# Unstable periodic orbits

Post-publication activity

Curator: Paul So

An unstable periodic orbit (UPO) is simply a periodic orbit which is dynamically unstable.

## Basic Properties

The following summary on the basic properties of an (unstable) periodic orbit is taken mainly from (Guckenheimer and Holmes 1983). Additional fundamental information for a periodic orbit can also be found in the following Scholarpedia entry (periodic orbit).

### Local Properties

For a smooth invertible map $$\mathbf{x}_{n+1}=\mathbf{M}(\mathbf{x}_n)$$ in $$\mathbb{R}^d\ ,$$ a periodic orbit with period $$k$$ is a sequence of $$k$$ distinct points $$\mathbf{p}_{j}=\mathbf{M}^j(\mathbf{p}_0), j=0,\cdots,k-1$$ with $$\mathbf{M}^k(\mathbf{p}_0)=\mathbf{p}_0\ .$$ (The dynamics of a flow can be analyzed as a smooth invertible map by considering the induced map on its Poincare surface of section.) The stability of this periodic orbit is determined by the eigenvalues $$\lbrace\Lambda_j\rbrace_{j=1}^d$$ of the Jacobian matrix $$\mathbf{DM}^k(\mathbf{p}_0)$$ which is the linearization of the k-th iterated map $$\mathbf{M}^k\ .$$ (Here, for simplicity, we assume that there is no multiplicity in the eigenvalues.)

• If $$\left |\Lambda_j\right |<1$$ for all eigenvalues, then the periodic orbit is a stable ‘‘sink’’.
• If $$\left |\Lambda_j\right |<1$$ for some eigenvalues and $$\left |\Lambda_j\right |>1$$ for others, then the periodic orbit is a ‘‘saddle’’.
• If $$\left |\Lambda_j\right |>1$$ for all eigenvalues, then the periodic orbit is an unstable ‘‘source’’.

Furthermore, if none of the eigenvalues has unit modulus, the periodic orbit is called hyperbolic. When a trajectory is near a hyperbolic orbit, the asymptotic behavior of the dynamical system is determined solely by the Jacobian matrix $$\mathbf{DM}^k$$ but if the periodic orbit is non-hyperbolic, the stability cannot be easily determined by $$\mathbf{DM}^k$$ alone. (Without the loss of generality and for simplicity in our continuing discussion, we will assume the periodic orbit to be of period-1 so that $$\mathbf{p}_0=\mathbf{p}$$ is simply a fixed point of the map $$\mathbf{M}\ .$$)

In terms of the set of independent eigenvectors of $$\mathbf{DM}\ ,$$ one can also define the following invariant subspaces in the tangent space about $$\mathbf{p}\ ,$$

• the stable subspace, $$E^s=span\lbrace v^1,\cdots,v^{n_s}\rbrace\ ,$$
• the unstable subspace, $$E^u=span\lbrace u^1,\cdots,u^{n_u}\rbrace\ ,$$
• the center subspace, $$E^c=span\lbrace w^1,\cdots,w^{n_c}\rbrace\ ,$$

where $$v^1,\cdots,v^{n_s}$$ are the $$n_s$$ eigenvectors with $$\left|\Lambda_j\right|<1\ ,$$ $$u^1,\cdots,u^{n_u}$$ are the $$n_u$$ eigenvectors with $$\left|\Lambda_j\right|>1\ ,$$ and $$w^1,\cdots,w^{n_c}$$ are the $$n_c$$ eigenvectors with $$\left|\Lambda_j\right|=1\ .$$ Since the tangent space about the periodic orbit is d-dimensional, $$n_s+n_u+n_c=d\ .$$ Most importantly, near a hyperbolic periodic orbit, the linearized (un)stable subspaces defined above can be extended to the nonlinear system through the Hartman-Grobman theorem and the Stable Manifold Theorem (see Guckenheimer and Holmes 1983 for a review). In particular, near a hyperbolic periodic orbit, one can find local stable and unstable manifolds $$W_{loc}^s(\mathbf{p})$$ and $$W_{loc}^u(\mathbf{p})$$ tangent to the eigenspaces $$E^s$$ and $$E^u$$ of $$\mathbf{DM}(\mathbf{p})$$ at $$\mathbf{p}\ .$$ Thus, an understanding of the linearized dynamics given by the Jacobian matrix $$\mathbf{DM}$$ provides a local approximation to the full nonlinear system $$\mathbf{M}$$ near the periodic orbit. An example of these invariant manifolds for a saddle periodic orbit is illustrated in (Figure 1). Although the discussion here is for a periodic orbit in a map, an analogous description can be made for continuous-time flows (see Guckenhiemer and Holmes 1983).

