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A phase curve is a plot of the solution to a set of equations of motion in a phase plane (or more generally, a phase space) as a function of time (Tabor 1989, p. 14). Phase curves are sometimes also known as level curves. The motion along a phase curve is known as phase flow. When multiple phase curves corresponding to different initial conditions are plotted in the same phase plane, the result is known as a phase portrait.

For a two-dimensional map with ,where are the Lyapunov characteristic exponents.

A dynamical system in which the phase space volume contracts along a trajectory. This means that the generalized divergence is less than zero,where Einstein summation has been used.

A phase curve (i.e., an invariant manifold) which meets a hyperbolic fixed point (i.e., an intersection of a stable and an unstable invariant manifold) or connects the unstable and stable manifolds of a pair of hyperbolic or parabolic fixed points. A separatrix marks a boundary between phase curves with different properties.For example, the separatrix in the equation of motion for the pendulum occurs at the angular momentum where oscillation gives way to rotation. There are also many systems that have pairs of connected fixed points, e.g., the flow in an open cavity, which has a separatrix that connects two parabolic points.

The period for a quasiperiodic trajectory to pass through the same point in a surface of section. If the rotation number is irrational, the trajectory will densely fill out a curve in the surface of section. If the rotation number is rational, it is called the map winding number, and only a finite number of points in the surface of section will be visited by the trajectory.

Let be a one-parameter family of map satisfyingthen there exist intervals , and such that 1. If , then has two fixed points in with the positive one being unstable and the negative one stable, and 2. If , then has no fixed points in . This type of bifurcation is known as a fold bifurcation, sometimes also called a saddle-node bifurcation or tangent bifurcation. An example of an equation displaying a fold bifurcation is(Guckenheimer and Holmes 1997, p. 145).

Mann's iteration is the dynamical system defined for a continuous function ,with . It can also be writtenThis iteration always converges to a fixed point of .

Let be a one-parameter family of maps satisfying(1)(2)(3)(4)Then there are intervals , , and such that 1. If , then has one unstable fixed point and one stable orbit of period two for , and 2. If , then has a single stable fixed point for . This type of bifurcation is known as a flip bifurcation.An example of an equation displaying a flip bifurcation is(5)

Let be a one-parameter family of maps satisfying (1)(2)(3)(4)Here, it turns out that condition (1) can be relaxed slightly, and the left-hand side of (2) has been corrected from the value of 1 given by Rasband (1990, p. 30).Then there are two branches, one stable and one unstable. This bifurcationis called a transcritical bifurcation.An example of an equation displaying a transcritical bifurcation is(5)(Guckenheimer and Holmes 1997, p. 145).

Given a Lyapunov characteristic exponent , the corresponding Lyapunov characteristic number is defined as(1)For an -dimensional linear map,(2)The Lyapunov characteristic numbers , ..., are the eigenvalues of the map matrix. For an arbitrary map(3)(4)the Lyapunov numbers are the eigenvalues of the limit(5)where is the Jacobian(6)If for all , the system is not chaotic. If and the map is area-preserving (Hamiltonian), the product of eigenvalues is 1.

The Fibonacci chain map is defined as(1)(2)where is the fractional part, is the sign, and is the golden ratio.

The Lyapunov characteristic exponent [LCE] gives the rate of exponential divergence from perturbed initial conditions. To examine the behavior of an orbit around a point , perturb the system and write(1)where is the average deviation from the unperturbed trajectory at time . In a chaotic region, the LCE is independent of . It is given by the Oseledec theorem, which states that(2)For an -dimensional mapping, the Lyapunov characteristic exponents are given by(3)for , ..., , where is the Lyapunov characteristic number.One Lyapunov characteristic exponent is always 0, since there is never any divergence for a perturbed trajectory in the direction of the unperturbed trajectory. The larger the LCE, the greater the rate of exponential divergence and the wider the corresponding separatrix of the chaotic region. For the standard map, an analytic estimate of the width of the chaotic zone by Chirikov (1979) finds(4)Since the Lyapunov characteristic..

A means of describing how one state develops into another state over the course of time. Technically, a dynamical system is a smooth action of the reals or the integers on another object (usually a manifold). When the reals are acting, the system is called a continuous dynamical system, and when the integers are acting, the system is called a discrete dynamical system. If is any continuous function, then the evolution of a variable can be given by the formula(1)This equation can also be viewed as a difference equation(2)so defining(3)gives(4)which can be read "as changes by 1 unit, changes by ." This is the discrete analog of the differential equation(5)

A two-dimensional map similar to the Hénon map but with the term replaced by . It is given by the equations (1)(2)The strange attractor illustrated above results from , .

