Abstract
We investigate the convergence towards periodic orbits in discrete dynamical systems. We examine the probability that a randomly chosen point converges to a particular neighborhood of a periodic orbit in a fixed number of iterations, and we use linearized equations to examine the evolution near that neighborhood. The underlying idea is that points of stable periodic orbit are associated with intervals. We state and prove a theorem that details what regions of phase space are mapped into these intervals (once they are known) and how many iterations are required to get there. We also construct algorithms that allow our theoretical results to be implemented successfully in practice.
Citation: San Martín J, Porter MA (2014) Convergence Time towards Periodic Orbits in Discrete Dynamical Systems. PLoS ONE 9(4):
e92652.
https://doi.org/10.1371/journal.pone.0092652
Editor: Mark R. Muldoon, Manchester University, United Kingdom
Received: December 26, 2013; Accepted: February 3, 2014; Published: April 15, 2014
Copyright: © 2014 San Martín, Porter. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The authors have no support or funding to report.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Periodic orbits are the most basic oscillations of nonlinear systems, and they also underlie extraordinarily complicated recurrent dynamics such as chaos [1]-[5]. Moreover, they occur ubiquitously in applications throughout the sciences and engineering. It is thus important to develop a deep understanding of periodic dynamics.
It is important and common to question how long it takes a point in phase space to reach a stable periodic orbit from an arbitrary initial condition. When studying synchronization and other forms of collective behavior, it is crucial to examine not only the existence of stable periodic orbits but also the time that it takes to converge to such dynamics in both natural and human-designed systems [6]-[8]. For example, it is desirable to know how long it will take an engineered system that starts from an arbitrary initial condition to achieve the regular motion at which it is designed to work [9], [10]. A system can also be perturbed from regular motion by accident, and it is important to estimate how long it will take to return to regular dynamics. Similar questions arise in physics [11], [12], biology [6], [13], [14], and many other areas. It is also important to consider the time to synchronize networks [15]-[17] and to examine the convergence properties of algorithms for finding periodic orbits [2], [18].
To study the problem of convergence time to periodic orbits, let's first consider the Hartman-Grobman Theorem [19], [20], which states that the flow of a dynamical system (i.e., a vector field) near a hyperbolic equilibrium point is topologically equivalent to the flow of its linearization near this equilibrium point. If all of the eigenvalues of the Jacobian matrix evaluated at an equilibrium have negative real parts, then this equilibrium point is reached exponentially fast when one is in a small neighborhood of it. To determine convergence time to a hyperbolic equilibrium, we thus need to calculate how long it takes to reach a neighborhood of the equilibrium from an arbitrary initial condition. After reaching the neighborhood, the temporal evolution is then governed by a linear dynamical system (which can be solved in closed form). An analogous result holds for hyperbolic periodic orbits in vector fields [21]. To turn periodic orbits in vector fields into fixed points in maps, one can use Poincaré return maps, which faithfully capture properties of periodic orbits. A Poincaré map can be interpreted as a discrete dynamical system, so the problem of determining how long it takes to reach a hyperbolic stable periodic orbit from arbitrary initial conditions in a vector field is reduced to the problem of determining how long it takes to reach the neighborhood of a hyperbolic fixed point in a discrete dynamical system.
Our work considers how long it takes to reach a periodic orbit of a differential equation—starting from an arbitrary point in phase space—by using a Poincaré return map of its associated vector field. For simplicity, suppose that a return map (which is built from a Poincaré section) is unimodal. If we approximate the unimodal Poincaré map by using a unimodal function 
, then we can use 
in our algorithm to estimate the convergence time to the periodic orbit. Periodic motion is ubiquitous in models (and in nature), and it is important to explore how long it takes to converge to such behavior.
In this paper, we prove a theorem for the rate of convergence to stable periodic orbits in discrete dynamical systems. Our basic strategy is as follows. We define the neighborhood 
of a hyperbolic fixed point, and we calculate what fraction 
of the entire phase space 
is mapped into 
after 
iterations. Using 
and 
, respectively, to denote the measures of 
and 
, a point that is selected uniformly at random from 
has a probability of 
to reach 
in 
iterations. To illustrate our ideas, we will work with a one-dimensional (1D) discrete dynamical system 
that is governed by a unimodal function 
and is parametrized by a real number 
. We focus on unimodal functions for two primary reasons: (i) many important results in dynamical systems are based on such functions; and (ii) it is simpler to illustrate the salient ideas using them than with more complicated functions.