Figure 1: Unstable and stable manifolds near a saddle. One should notice that trajectories (see iterates of $$\mathbf{x}$$ and $$\mathbf{y}$$) near the stable manifold will get closer together toward the saddle while trajectories near the unstable manifold will diverge in time.

### Global Properties

A saddle orbit as shown above is unstable but unlike an unstable sink where all nearby initial conditions will eventually move away. For a saddle orbit, not all nearby initial conditions will move away. In fact, the set of initial conditions on the stable manifold $$W^s(\mathbf{p})$$ are exactly those which will converge to $$\mathbf{p}$$ asymptotically in time. As one can see from (Figure 1), the local stable and unstable manifolds, $$W^s_{loc}$$ and $$W^u_{loc}\ ,$$ are gentle curves tangent to the linearized stable and unstable subspaces $$E^s$$ and $$E^u$$ at $$\mathbf{p}\ .$$ Global stable and unstable manifolds, $$W^s(\mathbf{p})$$ and $$W^u(\mathbf{p})$$ can be defined as the unions of backward and forward iterates of these local manifolds. The basic property is that points on the stable manifold will tend toward $$\mathbf{p}$$ under the forward iterations of the map $$\mathbf{M}$$ and points on the unstable manifold will tend toward $$\mathbf{p}$$ under the backward iterations of $$\mathbf{M}\ .$$

These global manifolds can have a very complicated structure and the structure of these global manifolds for a saddle can have important implications on the existence of chaos. In particular, the unstable and the stable manifold of a saddle orbit can intersect and result in a homoclinic orbit. A demonstration of a homoclinic orbit is illustrated in (Figure 2). The key point is that the existence of one of such homoclinic intersection $$\mathbf{h}_0$$ implies the existence of infinitely many of such intersections.

Figure 2: A schematic diagram of a homoclinic orbit.

To see this, let consider the situation when $$W^s(\mathbf{p})$$ and $$W^u(\mathbf{p})$$ intersect at a point $$\mathbf{h}_0\ .$$ Since the stable and the unstable manifolds are invariant under the dynamics of the map $$\mathbf{M}\ ,$$ both the forward iterates $$\lbrace \mathbf{h}_1,\mathbf{h}_2,\cdots\rbrace$$ and the backward iterates $$\lbrace \mathbf{h}_{-1},\mathbf{h}_{-2},\cdots\rbrace$$ of $$\mathbf{h}_0$$ must also be on the intersection of the two manifolds. Moreover, these points are on the stable and unstable manifolds of $$\mathbf{p}\ ,$$ they must also asymptotically converge to $$\mathbf{p}$$ itself both forward and backward in time, i.e., $$\mathbf{M}^n(\mathbf{h}_0)\to\mathbf{p}$$ as $$n\pm\infty\ .$$ As a result, the stable and unstable manifolds are forced to intertwine and produce the resulting homoclinic tangle. The unstable eigenvalues of the saddle orbit is the source of the system’s sensitive dependence on initial conditions and the existence of the homoclinic tangle provides the mechanism in spreading this germ of chaotic behavior away from the unstable saddle orbit.

Poincaré was the first to discover this mechanism to generate complex dynamics from an unstable saddle orbit when he corrected his entry to the King Oscar’s prize (Alligood et al. 1996). The understanding of this complex dynamics was further developed and explained when Smale in the 1960s (Smale 1967) demonstrated that the existence of a homoclinic points implies the necessary existence of a horseshoe. The Smale horseshoe is a geometric construction which exemplifies the two basic ingredients for all chaos dynamics, namely the repeated actions of stretching and folding of a given phase space volume. The Smale horseshoe also is the fundamental tool in understanding the symbolic dynamics intrinsic to all chaos systems.