A catastrophe which can occur for three control factors and one behavior axis. The swallowtail catastrophe is the universal unfolding of singularity with codimension 3, i.e., in three unfolding parameters, and is of the form . The equations(1)(2)(3)display such a catastrophe (von Seggern 1993, Nordstrand). The above surface uses and .

The Lorenz attractor is an attractor that arises in a simplified system of equations describing the two-dimensional flow of fluid of uniform depth , with an imposed temperature difference , under gravity , with buoyancy , thermal diffusivity , and kinematic viscosity . The full equations are(1)(2)Here, is a stream function, defined such that the velocity components of the fluid motion are(3)(4)(Tabor 1989, p. 205).In the early 1960s, Lorenz accidentally discovered the chaotic behavior of this system when he found that, for a simplified system, periodic solutions of the form(5)(6)grew for Rayleigh numbers larger than the critical value, . Furthermore, vastly different results were obtained for very small changes in the initial values, representing one of the earliest discoveries of the so-called butterfly effect.Lorenz included the terms(7)(8)(9)where is proportional to convective intensity, to the temperature difference..

The functionfor . The natural invariant is

States that for a nondissipative Hamiltonian system, phase space density (the area between phase space contours) is constant. This requires that, given a small time increment ,(1)(2)(3)(4)the Jacobian be equal to one:(5)(6)(7)(8)Expressed in another form, the integral of the Liouvillemeasure,(9)is a constant of motion. Symplectic maps of Hamiltonian systems must therefore be area preserving (and have determinants equal to 1).

The Smale horseshoe map consists of a sequence of operations on the unit square. First, stretch in the direction by more than a factor of two, then compress in the direction by more than a factor of two. Finally, fold the resulting rectangle and fit it back onto the square, overlapping at the top and bottom, and not quite reaching the ends to the left and right (and with a gap in the middle), as illustrated in the diagram above. The shape of the stretched and folded map gives the horseshoe map its name. Note that it is vital to the construction process for the map to overlap and leave the middle and vertical edges of the initial unit square uncovered.Repeating this generates the horseshoe attractor. If one looks at a cross section of the final structure, it is seen to correspond to a Cantor set.The Smale horseshoe map is the set of basic topological operations for constructing an attractor consist of stretching (which gives sensitivity to initial conditions)..

Consider the general system of two first-orderordinary differential equations(1)(2)Let and denote fixed points with , so(3)(4)Then expand about so(5)(6)To first-order, this gives(7)where the matrix is called the stability matrix.In general, given an -dimensional map , let be a fixed point, so that(8)Expand about the fixed point,(9)(10)so(11)The map can be transformed into the principal axis frame by finding the eigenvectors and eigenvalues of the matrix (12)so the determinant(13)The mapping is(14)When iterated a large number of times, only if for all , but if any . Analysis of the eigenvalues (and eigenvectors) of therefore characterizes the type of fixed point.

Codimension is a term used in a number of algebraic and geometric contexts to indicate the difference between the dimension of certain objects and the dimension of a smaller object contained in it. This rough definition applies to vector spaces (the codimension of the subspace in is ) and to topological spaces (with respect to the Euclidean topology and the Zariski topology, the codimension of a sphere in is ).The first example is a particular case of the formula(1)which gives the codimension of a subspace of a finite-dimensional abstract vector space . The second example has an algebraic counterpart in ring theory. A sphere in the three-dimensional real Euclidean space is defined by the following equation in Cartesian coordinates(2)where the point is the center and is the radius. The Krull dimension of the polynomial ring is 3, the Krull dimension of the quotient ring(3)is 2, and the difference is also called the codimension of the ideal(4)According..

The circle map is a one-dimensional map which maps a circleonto itself(1)where is computed mod 1 and is a constant. Note that the circle map has two parameters: and . can be interpreted as an externally applied frequency, and as a strength of nonlinearity. The circle map exhibits very unexpected behavior as a function of parameters, as illustrated above.It is related to the standard map(2)(3)for and computed mod 1. Writing as(4)gives the circle map with and .The one-dimensional Jacobian of the circle map is(5)so the circle map is not area-preserving.The unperturbed circle map has the form(6)If is rational, then it is known as the map map winding number, defined by(7)and implies a periodic trajectory, since will return to the same point (at most) every map orbits. If is irrational, then the motion is quasiperiodic. If is nonzero, then the motion may be periodic in some finite region surrounding each rational . This execution of periodic motion..

