule is often written as a system of first-order differential equations. To see how this works, consider again the linear differential equation
Let us rewrite this second-order differential equation as a system of two first-order differential equations by introducing the new variable v, velocity, defined by dx/dt = v, so that
The manifold in this example is the real plane, 
, which consists of
the ordered pair of variables (x, v). Each point in this plane represents an
individual state , or possible initial condition, of the
system. And the collection of all possible states is called the phase
space of the system. A process
is said to be deterministic if both its future and
past states are uniquely determined by its present state.
A process is called semideterministic
when only the future state, but not the past, is uniquely determined
by the present state.
Not all physical systems are
deterministic, as the bouncing ball system (which
is only semideterministic) of Chapter 1 demonstrates.
Nevertheless, full determinism is commonly assumed in the classical scientific
world view.
A system of first-order differential equations assigns to each point of the manifold a vector, thereby forming a vector field on the manifold (Fig.\ 2).
Figure 2: Examples of vector fields on different manifolds.
In our example each point of the phase plane (x, v) gets assigned a vector (v, -x), which forms rings of arrows about the origin (Fig. 3).
Figure 3: Vector field and flow for a linear differential equation.
A solution to a differential equation is called a trajectory or an integral curve , since it results from ``integrating'' the differential equations of motion. An individual vector in the vector field determines how the solution behaves locally. It tells the trajectory to ``go thataway.'' The collection of all solutions, or integral curves, is called the flow (Fig.\ 3).
When analyzing a system of differential equations it is important to present both the equations and the manifold on which the equations are specified. It is often possible to simplify our analysis by transferring the vector field to a different manifold, thereby changing the topology of the phase space (see section 3.4.3). Topology is a kind of geometry which studies those properties of a space that are unchanged under a reversible continuous transformation. It is sometimes called rubber sheet geometry. A basketball and a football are identical to a topologist. They are both ``topological'' spheres. However, a torus and a sphere are different topological spaces as you cannot push or pull a sphere into a torus without first cutting up the sphere. Topology is also defined as the study of closeness within neighborhoods. Topological spaces can be analyzed by studying which points are ``close to'' or ``in the neighborhood of'' other points. Consider the line segment between 0 and 1. The endpoints 0 and 1 are far away; they aren't neighbors. But if we glue the ends together to form a circle, then the endpoints become identical, and the points around 0 and 1 have a new set of neighbors.
In its grandest form, Poincaré's program to study the qualitative behavior of ordinary differential equations would require us to analyze the generic dynamics of all vector fields on all manifolds. We are nowhere near achieving this goal yet. Poincaré was inspired to carry out this program by his success with the Swedish mathematician Ivar Bendixson in analyzing all typical behavior for differential equations in the plane. As illustrated in Figure 4, the Poincaré-Bendixson Theorem says that typically no more than four kinds of motion are found in a planar vector field, those of a source , sink , saddle , and limit cycle . In particular, no chaotic motion is possible in time-independent planar vector fields. To get chaotic motion in a system of differential equations one needs three dimensions, that is, a vector field on a three-dimensional manifold.
Figure 4: Typical motions in a planar vector field: (a) source, (b) sink, (c)
saddle, and (d) limit cycle.
The asymptotic motions ( 
limit sets)
of a flow are characterized by four general types
of behavior.
In order of increasing complexity
these are equilibrium points, periodic solutions,
quasiperiodic solutions, and chaos.
An equilibrium point of a flow is a constant, time-independent solution. The equilibrium solutions are located where the vector field vanishes. The source in Figure 0.4(a) is an example of an unstable equilibrium solution. Trajectories near to the source move away from the source as time goes by. The sink in Figure 0.4(b) is an example of a stable equilibrium solution. Trajectories near the sink tend toward it as time goes by.
A periodic solution of a flow is a time-dependent trajectory that precisely returns to itself in a time T, called the period . A periodic trajectory is a closed curve. Like an equilibrium point, a periodic trajectory can be stable or unstable, depending on whether nearby trajectories tend toward or away from the periodic cycle. One illustration of a stable periodic trajectory is the limit cycle shown in Figure 0.4(d). A quasiperiodic solution is one formed from the sum of periodic solutions with incommensurate periods. Two periods are incommensurate if their ratio is irrational. The ability to create and control periodic and quasiperiodic cycles is essential to modern society: clocks, electronic oscillators, pacemakers, and so on.