## UPOs and Chaos

It was originally recognized by Poincaré and decades later by many founders of modern dynamical system theory that periodic orbits play an important role in understanding the rich structures in a dynamical system. Its basic properties has been briefly discussed above and for a chaotic system, the set of unstable periodic orbits can also be thought of as the skeleton for the dynamics. Borrowing the language from statistical mechanics in physics, the set of unstable periodic orbits can be viewed as the microstates from which macroscopic description of the system can be calculated. In fact, one of the common characterizations of chaos is the positivity of the topological entropy and the topological entropy of a system is related to the exponential growth rate of the number of UPOs embedded within the attractor as one enumerates the orbits by their lengths (see, e.g. (Katok 2003), (Ott, 1993) and (Smale 1967)). Furthermore, many dynamical averages, such as, the natural measure, the Lyapunov exponents, the fractal dimensions, and the entropy, can by efficiently expressed in terms of a sum over the unstable periodic orbits (Auerbach 1987, Cvitanovic 1988, Artuso 1990a,b, Ott 1993).

Figure 3: An abstract “dynamical landscape” with peaks representing the UPOs. A chaotic trajectory can be visualized as the motion of a ball rolling on this abstract dynamical landscape. The motion of the ball will be strongly affected by the sharpness of the peak (the stability of the UPOs) and the landscape is dense populated by these speed bumps. The relative height of the peaks is related to the natural measure of the indicated UPOs and the straight lines are just visual aids in identifying the corresponding cyclic pieces of the period-2 cycles in the diagram.

A chaotic system can be represented and approximated by its periodic orbits. The periodic orbits are typically hierarchically organized, and an increasing accuracy can be achieved by including increasingly longer orbits into the hierarchy. Conceptually, one can visualize the chaotic trajectories of a dynamical system to live on an abstract “dynamical landscape” where the peaks represent the set of UPOs. (UPOs in chaotic set can be saddles as well as repellers but they are all cartoonishly represented by peaks in this abstract dynamical landscape.) Since all of the periodic orbits within this abstract dynamical landscape are unstable (the system is chaotic), a trajectory will never settle down to any one of them. However, since the set of UPOs is dense within the chaotic set, a typical trajectory will wanders incessantly in a sequence of close approaches to these orbits. The more unstable an orbit, the less time that a trajectory will spend near it. Most importantly, with a knowledge of the local linearized dynamics of the UPOs, the full dynamics on the chaotic system can be approximated by tessellating this smooth abstract dynamical landscape by a piece-wise linear one in which the local dynamics on each flat panel is governed by the UPO located at that position.

### UPOs and the Natural Measure

For a general dynamical system $$\mathbf{M}(\mathbf{x})\ ,$$ the natural measure of a set $$S$$ is the probability for it to remain in the set $$S\ .$$ In the discrete time case, it is given by (Kandanoff 1984), $\mu(S)=\lim_{n\rightarrow\infty} \int_S d\mathbf{x} \delta(\mathbf{x}-\mathbf{M}^{(n)}(\mathbf{x})),$ and this integrates to $\tag{1} \mu(S)=\lim_{n\rightarrow\infty}\sum_{\mathbf{x}\in Fix \mathbf{M}^n}\frac{1}{|det(\mathbf{1}-\mathbf{DM}^{(n)}(\mathbf{x}))|},$

where the sum is over all fixed points of the map $$\mathbf{M}^n\ .$$ (The continuous time situation can be found in (Cvitanovic 1991).)

This shows the intimate relation between the invariant density of chaotic sets and periodic orbits. For strongly unstable systems, the determinants can be expanded and only the unstable directions contribute. Then Eq. (1) becomes approximately $\tag{2} \mu(S)= \lim_{n \to \infty}\sum_{\mathbf{x}\in Fix \mathbf{M}^n} \frac{1}{\vert\Lambda_u(\mathbf{x})\vert} \ ,$

where the summation is taken over all fixed points of $$M^n$$ in $$S\ .$$ $$\Lambda_u$$ indicates the largest expanding eigenvalue of $$\mathbf{DM}^n\ .$$ One should note that the general result given in Eq. (1) is exact and it reduces to the expression given in Eq. (2) to leading order (Russberg 1993).