To determine the set that is mapped into 
, we take advantage of the fact that points in periodic orbits are repeated periodically, so their corresponding neighborhoods must also repeat periodically. In theory, an alternative procedure would be to iterate backwards from 
, but this does not work because one cannot control successive iterations of 
. The function 
is unimodal, so it is not bijective and in general one obtains multiple sets for each backward iteration of a single set. The number of sets grows geometrically, and one cannot in general locate them because an analytical expression for 
is not usually available.
To explain the main ideas of this paper and for the sake of simplicity, consider a stable periodic orbit 
of period 
that is born in 
saddle-node bifurcations of 
. Every point 
(with 
of 
has a sibling point 
that is born in the same saddle-node bifurcation. Because 
and 
, it follows that 
, where 
. That is, 
, 
, and 
all repeat periodically. Roughly speaking, we will build the interval 
from the interval 
.
Consider a plot in which points along the horizontal axis are mapped via 
to points along the vertical axis (as is usual for 1D maps). The orbit 
is periodic with period 
, so 
implies that 
yields 
periodic points with a horizontal axis location of 
. We say that these points are located in the ’’column" 
. Because 
for some 
, we obtain 
points located in the same column 
. These points are given by 
. As we have indicated above, each point 
is associated with an interval 
. No matter how many iterations we do, the fact that the orbit is periodic guarantees that there are exactly 
intervals in the same place (where the points 
are located). We thereby know the exact number and locations of all intervals.
To complete the picture, we must also take into account that if there exists an interval 
such that 
, then any point of 
will reach a point of 
in at most 
iterations. The geometric construction above yields the interval 
, as one can see by drawing a pair of parallel line segments that intersect both 
and the endpoints of the interval 
. We will approximate 
by a set of such line segments so that we can easily calculate the intersection points.
The remainder of this paper is organized as follows. First, we give definitions and their motivation. We then prove theorems that indicate how long it takes to reach the interval 
from an arbitrary initial condition. We then construct algorithms to implement the results of the theorems. Finally, we discuss a numerical example and then conclude.
Definitions
Consider the discrete dynamical system 
(1)
where 
is a one-parameter family of unimodal functions with negative Schwarzian derivative and a critical point at 
. Without loss of generality, we suppose that there is a (both local and global) maximum at 
. At a critical point of a map 
, either 
(as in the logistic map) or 
does not exist (as in the tent map). Some of the results of this paper related with critical points only require continuous functions, which is a much weaker condition than the requirement of a negative Schwarzian derivative.
Remark 1 Because 
has a negative Schwarzian derivative, 
does as well (because it is a composition of functions with negative Schwarzian derivatives). By using the chain rule, we obtain 
only at extrema. Therefore, 
between consecutive extrema. The Minimum Principle [22] for a function with negative Schwarzian derivative then guarantees that there is only one point of inflection between two consecutive extrema of 
. If there were more than one point of inflection, then 
at least two points. One of them would be a maximum of 
, and the other one would be a minimum. This contradicts the Minimum Principle. Consequently, the graph of 
between two consecutive extrema has a sigmoidal shape (i.e., it looks like ∫ or 
), which becomes increasingly steep as 
becomes larger. This fact makes it possible to approximate 
between two consecutive extrema by a line segment near the only point of inflection that is located between two consecutive extrema.
Because the Schwarzian derivative of 
is negative, Singer's Theorem [23] ensures that the system (1) has no more than one stable orbit for every fixed value of the parameter 
. Additionally, the system (1) exhibits the well-known Feigenbaum cascade [24]-[26], which we show in the bifurcation diagram in Fig. 1.
For a particular value of the parameter 
, the map 
has 
simultaneous saddle-node (SN) bifurcations, which result in an SN 
-periodic orbit. As 
is varied, the SN orbit bifurcates into a stable orbit 
and an unstable orbit 
. The points 
and 
are, respectively, the node and the saddle generated in an SN bifurcation, so 
is the nearest unstable point to 
(see Fig. 2). In other words, the points in the stable orbits (called "node orbits") are node points, whereas the points in the unstable orbit (called "saddle orbits") are saddle points. From Remark 1, we know that the neighborhoods of these points are concave or convex.
