The equations of motion of a mechanical system are usually of second order, and they determine the entire future from the initial state, which consists of both positions and velocities. The space of all states is called the phase space. For the mathematical pendulum (a point mass in the plane at one end of a massless rod whose other end is fixed), the configuration space is a circle, at each point of which one can choose any tangent vector as the velocity. Therefore, the phase space is a circle with a line attached to each point, that is, a cylinder. The pendulum equation for the position x and velocity υ, converts to so the dynamics is described by integral curves of the vector field R(x, υ)=(υ, −sin x) on the cylinder.
That the notion of phase space is natural is also suggested by the Liouville theorem: the skew-symmetry of the Hamilton equation makes the Hamiltonian vector field divergence-free, and accordingly, a Hamiltonian flow preserves the volume on phase space.
Generally, a dynamical system consists of a phase space and a time-evolution of first order. The phase space is a set with some structure, such as differentiable (in the case of differential equations, this belongs to smooth dynamics), topological (one then speaks of topological dynamics), or measurable (this is the subject of ergodic theory, which arose from the Liouville theorem), and the time evolution is a one-parameter family of transformations that preserve this structure and that map initial states to states at another time. The time parameter may run through real numbers (continuous-time system) or integers (discrete-time system, iterations of a map and possibly its inverse). Specifically, a continuous-time system is given by a family of maps. If ft(fs(x))=ft+s(x) for every s, t, x, then we say that this family is a flow. In the discrete-time case, one considers the iterates where f0(x)=x, fn+1(x)=f(fn(x)) for n≥0, and fn(x)=(f−1)n(x) for n<0, or if the map is not invertible, only positive iterates The maps fa(x)=ax(1−x) are a popular example of the latter (the so-called logistic map).
The long-term behavior of flows in the plane is well understood (SeePhase plane): In the long run, any orbit either approaches fixed points or is asymptotically periodic (Poincaré-Bendixson theorem). This is ultimately due to the fact that a closed curve, such as a periodic orbit, divides the plane into separate regions. Already in three-space, one gets chaotic behavior, such as in the Lorenz attractor.
Qualitative Theory of Differential Equations and Dynamical Systems
On the phase space of a smooth continuous-time dynamical system, the time evolution is given by a first-order differential equation Suppose the right-hand side R satisfies a Lipschitz condition in x. This means that there is a constant M such that d(R(x, t), R(x′, t))≤Md(x, x′) for all x, x′. The basic Picard theorem then guarantees existence and uniqueness of solutions for any initial condition. Otherwise solutions may not be unique has infinitely many solutions with initial value 0) or may not exist for any uniform amount of time (the solutions of have singularities).
If solutions x(t) exist for all time, as we henceforth assume, they define time-t-maps by ft(x(0))=x(t). Each map ft is as smooth as the right-hand side R of the differential equation (smooth dependence on initial conditions).
If R is independent of t, then the differential equation is said to be “autonomous,” and R gives a vector field on the phase space that prescribes the velocity vectors of solutions. The family of time-t-maps is then a flow. The iterates of the time-1-map of a flow produce a discrete-time dynamical system whose study may yield useful information about the flow. If R depends on t, the system is said to be non-autonomous. Explicit time dependence can, for example, arise from forcing terms (forced pendulum ) or from varying parameters (driving of a swing by parametric forcing, ).
An “orbit” or trajectory of a continuous-time system is a parametrized curve An orbit or trajectory of a map consists of the sequence of images of a point under iteration of the map:
A singular point of a differential equation is a point x for which the right-hand side is zero for all t, that is, an equilibrium, or constant solution, or fixed point. Fixed points of a map f are those points x for which f(x)=x. A periodic point is a state that repeats at some positive time. For differential equations, this corresponds to solutions that are periodic functions of time; for maps, these are fixed points of an iterate. For the flow on the cylinder generated by all but four orbits are periodic; the point is two-periodic for the map 4x(1−x).
Fixed and periodic points can be anchors for the study of the global orbit structure, and therefore, it is important to understand the behavior of nearby orbits. A fixed point is said to be attracting if orbits of nearby
Figure 1. Stable and unstable manifold of a hyperbolic fixed point.
points stay nearby (Poisson stability) and converge to it for large positive time (asymptotic stability). (The example of a circle map like this with a fixed point at the top illustrates that the second condition does not imply the first.) This is the case if the differential of the map (or time-1-map in the case of a flow) at that point has only eigenvalues of absolute value less than 1. If all eigenvalues have absolute value greater than 1 then the point is repelling: There is a neighborhood which every other point leaves in positive time. The map f2(x)=2x(1−x) has 0 and as fixed points. 0 is repelling and is attracting. In fact, is superattracting: and orbits near approach faster than exponentially.
If eigenvalues of the differential are allowed to lie both inside the unit circle and outside it but not on it, then the fixed point is said to be “hyperbolic.” In this case the Hartman-Grobman theorem states that there is a continuous coordinate change that maps orbits near the fixed point to orbits of the linearized map. Moreover, tangent to the contracting and expanding subspaces of the linearization, there are the stable and unstable manifold of points positively and negatively asymptotic to the fixed point, respectively. These are smooth subspaces without self-intersections, but they may be packed into the phase space in a complicated way.