Figure 4: A Small Cell used in Partitioning Phase Space

Nonetheless, the simpler expression in Eq. (2) is useful and it has a very intuitive explanation (Grebogi 1988 and Ott 1990). For concreteness, let say $$\mathbf{M}(\mathbf{x})$$ is an invertible hyperbolic chaotic map. Consider a small cell $$C_k$$ within the two dimensional phase space partitioned by segments of the stable and unstable invariant manifolds. If $$C_k$$ is chosen small enough, one can envision it being a parallelogram with the stable (unstable) manifolds as its boundaries (see Figure 4). One then sprinkles a large number of initial conditions within this cell according to the natural measure of the attractor. After n time steps forward, most of these initial conditions will leave the cell and a small fraction of it will return to $$C_k\ .$$ In the large n limit, the natural measure of the cell $$C_k\ ,$$ $$\mu(C_k)\ ,$$ will exactly be the fraction of initial conditions remain in the cell $$C_k\ .$$

Let consider one of these initial conditions $$\mathbf{x}_0$$ which returns on its nth iterates and for clarity, the small cell $$C_k$$ is schematically straightened out into a small rectangle with its stable manifold (contracting direction) along the horizontal axis and its unstable manifold (expanding direction) along the vertical axis (see Figure 5).

Figure 5: Dynamics within the cell $$C_k$$

With the assumption of hyperbolicity, a sufficiently thin horizontal strip will return to the cell after n iterates. In particular, one can choose the dimension of the cell $$C_k$$ such that a thin long rectangle containing $$\mathbf{x}_0$$ will map onto a tall narrow rectangle containing $$\mathbf{x}_n$$ as shown. At the intersection of these two rectangles, there must exist a fixed point ($$\mathbf{x}_{jn}$$) of the n iterated map $$M^n\ .$$ If one knows the expanding and contracting eigenvalues of this fixed point, $$\Lambda_u(\mathbf{x}_{jn},n), \Lambda_s(\mathbf{x}_{jn},n)\ ,$$ then one can estimate the sizes of these long thin rectangles (see Figure 5). Since the attractor’s natural measure is smooth along its unstable direction, one can assume that $$\mu(C_k)$$ is uniform in the vertical direction and the measure of the thin horizontal strip containing the fixed point $$\mathbf{x}_{jn}$$ is simply proportional to the ratio of the height of this thin horizontal strip to the full height of the cell, i.e., $$1/\vert\Lambda_u(\mathbf{x}_{jn},n)\vert\ .$$ Then, in the large n limit, the natural measure of the whole cell $$\mu(C_k)$$ is given by the sum over all periodic points within $$C_k\ ,$$ $\mu(C_k)= \lim_{n \to \infty}\sum_{\mathbf{x}_{jn}\in C_k} \frac{1}{\vert\Lambda_u(\mathbf{x}_{jn},n)\vert}.$

Finally, for a sufficiently well-behaved hyperbolic system, any set $$S$$ within the attractor can be covered by such small cells and the claimed result follows.

This is an instructive example since a number of dynamical averages for this hyperbolic chaotic map can be estimated in terms of a weighted average with respect to the nature measure of the attractor (Ott 1993, Cvitanovic 1988, Cvitanovic 1995).

#### Escape rates

If the set $$S$$ is the entire attractor, the above result gives the following identity (Hannay 1984) $\lim_{n \to \infty}\sum_{j} \frac{1}{\vert\Lambda_u(\mathbf{x}_{jn},n)\vert}=1 \ ,$ where the sum is over all periodic orbits embedded within the attractor. On the other hand, if the chaotic set is nonattracting, a similar derivation gives the average escape rate $$\gamma$$ (Grebogi 1988), $\lim_{n \to \infty}e^{n\gamma}\sum_{j} \frac{1}{\vert\Lambda_u(\mathbf{x}_{jn},n)\vert}=1 \ .$

#### Lyapunov exponents

For the two dimensional hyperbolic chaotic map, the two Lyapunov exponents can be expressed as the following sum (Ott 1993), $\lambda_{u,s}=\lim_{n \to \infty} \frac{1}{n} \langle\ln\vert\Lambda_{u,s}(\mathbf{x}_{jn},n)\vert \rangle =\lim_{n \to \infty} \frac{1}{n} \sum_{j} \frac{1}{\vert\Lambda_{u,s}(\mathbf{x}_{jn},n)\vert}\ln\vert\Lambda_{u,s}(\mathbf{x}_{jn},n)\vert,$ where $$\langle \rangle$$ indicates an average with respect to the natural measure of the chaotic set and "u" and "s" are the labels for the unstable and stable directions in our two dimensional hyperbolic example.