The derivative of 
is 1 at the fixed point where the SN bifurcation takes place. As one varies 
, the derivative evaluated at that bifurcation point changes continuously from 
to 
. When the derivative is 
, the stable orbit (i.e., the node orbit) undergoes a period-doubling bifurcation. As a result, the stable orbit becomes unstable (yielding the orbit 
) and two new stable orbits (
and 
) appear. The points 
and 
are nodes, and the point 
is a saddle. From our geometric approach, the intervals 
and 
that are generated via the period-doubling bifurcation behave in the same way as the interval 
that was generated in the SN bifurcation. Therefore, we can drop the indices "1" and "2" and write 
for the orbits that arise from both the SN bifurcation and the period-doubling bifurcation.
Notation 1 Let 
denote the nearest point to 
that results from the intersection of the line 
with 
(see Figs. 2, 3, and 4).
Definition 1 Consider the points 
and 
that satisfy 
and 
. If 
is concave (respectively, convex) in a neighborhood of 
, we say that 
[respectively, 
] is the 
th capture interval of the stable 
-periodic orbit 
and that 
is the (aggregate) capture interval of the stable 
-periodic orbit 
.
Notation 2 Let 
denote the subinterval 
that contains the critical point 
.
From Definition 1, we see for all 
that 
and 
as 
. Iterations of points 
are repelled from 
and 
, and they are attracted to 
. The system (1) is linearizable around the fixed points 
and 
. (Observe that 
, so we also have control over this point.) Consequently, the convergence of iterations of 
to 
is governed by the eigenvalues of the Jacobian matrix 
.
Because we can control the evolution inside 
, we can examine how long it takes to reach 
starting from an arbitrary point 
. As we will see below, to obtain this result, we need to discern which subintervals of 
are mapped by 
into 
for arbitrary 
. The first step in this goal is to split the interval 
in which 
is defined into subintervals in which 
is monotonic.
Definition 2 Let 
be the set of points at which 
has extrema. Let 
, and we recall that we are considering the interval 
. We will call 
the partition of monotonicity of 
. We will call 
(where 
) the 
th interval of monotonicity of 
.
By construction, 
, and 
is monotonic in 
. As we will explain below, one can calculate intervals of monotonicity 
easily by using Lemmas 1 and 2.
Once we know the intervals in which 
is monotonic, it is easy to obtain subintervals of 
that are mapped by 
into 
.
We proceed geometrically (see Figs. 2, 4, and 5):
- draw parallel lines through the points
and 
(i.e., through the endpoints of 
);
- obtain the points at which the lines intersect
;
- calculate which points are mapped by
into the intersection points of (ii), for which one uses the fact that 
is monotonic in 
;
- determine, using the points obtained in (iii), the interval that is mapped by
into 
.
Using this geometric perspective, we make the following definitions.
Definition 3 Let 
(2)
where the points 
, and they satisfy 
and 
.
(i) If 
does not have extrema in 
(see Figs. 2 and 4), then we let 
(3)
(ii) If 
has extrema in 
(see Figs. 2 and 5), then we let
(4)
Remark: If we did not take point (ii) into account, then 
would not be monotonic in 
.
By construction, all points 
reach 
in at most 
iterations (see Figs. 2, 4, and 5). That is, 
for 
.
Definition 4 We call 
the 
-capture interval of 
, as 
is captured after at most 
iterations. The interval 
is then the 
-capture interval of the orbit 
.
Observe that 
can be the empty set for some values of 
.
Theorems
Once we know 
, we can calculate the probability that a point picked uniformly at random from phase space is located in 
. We can then calculate the probability that that point reaches a capture interval of 
in at most 
iterations. We let 
denote the measure of 
, and we have the following theorem.