The nonlinear three-dimensional map (1)(2)(3)whose strange attractor is show above for , , and .

The condition for isoenergetic nondegeneracy for a Hamiltonianiswhich guarantees the existence on every energy levelsurface of a set of invariant tori whose complement has a small measure.

In homogeneous coordinates, the first positive quadrant joins with by "points" , and is mapped onto the hyperbolic line by the correspondence . Now define(1)Let be any bounded linear transformation of a Banach space that maps a closed convex cone of onto itself. Then the -norm of is defined by(2)for pairs with finite . Birkhoff's inequality then states that if the transform of under has finite diameter under , then(3)(Birkhoff 1957).

If an integrable quasiperiodic system is slightly perturbed so that it becomes nonintegrable, only a finite number of -map cycles remain as a result of mode locking. One will be elliptical and one will be hyperbolic.Surrounding the elliptic fixed point is a region of stable map orbits which circle it, as illustrated above in the standard map with . As the map is iteratively applied, the island is mapped to a similar structure surrounding the next point of the elliptic cycle. The map thus has a chain of islands, with the fixed point alternating between elliptic (at the center of the islands) and hyperbolic (between islands). Because the unperturbed system goes through an infinity of rational values, the perturbed system must have an infinite number of island chains.

Let be an ergodic endomorphism of the probability space and let be a real-valued measurable function. Then for almost every , we have(1)as . To illustrate this, take to be the characteristic function of some subset of so that(2)The left-hand side of (1) just says how often the orbit of (that is, the points , , , ...) lies in , and the right-hand side is just the measure of . Thus, for an ergodic endomorphism, "space-averages = time-averages almost everywhere." Moreover, if is continuous and uniquely ergodic with Borel measure and is continuous, then we can replace the almost everywhere convergence in (1) with "everywhere."

where is an -dimensional ball of radius centered at and is the probability measure.

An invariant set is said to be a () invariant manifold if has the structure of a differentiable manifold (Wiggins 1990, p. 14).When stable and unstable invariant manifolds intersect, they do so in a hyperbolic fixed point (saddle point). The invariant manifolds are then called separatrices. A hyperbolic fixed point is characterized by two ingoing stable manifolds and two outgoing unstable manifolds. In integrable systems, incoming and outgoing manifolds join up smoothly.

In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. The illustration above shows bifurcations (occurring at the location of the blue lines) of the logistic map as the parameter is varied. Bifurcations come in four basic varieties: flip bifurcation, fold bifurcation, pitchfork bifurcation, and transcritical bifurcation (Rasband 1990).More generally, a bifurcation is a separation of a structure into two branches or parts. For example, in the plot above, the function , where denotes the real part, exhibits a bifurcation along the negative real axis and .

Consider an -dimensional deterministic dynamical systemand let be an -dimensional surface of section that is traverse to the flow, i.e., all trajectories starting from flow through it and are not parallel to it. Then a Poincaré map is a mapping from to itself obtained by following trajectories from one intersection of the surface to the next. Poincaré maps are useful when studying swirling flows near periodic solutions in dynamical systems.

Let be a one-parameter family of maps satisfying (1)(2)(3)(4)(Rasband 1990, p. 31), although condition (1) can actually be relaxed slightly. Then there are intervals having a single stable fixed point and three fixed points (two of which are stable and one of which is unstable). This type of bifurcation is called a pitchfork bifurcation.An example of an equation displaying a pitchfork bifurcation is(5)(Guckenheimer and Holmes 1997, p. 145).

Refer to the above figures. Let be the point of intersection, with ahead of on one manifold and ahead of of the other. The mapping of each of these points and must be ahead of the mapping of , . The only way this can happen is if the manifold loops back and crosses itself at a new homoclinic point. Another loop must be formed, with another homoclinic point. Since is closer to the hyperbolic point than , the distance between and is less than that between and . Area preservation requires the area to remain the same, so each new curve (which is closer than the previous one) must extend further. In effect, the loops become longer and thinner. The network of curves leading to a dense area of homoclinic points is known as a homoclinic tangle or tendril. Homoclinic points appear where chaotic regions touch in a hyperbolic fixed point.The homoclinic tangle is the same topological structure as the Smalehorseshoe map...