#### Entropy

A generalization of the topological and metric entropy for a hyperbolic attractor in two dimensions can be defined in terms of the $$q$$-order entropy spectrum (Fujisaka 1983) and this quantity can be written as a sum over the periodic orbits as, $H_q=\frac{1}{1-q}\lim_{n \to \infty} \frac{1}{n} \ln \langle e^{(1-q)\ln\vert\Lambda_u(\mathbf{x}_{jn},n)\vert}\rangle =\frac{1}{1-q}\lim_{n \to \infty} \frac{1}{n} \ln\lbrace\sum_{j}\vert\Lambda_{u}(\mathbf{x}_{jn},n)\vert^{-q}\rbrace.$ In this formalism, $$H_0$$ is the topological entropy and $$H_1$$ is the metric entropy.

#### Dimensions

The dimension spectrum for a hyperbolic chaotic set can be defined using the Lyapunov partition function and this function can again be written in terms of a sum over the periodic orbits (Ott, 1993), $\Gamma_q(D,n)=\langle\lbrack\vert\Lambda_s(\mathbf{x}_{jn},n)\vert^{D-1}\vert\Lambda_u(\mathbf{x}_{jn},n)\vert\rbrack^{1-q}\rangle =\sum_{j} \frac{\vert\Lambda_s(\mathbf{x}_{jn},n)\vert^{(D-1)(1-q)}}{\vert\Lambda_{u}(\mathbf{x}_{jn},n)\vert^q}.$ For various $$q$$ values, $$D_q$$ gives the box-counting ($$q=0$$), information ($$q=1$$), and correlation ($$q=2$$) dimensions. Using the Lyapunov partition function, $$D_q$$ is extracted from the transition point where $$\lim_{n \to \infty}\Gamma_q(D,n)$$ changes from 0 to $$\infty\ .$$

Since $$\Lambda_u(\mathbf{x}_{jn},n)$$ tends to grow exponentially with increasing $$n\ ,$$ the sum in Eq. (2) is typically dominated by terms from the short periodic orbits and a good estimate of $$\mu(S)$$ can typically be achieved with considerations of the short orbits.

### Cycle Expansions

A more formal way to express this idea is through the technique of cycle expansions (Cvitanovic, 1988, Artuso, 1990a,b). Again, the system is assumed to be hyperbolic. Using the escape rate as an example, one again has the following equality, $\lim_{n \to \infty}e^{n\gamma}\sum_{j} \frac{1}{\vert\Lambda_u(\mathbf{x}_{jn},n)\vert}=1.$ Intuitively, in order for the equality to be valid, the factor with the sum over the periodic orbits must decrease exponentially fast (enough) so that it balances out the $$e^{n\gamma}$$ term in the large $$n$$ limit to produce a finite result, i.e., $\sum_{j} \frac{1}{\vert\Lambda_u(\mathbf{x}_{jn},n)\vert} \to e^{-n\gamma}.$ Motivated by the above observation, one can formally define the following sum over all periodic orbits, $\Omega(z)=\sum_{n=1}^{\infty}z^n\sum_{j}^{(n)}\frac{1}{\vert\Lambda(\mathbf{x}_{jn},n)\vert}.$ For sufficiently small $$z\ ,$$ this infinite sum converges to zero and the average escape rate $$\gamma$$ is determined by the smallest $$z=e^{\gamma}$$ such that the sum does not vanish. One can view z as a parametric probe for the asymptotic behavior of the sum $$\sum_{j}^{(n)}\frac{1}{\vert\Lambda(\mathbf{x}_{jn},n)\vert}\ .$$

The sum in $$\Omega(z)$$ can also be rewritten in terms of the prime cycles only, $\Omega(z)=\sum_{p}n_p\sum_{r=1}^{\infty}\left (\frac{ z^{n_p}}{\vert\Lambda_p\vert}\right )^r.$ Here, $$p$$ indicates a prime cycle and $$r$$ is the number of times that an orbit retraces itself over the prime cycle. Since the stability of a prime cycle ($$\Lambda_p$$) is the same everywhere along the cycle, a prime cycle of length $$n_p$$ contributes $$n_p$$ terms to the above sum. Lastly, if one uses the geometric series formula to write out the interior sum, one arrives at, $\Omega(z)=\sum_{p} \frac{n_p z^{n_p}\vert\Lambda_p ^{-1}\vert }{1-z^{n_p}\vert\Lambda_p^{-1}\vert }.$ Expressing $$\Omega(z)$$ in this form, one can then interpret $$\Omega(z)$$ as the logarithmic derivative of the following infinite product, $1/\zeta(z) = \prod_{p}\left (1-z^{n_p}\vert\Lambda_p^{-1}\vert\right ).$ This is example of a dynamical $$\zeta$$ function commonly used in the thermodynamic formalism (or partition function formalism) in statistical mechanics (Ruelle, 1978). Since the average escape rate corresponds to the smallest $$z$$ where $$\Omega(z)\not\to 0\ ,$$ it also corresponds to the smallest $$z$$ where $$1/\zeta(z)=0\ .$$ In other words, the average escape rate (as well as other thermodynamic averages) can be calculated by looking for the leading root of the appropriate dynamical $$\zeta$$ function.