Theorem 1 Let 
be a stable 
-periodic orbit of the system (1). Given an arbitrary point 
, the probability to reach a capture interval of 
after at most 
iterations is
(5)
Proof 1 From the definition (4) of 
, all 
satisfy 
for 
. There always exist values of 
such that 
because extrema of 
that satisfy 
are necessarily also extrema of 
, and points belonging to the latter set of extrema reach a capture interval of 
after at most 
iterations (see Lemma 1 below). Consequently, one reaches 
from 
after at most 
iterations (and we note that it need not be exactly 
iterations). Thus, the probability to reach 
from an arbitrary point 
after at most 
iterations (i.e., the probability that 
) is 
(6)
Corollary 1 With the hypotheses of Theorem 1, the probability to reach a capture interval of 
in exactly 
iterations is 
This answers the question of how long it takes to reach a capture interval of a 
-periodic orbit from an arbitrary point. However, we also need to calculate 
. To do this, we need to understand the structure of 
. As the following lemma indicates, some of these subintervals are located where 
is monotonic and others contain extrema of 
.
Lemma 1 If 
is an unimodal 
function with a critical point at 
, then 
has extrema
(i) at points for which 
;
(ii) at the same points at which 
has extrema.
Proof 2
(i)For all 
such that 
, we know that 
. Therefore, 
has an extremum because 
has an extremum.
(ii)Write 
, where 
and 
, so 
is a monotonic function on the intervals 
and 
.
(ii.a) If 
or 
and the function 
has an extremum, then we know that 
is a monotonically increasing function on one side of 
and a monotonically decreasing function on the other. Consequently, 
is the composition of two monotonic functions (
and 
), both of which are increasing (or decreasing) on one side of 
. On the other side of 
, one of them is increasing and the other is decreasing. Therefore, there is an extremum at 
.
(ii.b) Otherwise, if 
, then we see straightforwardly that 
has an extremum.
We have just seen how to determine the locations of extrema of 
. We also need to know the values that 
takes at these extrema.
As we will see below, if the system (1) has a stable 
-periodic orbit and 
, then the values that 
takes at its extrema are the same as those that 
takes at its extrema. This makes it possible to calculate the subintervals 
that are associated with extrema of 
by using 
and the derivative of 
.
Lemma 2 Let 
be a stable 
-periodic orbit of the system (1). The coordinates of the extrema of 
(where 
) are 
, where 
denotes the points 
such that 
, the index 
takes values of 
(where we note that 
is the identity map), and 
.
Proof 3 According to Lemma 1, the extrema of 
are
(i) 
such that 
;
(ii) 
such that 
is an extremum.
It thus follows that the extrema of 
are
(ia) 
such that 
;
(iib) 
such that 
is an extremum.
Repeating the process, we obtain that extrema of 
are located at 
, where 
. The value of 
at 
is 
.
Because 
is a stable 
-periodic orbit, there exists one point of 
near 
that is repated periodically after 
iterations. Consequently, 
is a periodic sequence and 
.
Algorithms
As we discussed above, Lemmas 1 and 2 determine intervals of monotonicity (see definition 2), and they also make it possible to construct algorithms for calculating 
.
For these algorithms, we approximate 
by line segments in the subintervals in which 
is monotonic. This approximation is very good unless one is extremely close to an extremum (see Fig. 6), and this is already the case even for relatively small 
(as we will demonstrate below). Additionally, recall that 
is determined by the intersection points of 
with line segments. Therefore, once we have approximated 
by a set of line segments, it is straightforward to calculate those intersection points.
Algorithm 1 (Calculating coordinates for extrema of 
) Suppose that we know the coordinates of the extrema of 
. According to Lemma 1, the extrema of 
are located at the points
(i) 
such that 
and 
is not an extremum;
(ii) 
such that 
is an extremum.
We know the extrema in (ii) by hypothesis. To find the extrema in (i), we need to calculate the points 
that satisfy 
. Because we know the coordinates of extrema of 
, we construct the lines that connect two consecutive extrema (see Fig. 7). Let 
be the equation for such a line. We solve 
to obtain a seed that we can use in any of the many numerous numerical methods for obtaining roots of nonlinear algebraic equations. Observe that 
is monotonic in the interval in which the line 
is defined. This circumvents any problem that there might otherwise be in obtaining a good seed to ensure convergence of the root solver. Moreover, we have as many seeds as there are points 
that satisfy 
. Note that we need to construct both the line that connects 
with the first extremum of 
and the line that connects 
with the last extremum of 
.