The best known example of an Anosov diffeomorphism.It is given by the transformation(1)where and are computed mod 1. The Arnold cat mapping is non-Hamiltonian, nonanalytic, and mixing. However, it is area-preserving since the determinant is 1. The Lyapunov characteristic exponents are given by(2)so(3)The eigenvectors are found by plugging into the matrix equation(4)For , the solution is(5)where is the golden ratio, so the unstable (normalized) eigenvector is(6)Similarly, for , the solution is(7)so the stable (normalized) eigenvector is(8)

A point where a stable and an unstable separatrix (invariant manifold) from the same fixed point or same family intersect. Therefore, the limitsandexist and are equal.A small disk centered near a homoclinic point includes infinitely many periodic points of different periods. Poincaré showed that if there is a single homoclinic point, there are an infinite number. More specifically, there are infinitely many homoclinic points in each small disk (Nusse and Yorke 1996).

The definition of an Anosov map is the same as for an Anosov diffeomorphism except that instead of being a diffeomorphism, it is a map. In particular, an Anosov map is a map f of a manifold to itself such that the tangent bundle of is hyperbolic with respect to .A trivial example is to map all of to a single point of . Here, the eigenvalues are all zero. A less trivial example is an expanding map on the circle , e.g., , where is identified with the real numbers (mod 1). Here, all the eigenvalues equal 2 (i.e., the eigenvalue at each point of ). Note that this map is not a diffeomorphism because , so it has no inverse.A nontrivial example is formed by taking Arnold's cat map on the 2-torus , and crossing it with an expanding map on to form an Anosov map on the 3-torus , where denotes the Cartesian product. In other words,..

An Anosov diffeomorphism is a diffeomorphism of a manifold to itself such that the tangent bundle of is hyperbolic with respect to . Very few classes of Anosov diffeomorphisms are known. The best known is Arnold's cat map.A hyperbolic linear map with integer entries in the transformation matrix and determinant is an Anosov diffeomorphism of the -torus. Not every manifold admits an Anosov diffeomorphism. Anosov diffeomorphisms are expansive, and there are no Anosov diffeomorphisms on the circle.It is conjectured that if is an Anosov diffeomorphism on a compact Riemannian manifold and the nonwandering set of is , then is topologically conjugate to a finite-to-one factor of an Anosov automorphism of a nilmanifold. It has been proved that any Anosov diffeomorphism on the -torus is topologically conjugate to an Anosov automorphism, and also that Anosov diffeomorphisms are structurally stable...

A hyperbolic linear map with integer entries in the transformation matrix and determinant is an Anosov diffeomorphism of the -torus, called an Anosov automorphism (or hyperbolic automorphism). Here, the term automorphism is used in the group theory sense.

For an -dimensional map, the Lyapunov characteristic exponents are given byfor , ..., , where is the Lyapunov characteristic number.

Consider a one-dimensional Hamiltonian map ofthe form(1)which satisfies Hamilton's equations(2)(3)Now, write(4)where(5)(6)Then the equations of motion become(7)(8)Note that equations (7) and (8) are not area-preserving, since(9)(10)(11)However, if we take instead of (9) and (10),(12)(13)(14)(15)which is area-preserving.

A point in a manifold is said to be nonwandering if, for every open neighborhood of , it is true that for a map for some . In other words, every point close to has some iterate under which is also close to . The set of all nonwandering points is denoted , which is known as the nonwandering set of .

Let be the fraction of time a typical dynamical map orbit spends in the interval , and let be normalized such thatover the entire interval of the map. Then the fraction the time an map orbit spends in a finite interval , is given byThe natural invariant is also called the invariant density or natural density.

For a system of first-order ordinary differential equations (or more generally, Pfaffian forms), the -dimensional space consisting of the possible values of () is known as its phase space.If , the phase space is known as a phase plane.

A phase portrait is a plot of multiple phase curves corresponding to different initial conditions in the same phase plane (Tabor 1989, p. 14). Phase portraits for simple harmonic motion(1)and pendulum(2)are illustrated above.

A surface (or "space") of section, also called a Poincaré section (Rasband 1990, pp. 7 and 93-94), is a way of presenting a trajectory in -dimensional phase space in an -dimensional space. By picking one phase element constant and plotting the values of the other elements each time the selected element has the desired value, an intersection surface is obtained.The above surface of section is for the Hénon-Heiles equation with energy plotting vs. at values where .If the equations of motion can be formulated as a map in which an explicit formula gives the values of the other elements at successive passages through the selected element value, the time required to compute the surface of section is greatly reduced.

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