The essential ingredient in the cycle expansion technique is the recognition that the dynamical $$\zeta$$ function can typically be written down as a sum of a few terms involving the (short) fundamental cycles plus a sum of small curvature corrections that can be controlled with a good understanding of the system’s symbolic dynamics, $1/\zeta(z) = 1- \sum_{f}t_f - \sum_{n}c_n$ where the first sum is over the fundamental cycles and $$t_f = z^{n_f}\vert\Lambda_f^{-1}\vert$$ in the calculation for the escape rate. (Other dynamical averages in other applications will have a different expression for $$t_f\ .$$) The second sum is over the curvature corrections of a given cycle length $$n\ .$$ For a given cycle length $$n\ ,$$ $$c_n$$ consists of terms in the following form, $$t_{p_{1\cdots n}}-t_{\lbrace p_1\cdots p_n\rbrace}\ ,$$ where $$p_{1\cdots n}$$ indicates a prime cycle of length $$n$$ and $${\lbrace p_1\cdots p_n\rbrace}$$ indicates a pseudo-orbit of length $$n$$ composed of a combination of fundamental cycles of shorter lengths which shadows the prime cycle $$p_{1\cdots n}\ .$$ Intuitively, these curvature corrections correspond to the topological deviations in balancing long orbits by piecing together pseudo-orbits from shorter fundamental cycles. A firm control on these corrections can be achieved by a good understanding of the symbolic dynamics of the system. However, one should caution that this zeta function formalism might not behave well if the symbolic dynamics is not complete. In this case, a supplemental pruning grammar must be given. A very accessible discussion on the technique of cycle expansion and its applicability can be found in Cvitanovic et al.'s webbook.

## UPOs and Applications

UPOs are the "skeleton" of a dynamical system and a knowledge of a hierarchy of them (and typically from only a few short fundamental cycles) can give one’s access to many of the system’s dynamical averages. In addition to this fundamental conceptual linkage, UPOs are also important tools in affecting the behaviors of dynamical systems.

In a chaotic system, UPOs with a range of dynamical properties are dense within the attractor. Many design features corresponding to different UPOs are possible within a single system. The ergodic property of the attractor also guarantees that all orbits are (in principle) accessible. In most situations, it has been conjectured that optimal performance is typically achieved by the set of the lower-order periodic orbits (Hunt, 1996). From an engineering point of view, UPOs (especially the most experimentally accessible short orbits) are useful tools in system designs.

### Detecting UPOs in Experimental Data

Figure 6: Detection of UPOs using the Periodic-Orbit Transform

In most experimental situations, a time series data is usually the only available information from a dynamical system. To further analyze the system, one typically needs to dynamically reconstruct its phase space (Takens, 1981, Sauer, 1991). A basic scheme in detecting UPOs from this reconstructed data set is to look for recurrences of the $$n$$ iterated reconstructed map (Auerbach 1987, Lathrop 1989, Pierson 1995). The standard procedure is to look for peaks in a histogram of recurred points as a function of their recurring periods. The sensitivity of this method in finding UPOs will naturally depend on the natural measure of the UPOs.

An enhancement of the standard recurrence methods was proposed later (So 1996, So 1998). In this method, experimentally extracted linear dynamics near each state point was incorporated into a periodic-orbit transform that take experimental data into a space where the probability measure at the UPOs are enhanced and at other non-recurring points are dispersed. Similar to the previous recurrent methods, an experimenter detects UPOs by looking for peaks in the transformed space. An example of this is given in the following graph (Figure 6) for a 1D chaotic map shown in the inset of the top panel. The top panel is a graph of the natural measure of the 1D map and the middle and bottom panels show the histograms from the period-1 transform with and without the dispersing factor. The true fixed point can be easily identified in the middle panel.