For periodic points of maps, the analysis of stability can be carried out by studying the appropriate iterate; for a flow, one likewise studies Poincaré return maps as follows. Take a small hypersurface through the periodic point transverse (for example, orthogonal) to the flow. The orbit of every point sufficiently near the periodic point returns to this surface at a time close to the period, and this defines a map from a neighborhood of the periodic point in the hypersurface into the hypersurface, with the original periodic point as a fixed point.
If, for example, a periodic orbit is an attracting fixed point for the return map, then it is a limit cycle: All nearby orbits are asymptotic to it. For the mathematical pendulum from the introduction, the circle {υ=0} is a
Figure 2. A section.
Figure 3. Two attracting points.
section of the cylinder, and the return map is defined for all of its points. It has two fixed points, and all other points are two-periodic.
A property complementary to stability of a fixed or periodic point as defined in terms of the behavior of nearby orbits (i.e., perturbations of the initial condition) is that of stability under perturbations of the dynamical system. An easy way to guarantee this is transversality, which is weaker than hyperbolicity: A fixed point x=f(x) is said to be transverse if the derivative of f at x does not have 1 as an eigenvalue. (This implies that there are no other fixed points nearby.) In this case, any C1-perturbation of f (i.e., one that changes derivatives only a little) also has a (transverse) fixed point near x. The origin is a nontransverse fixed point of x(1−x), and indeed, it is absent for x(1−x)+ε with ε<0. The creation of two (hyperbolic) fixed points as ε changes from negative to positive is a basic local bifurcation. For differential equations, transversality corresponds to invertibility of the differential of the right-hand side.
An “invariant set” is a union of orbits; for example, [0, 1] is invariant under 4x(1−x). It is a repeller if it has a neighborhood in which only orbits of points in the invariant set stay for all positive time. It is an “attractor” if there is a neighborhood that is mapped into itself and the intersection of whose positive-time iterates is the invariant set. (Usually, one also requires that there is no proper subset with the same property. Thus, Figure 3 shows two attracting fixed points, and the interval with these as endpoints is not considered an attractor.)
The “basin of attraction” is the set of points that are asymptotic to the attractor. For example, the interval (0, 1) is the basin of attraction of for the map 2x(1−x).
Figure 4. The Birkhoff-Smale theorem.
If two hyperbolic fixed points (saddles) in the plane are connected by a curve segment that lies in the unstable manifold of one of them and in the unstable manifold of the other, then this segment is called a “separatrix” (because it often separates two basins of attraction). More generally, the intersection of the stable manifold of one hyperbolic point with the unstable manifold of another is called a “heteroclinic intersection,” and the orbit of every intersection point is called a “heteroclinic orbit.” If the two fixed points coincide then one uses the terms “homoclinic intersection” and homoclinic orbit instead. The Birkhoff-Smale theorem asserts that if a homoclinic intersection is transverse (or if there is a pair of transverse heteroclinic intersections, that is, two hyperbolic points such that the unstable manifold of each of them intersects the stable manifold of the other point transversely), then there is a “horseshoe,” that is, a rectangle that (under an iterate) gets mapped across itself in a horseshoe-like fashion as illustrated in Figure 4 and in Anosov and Axiom-A systems. This implies directly that there is an invariant Cantor set on which the dynamical system exhibits deterministic chaos.
There are several ingredients that make up chaotic behavior. One of these is recurrence. There are several recurrence properties. A point is said to be recurrent if it returns arbitrarily near to its initial condition. For a rigid rotation of a circle by an irrational number of degrees, all points have this property. By contrast, a point is said to be transient or wandering if it has a neighborhood all of whose images are pairwise disjoint. For the circle map with a fixed point on top, all nonfixed points are wandering. The set of nonwandering points is called the nonwandering set. Nonwandering orbits can be closed by a localized C1 perturbation of the map (Pugh closing lemma). A dynamical system is said to be “topologically transitive” if it has a dense orbit and “minimal” if every orbit is dense. Irrational circle rotations have both properties. Minimality does not reflect chaotic behavior. Existence of a dense orbit is equivalent to the condition that for any two open sets there are arbitrarily large times at which the image of one of these sets overlaps the other. A strengthening of this property is that such overlap occurs for all sufficiently large times; this is called topological mixing and implies sensitive dependence on initial conditions. Following Devaney, one can say that a dynamical system is chaotic if it is topologically transitive and the set of periodic points is dense. This also implies sensitive dependence. A condition stronger than sensitive dependence is “expansivity”: There is a universal positive constant by which the images of any two points, no matter how close initially, are separated at some time. The cat map and horseshoes are good examples of dynamical systems with these properties.