An asymptotic motion that is not an equilibrium point, periodic, or quasiperiodic is often called chaotic. This catchall use of the term chaos is not very specific, but it is practical. Additionally, we require that a chaotic motion is a bounded asymptotic solution that possesses sensitive dependence on initial conditions: two trajectories that begin arbitrarily close to one another on the chaotic limit set start to diverge so quickly that they become, for all practical purposes, uncorrelated. Simply put, a chaotic system is a deterministic system that exhibits random (uncorrelated) behavior. This apparent random behavior in a deterministic system is illustrated in the bouncing ball system (see section 1.4.5). A more rigorous definition of chaos is presented in section {4.10.
All of the stable asymptotic motions (or limit sets) just described (e.g., sinks, stable limit cycles), are examples of attractors . The unstable limit sets (e.g., sources) are examples of repellers . The term strange attractor (strange repeller) is used to describe attracting (repelling) limit sets that are chaotic. We will get our first look at a strange attractor in a physical system when we study the bouncing ball system in Chapter 1.
Maps are the discrete time analogs of flows. While flows are specified by differential equations, maps are specified by difference equations. A point on a trajectory of a flow is indicated by a real parameter t, which we think of as the time. Similarly, a point in the orbit of a map is indexed by an integer subscript n, which we think of as the discrete analog of time. Maps and flows will be the two primary types of dynamical systems studied in this book.
Maps (difference equations) are easier to solve numerically than flows (differential equations). Therefore, many of the earliest numerical studies of chaos began by studying maps. A famous map exhibiting chaos studied by the French astronomer Michel Hénon (1976), now known as the Hénon map , is
where n is an integer index for this
pair of nonlinear coupled difference equations,
with 
and 
being the parameter values
most commonly studied.
The Hénon map carries a point in the plane, 
, to
some new point, 
. An orbit of a map
is the sequence of points generated by some initial condition
of a map. For instance, if we start the Hénon map at the point
, we find that the orbit for this
pair of initial conditions is
and so on to generate 
, 
, etc.
Unlike planar differential
equations, this two-dimensional difference equation
can generate chaotic orbits. In fact, in Chapter 2 we will study a
one-dimensional difference equation called the quadratic map, which can also
generate chaotic orbits.
The Hénon map is
an example of a diffeomorphism of a manifold (in
this case the manifold is the plane ).
A map is a
homeomorphism if it is bijective (one-to-one and onto), continuous, and has
a continuous inverse.
A
diffeomorphism
is a differentiable homeomorphism.
A map with
an inverse is called invertible . A map without an
inverse is called noninvertible .
Maps exhibit similar types of asymptotic behavior as flows: equilibrium points, periodic orbits, quasiperiodic orbits, and chaotic orbits. There are many similarities and a few important differences between the theory and language describing the dynamics of maps and flows. For a detailed comparison of these two theories see Arrowsmith and Place, An introduction to dynamical systems.
The dynamics of flows and maps are closely related. The study of a flow can often be replaced by the study of a map. One prescription for doing this is the so-called Poincaré map of a flow. As illustrated in Figure 5, a cross section of the flow is obtained by choosing some surface transverse to the flow.
Figure 5: A Poincaré map for a three-dimensional flow with a
two-dimensional cross section.
A
cross section for a three-dimensional flow is shown in
the illustration and is obtained by choosing the x-y plane, (x, y, z=0).
The flow defines a map of this cross section to itself, and this map
is an example of a Poincaré map (also called a first return
map ). A trajectory
of the flow carries a point 
into a new point 
. And this in turn goes to the point 
. In this way
the flow generates a map of a portion of the plane, and an orbit of this map
consists of the sequence of points
and so on.
There is another reason for studying maps. To quote
Steve Smale
on the ``diffeomorphism problem,''
[T]here is a second and more important reason for studying the diffeomorphism problem (besides its great natural beauty). That is, the same phenomena and problems of the qualitative theory of ordinary differential equations are present in their simplest form in the diffeomorphism problem. Having first found theorems in the diffeomorphism case, it is usually a secondary task to translate the results back into the differential equations framework.The first dynamical system we will study, the bouncing ball system, illustrates more fully the close connection between maps and flows.