There are also other more recent UPO detection methods for experimental data. Among them, there are a couple of notable ones:

• Another transformation based method developed by Schmelcher and Diakonos (Dikonos 1997).
• An adaptive control-based detection method developed by Christini and Kaplan (Christini 2000).

Figure 7: Short Term Prediction Using UPOs

### Prediction, Tracking, and Control

A knowledge of the locations of the periodic orbits in an experimental reconstructed attractor and their associated stabilities gives experimentalists the ability to short-term predict the future state of the system (Pawelzik, 1991, So, 1998). Figure 7 is a demonstration on how short term prediction can be achieved using a time series from a rat brain slice in a neuronal experiment (So, 1998).

Three unstable periodic orbits were extracted from an experimental time series. The top row shows the data points (training set) used to identify the UPOs. The color lines indicate the eigendirections of the linearized local dynamics near the UPOs. Red and green lines indicate respectively the unstable and stable directions. The second and third rows show how subsequent experimental data points follow the extracted linear dynamics. The action of the contracting and expanding dynamics near the UPOs is shown in the second row by pairs of joined data points. One can visualize the projection of the dumbbell along the stable direction is being contracted and the projection of the dumbbell along the unstable direction is being stretched as the data points move around the UPOs. In the last row, all sequences of data points with good prediction for at least two time steps are shown. (Δ initial points of the testing sequence; O subsequent points of the testing sequence; + predicted positions of the circles.)

Figure 8: Tracking Time Variations in EEG Data using UPOs

With the ability to detect UPOs quickly from an experimental setting also gives experimentalists a dynamical tool in tracking mild non-stationarity of the system. It can serve as a diagnostic probe for the time variations of some internal parameters of the system. The following graph (Figure 8) is an example from an archived human EEG data (So, 1998). Within the total duration of the data set, three prominent period-1 UPOs were found. These UPOs appear, disappear and reappear as the system evolves. They seem to correspond to three distinct physiological epochs in the EEG data (see the inserted raw data).

The dense UPO structure within a chaotic attractor and the exponential sensitivity of a chaotic system also afford a very power mean in utilizing chaos for control as well as controlling chaos. As it has been demonstrated by many experimental examples, a more efficient control scheme was achieved by astronomers to target spacecraft trajectories using the idea of the butterfly effect in chaotic systems (Farquhar, 1985); designs for more powerful lasers were developed utilizing chaos control strategies (Roy, 1992); cardiac arrhythmias were successful tamed by a chaos stabilization method (Garfinkel, 1992); and seizure activities were controlled by judiciously timed electric pulses (Schiff, 1994).

There are other notable results for modelling physical and biological systems using UPOs:

• A laser system (Flepp 1991): The authors started with a very good idea on the equations which describe their laser system. Then, by comparing the periodic orbits extracted from the experiment and from the model, they determined the values of certain coupling constants and identified the significance of a nonlinear term.
• A sensory system and noise (Pei 1996 and Pierson 1995): UPOs were used to characterize the dynamics of the caudal photoreceptors in a crayfish and their nonlinear responses to noise were examined.

## Other Roles of UPOs

UPOs also play significant roles in many dynamical processes such as bifurcations, period doubling, intermittency, crises, desychronization of chaotic systems, riddle basin and others. It is also an indispensable theoretical tool in studying quantum chaos.