The Poincaré return map is not the only construction that produces a new dynamical system with a different phase space. Another straightforward one is the product of two dynamical systems. For example, the flow of rotations of the unit circle given by x1=α cos t, x2=α sin t can be combined with a similar flow y1=ω cos t, y2=ω sin t to a flow on the two-torus in defined by all four equations. (Note that the plane defined by y1=y2=0 is a section on which the return map is a time-2π/ω-map of the first flow.) If one projects this to the x1y1-plane, one gets Lissajous figures. In some applications, these readily show whether modes in weakly nonlinear oscillators are locked together.
Arnol’d, V.I. 1989. Mathematical Methods of Classical Mechanics, 2nd edition, Berlin and New York: Springer (original Russian edition 1974)
Arnol’d, V.I. 1992. Ordinary Differential Equations, Berlin and New York: Springer (original Russian edition 1971, translated from 3rd edition 1984)
Fielder, B. (editor). 2002. Handbook of Dynamical Systems, vol. 2, Amsterdam and New York: Elsevier
Hasselblatt, B. & Katok, A. (editors). 2002. Handbook of Dynamical Systems, vol. IA, Amsterdam and New York: Elsevier
Hasselblatt, B. & Katok, A. 2003. Dynamics: A First Course (see also vol. 1B, 2005) Cambridge and New York: Cambridge University Press
Katok, A. & Hasselblatt, B. 1995. Introduction to the Modern Theory of Dynamical Systems, Cambridge and New York: Cambridge University Press
Katok, A., de la Llave, R., Pesin, Y. & Weiss, H. (editors). 2001. Smooth Ergodic Theory and Its Applications (Summer Research Institute Seattle, WA, 1999), Proceedings of Symposia in Pure Mathematics, vol. 69. Providence, RI: American Mathematical Society
converts to 
so the dynamics is described by integral curves of the vector field R(x, υ)=(υ, −sin x) on the cylinder.
of maps. If ft(fs(x))=ft+s(x) for every s, t, x, then we say that this family is a flow. In the discrete-time case, one considers the iterates 
where f0(x)=x, fn+1(x)=f(fn(x)) for n≥0, and fn(x)=(f−1)n(x) for n<0, or if the map is not invertible, only positive iterates 
The maps fa(x)=ax(1−x) are a popular example of the latter (the so-called logistic map).
Suppose the right-hand side R satisfies a Lipschitz condition in x. This means that there is a constant M such that d(R(x, t), R(x′, t))≤Md(x, x′) for all x, x′. The basic Picard theorem then guarantees existence and uniqueness of solutions for any initial condition. Otherwise solutions may not be unique 
has infinitely many solutions with initial value 0) or may not exist for any uniform amount of time (the solutions of 
have singularities).
) or from varying parameters (driving of a swing by parametric forcing, 
).
An orbit or trajectory of a map consists of the sequence of images of a point under iteration of the map: 
is a point x for which the right-hand side is zero for all t, that is, an equilibrium, or constant solution, or fixed point. Fixed points of a map f are those points x for which f(x)=x. A periodic point is a state that repeats at some positive time. For differential equations, this corresponds to solutions that are periodic functions of time; for maps, these are fixed points of an iterate. For the flow on the cylinder generated by 
all but four orbits are periodic; the point 
is two-periodic for the map 4x(1−x).
with a fixed point at the top illustrates that the second condition does not imply the first.) This is the case if the differential of the map (or time-1-map in the case of a flow) at that point has only eigenvalues of absolute value less than 1. If all eigenvalues have absolute value greater than 1 then the point is repelling: There is a neighborhood which every other point leaves in positive time. The map f2(x)=2x(1−x) has 0 and 
as fixed points. 0 is repelling and 
is attracting. In fact, 
is superattracting: 
and orbits near 
approach 
faster than exponentially.
for the map 2x(1−x).
with a fixed point on top, all nonfixed points are wandering. The set of nonwandering points is called the nonwandering set. Nonwandering orbits can be closed by a localized C1 perturbation of the map (Pugh closing lemma). A dynamical system is said to be “topologically transitive” if it has a dense orbit and “minimal” if every orbit is dense. Irrational circle rotations have both properties. Minimality does not reflect chaotic behavior. Existence of a dense orbit is equivalent to the condition that for any two open sets there are arbitrarily large times at which the image of one of these sets overlaps the other. A strengthening of this property is that such overlap occurs for all sufficiently large times; this is called topological mixing and implies sensitive dependence on initial conditions. Following Devaney, one can say that a dynamical system is chaotic if it is topologically transitive and the set of periodic points is dense. This also implies sensitive dependence. A condition stronger than sensitive dependence is “expansivity”: There is a universal positive constant by which the images of any two points, no matter how close initially, are separated at some time. The cat map and horseshoes are good examples of dynamical systems with these properties.
defined by all four equations. (Note that the plane defined by y1=y2=0 is a section on which the return map is a time-2π/ω-map of the first flow.) If one projects this to the x1y1-plane, one gets Lissajous figures. In some applications, these readily show whether modes in weakly nonlinear oscillators are locked together.