To calculate the points 
for which 
is an extremum, we apply this algorithm recursively, and we note that we know by hypothesis that 
has an extrememum at 
. We first build the line segments that connect 
with 
and 
with 
. These two line segments give seeds from which to determine the points 
that satisfy 
. We thereby obtain the coordinates for the extrema of 
. We then use the same procedure to obtain the coordinates for extrema of 
, 
, 
, 
.
We will see below that if the system (1) has a stable 
-periodic orbit and 
, then the points 
with 
are given to a very good approximation by the intersection points of two lines. Moreover, as one can see in Fig. 6, the value 
is already large enough to approximate 
very successfully by a set of line segments when 
is the logistic map.
Algorithm 2 (Calculation of 
in the system (1)) Suppose that we know the coordinates of the extrema of 
(e.g., by computing them using Algorithm 1). We want to obtain 
from the definition (3), where
(7)
and the 
th capture interval is 
(8)
To determine the points 
and 
, we first approximated them by replacing 
by line segments that connect consecutive extrema of 
(i.e., by the same procedure that we use in Algorithm 1 to obtain approximations of points). Using the approximations of 
and 
, we construct the interval 
and then check if there is an extremum of 
in 
. (This is trivial because we know the coordinates of the extrema of 
.) We need to consider two cases.
(i)The map 
has no extrema in 
. This is equivalent to case (i) of Algorithm 1. We use the approximations of 
and 
as seeds in a numerical root-finding method.
(ii)The map 
has extrema in 
. This is equivalent to case (ii) of Algorithm 1.
If there is an extremum of 
in 
, then that extremum is necessarily one of the extrema given by Lemma 2: 
. Because 
and 
is a continuous function, there must exist an interval 
such that 
and 
Taking into account that 
is known, we construct the sequence 
where 
, 
, and 
Let 
be the linear map whose graph is the line of slope 
that intersects the point 
. If the period 
of the orbit is sufficiently large, then we can approximate 
near 
(where 
) by the linear map 
. Thus, instead of iterating 
with the map 
to obtain 
, we iterate 
with the linear map 
that approximates 
. That is, 
Because each 
is a linear map, it is straightforward to compute 
and hence to compute 
At the end of this section, we will discuss the error that is introduced by this approximation.
The interval 
that we have just constructed is the interval 
(9)
that we seek.
In Algorithm 1, we constructed line segments that connect two consecutive extrema of 
. They are located in the intervals 
and 
, respectively. We now have intervals 
and 
that contain these two consecutive extrema of 
, so we construct the line segment that connects the upper endpoint of 
to the lower endpoint of 
. (Note that we do not connect the two extrema directly via a line segment.) For 
, this line segment approximates 
outside of the intervals 
and 
. See Fig. 6, which illustrates (for the case when 
is the logistic map) that we can approximate 
by a set of line segments for 
. We can then use these line segments in Algorithm 1, and we do not need numerical computations to find the intersection points.
As we discussed previously, we can replace 
by linear expressions to approximate the intersection points when determining 
in Algorithms 1 and 2. Replacing 
by a linear approximation simplifies operations and reduces the amount of calculation. To determine the desired intersection points, we have thereby replaced a numerical method for obtaining roots of nonlinear algebraic equations by an analytical calculation that uses a system of two linear equations. We now estimate the error of replacing 
by lines segments. The line segments that replace the function 
intersect 
very close to the unique point of inflection between a pair of consecutive extrema of 
(see Remark 1 and Fig. 7). The Taylor polynomial of degree 3 of 
around the inflection point 
is 
Consequently, the error of approximating 
by the line 
is 
(10)
where we have taken into account that there are more than 
local extrema of 
in the interval 
. The exponential growth of 
enforces a fast decay in the error. Consequently, using line segments to approximate 
is an effective procedure with only a small error.
Numerical Example
Algorithms 1 and 2 are based on the same procedure: approximate 
by a line 
and solve 
to obtain an approximation of the 
(instead of solving 
directly). In this section, we consider an example application of Algorithm 1.