## References

• KT Alligood, TD Sauer and JA Yorke (1996) Chaos: An Introduction to Dynamical Systems (Springer-Verlag, New York).
• R Artuso, E Aurell and P. Cvitanovic (1990a) Recycling of Strange Sets: I. Cycle Expansions, Nonlinearity 3: 325.
• R Artuso, E Aurell and P. Cvitanovic (1990b) Recycling of Strange Sets: II. Applications, Nonlinearity 3: 361.
• D Auerbach, P Cvitanovic, J-P Eckmann and G. Gunaratne (1987) Exploring Chaotic Motion Through Periodic Orbits, Phys. Rev. Letts. 23: 2387.
• DJ Christini and DT Kaplan (2000) Adaptive Estimation and Control Method for Unstable Periodic Dynamics in Spike Trains, Phy. Rev. E 61: 5149.
• P Cvitanovic (1988) Invariant Measurement of Strange Sets in Terms of Cycles, Phys. Rev. Letts. 24: 2729.
• P Cvitanovic and B Eckhardt (1991) Periodic orbit expansions for classical smooth flows. Journal of Physics A, 24: L237.
• P Cvitanovic (1995) Dynamical Averaging in Terms of Periodic Orbits, Physica D 83: 109.
• P Cvitanovic, R Artuso, R Mainieri, G Tanner and G Vattay (2005) Chaos: Classical and Quantum, ChaosBook.org (Niels Bohr Institute, Copenhagen).
• R Farquhar, D Muhonen and SA Davis (1985) Trajectories and Orbit Maneuvers for the ISEE-3/ICE Comet Mission (1985) J. Astronaut. Sci. 33:235.
• L Flepp, T Holzner, E Brun, M Finardi and R Badii (1991) Model identification by periodic-orbit analysis for NMR-Laser chaos, Phys. Rev. Letts. 67: 2244.
• H Fujisaka (1983) Statistical Dynamical Generated by Fluctuations of Local Lyapunov Exponents, Prog. Theor. Phys. 70: 1264.
• A Garfinkel, ML Spano, WL Ditto, and JN Weiss (1992) Controlling Cardiac Chaos, Science 257: 1230.
• C Grebogi, E Ott and JA Yorke (1988) Unstable Periodic Orbits and the Dimensions of Multifractal Chaotic Attractors, Phys. Rev. A 37: 1711.
• J Guckenheimer and P Holmes (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York).
• JH Hannay and AM Ozorio de Almeida (1984) Periodic Orbits and a Correlation Function for the Semiclassical Density of States, J. Phys. A 17: 3429.
• B Hasselblatt and A Katok (2003) A First Course in Dynamics, Cambridge University Press, Cambridge.
• B Hunt and E Ott (1996) Optimal Periodic Orbits of Chaotic Systems, Phys. Rev. Letts. 76: 2254.
• LP Kadanoff and C Tang (1984) Escape from strange repellors, PNAS 81, 1276.
• E Ott (1993) Chaos in Dynamical Systems, 2nd ed. Cambridge University Press, Cambridge.
• K Pawelzik and HG Schuster (1991) Unstable Periodic Orbits and Prediction, Phys. Rev. A 43: 1808.
• X Pei and F Moss (1996) Characterization of low-dimensional dynamics in the crayfish caudal photoreceptor, Nature 379: 618.
• D Pierson and F Moss (1995) Detecting Periodic Unstable Points in Noisy Chaotic and Limit Cycle Attractors with Applications to Biology Phys. Rev. Letts. 75: 2124.
• Rajarshi Roy, TW Murphy, Jr., TD Maier, and Z Gills (1992) Dynamical Control of a Chaotic Laser: Experimental Stabilization of a Globally Coupled System, Phys. Rev. Letts. 68: 1259.
• G Russberg and B Eckhardt (1993) Resummation of classical and semiclassical periodic orbit expressions, Phys. Rev. E 47, (1993) 1578.
• T Sauer, JA Yorke and M Casdagli (1991) Embedology, J. Statist. Phys. 65: 579.
• SJ Schiff, K Jerger, DH Duong, T Chang, ML Spano and WL Ditto (1994) Controlling Chaos in the Brain, Nature 370: 615.
• P Schmelcher and FK Diakonos (1997) Detecting Unstable Periodic Orbits of Chaotic Dynamical Systems, Phys. Rev. Letts. 78: 4733.
• S. Smale (1967) Differentiable dynamical systems. Bulletin of the American Mathematical Society, 73: 747–817.
• P So, JT Francis, TI Netoff, BJ Gluckman and SJ Schiff (1998) Periodic Orbits: A New Language for Neuronal Dynamics, Biophys. J 74: 2776.
• P So, E Ott, SJ Schiff, DT Kaplan, T Sauer and C Grebogi (1996) Detecting Unstable Periodic Orbits in Chaotic Experimental Data, Phys. Rev. Letts. 76: 4705.
• P So, E Ott, T Sauer, BJ Gluckman, C Grebogi and SJ Schiff (1997) Extracting Unstable Periodic Orbits from Chaotic Time Series Data, Phys. Rev. E 55: 5398.
• F Takens (1981) Dynamical Systems and Turbulence, D Rand and LS Young eds., Springer-Verlag, Berlin.

Internal references