To obtain the critical points of 
, we need to calculate the points that satisfy 
. Suppose that 
(and again see Fig. 6 for an illustration of the line-segment approximation with 
for the logistic map). The biggest distance between consecutive extrema occurs near the critical point 
, so we approximate 
by a line segment in this region to obtain an upper bound for the error. The extrema are located at 
and 
, and they are connected by the line 
, from which we obtain the approximation 
for the solution of 
. From direct computation, the value of 
that satisfies 
is 
. The relative error is 
, and this is the largest error in this example from all of the approximating lines segments. As we showed in equation (10), the error decreases exponentially. Hence, when we approximate 
using line segments, the relative error will be bounded above by 
. One can observe this decrease in error in Fig. 7, in which we plot both 
and 
for the logistic map and the same parameter value 
. Observe that several extrema of 
lie between onsecutive extrema of 
, so using the line-segment approximation in 
induces a much smaller error than using it in 
.
Conclusions and Discussion
When studying dynamical systems, it is important to consider not only whether one converges to periodic orbits but also how long it takes to do so. We show how to do this explicitly in one-dimensional discrete dynamical systems governed by unimodal functions. We obtain theoretical results on this convergence and develop practical algorithms to exploit them. These algorithms are both fast and simple, as they are linear procedures. One can also apply our results to multimodal one-dimensional maps by separately examining regions of parameter space near each local extremum.
Although we have focused on periodic dynamics, the ideas that we have illustrated in this paper can also be helpful for trying to understand the dynamics of chaotic systems. Two important properties of a chaotic attractor are that (i) its skeleton can be constructed (via a ’’cycle expansion") by considering a set of infinitely many unstable periodic orbits; and (ii) small neighborhoods of the unstable orbits that constitute the skeleton are visited ergodically by dynamics that traverse the attractor [4]. In Refs. [21], [22], Schmelcher and Diakonos developed a method to detect unstable periodic orbits of chaotic dynamical systems. They transformed the unstable periodic orbits into stable ones by using a universal set of linear transformations. One could use the results of the present paper after applying such transformations. Moreover, the smallest-period unstable periodic orbits tend to be the most important orbits for an attractor's skeleton [4], so our results should provide a practical tool that can be used to help gain insights on chaotic dynamics.
Once unstable orbits has been transformed into stable ones we can use results of this paper to answer the above question.
Acknowledgments
We thank Erik Bollt, Takashi Nishikawa, Adilson Motter, Daniel Rodríguez, and Marc Timme for helpful comments.
Author Contributions
Analyzed the data: JSM MAP. Wrote the paper: JSM MAP. Proved theorems: JSM MAP. Designed algorithms: JSM MAP. Wrote code and performed numerical simulations: JSM MAP.
References
- 1.
Moehlis J, Josić K, Shea-Brown ET (2006) Periodic orbit. Scholarpedia 1: 1358.
- 2.
Cvitanović P, Artuso R, Mainieri R, Tanner G, Vattay G, et al. (2012) Chaos: Classical and Quantum. Version 14. Available: http://chaosbook.org.
- 3.
Poincaré H (1892–1899) Les méthodes nouvelles de la méchanique céleste. Paris, France.
- 4.
Auerbach D, Cvitanović P, Eckmann JP, Gunarathe G, Procaccia I (1987) Exploring chaotic? motions through periodic orbits. Physical Review Letters 58: 2387–2389.
- 5.
Artuso R, Aurell E, Cvitanović P (1990) Recycling of strange sets: I. cycle expansions. Nonlinearity 3: 325–359.
- 6.
Ermentrout GB, Terman DH (2010) Mathematical Foundations of Neuroscience. New York, NY, USA: Springer-Verlag.
- 7.
Strogatz SH (2000) From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143: 1–20.
- 8.
Strogatz SH (1994) Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering. New York, NY, USA: Perseus Books Publishing.
- 9.
Lellis PD, di Bernardo M, Garofalo F (2013) Adaptive synchronization and pinning control of networks of circuits and systems in Luré form. IEEE Transactions on Circuits and Systems I in press.
- 10.
Yu W, Lellis PD, di Bernardo M, Kurths J (2012) Distributed adaptive control of synchronization in complex networks. IEEE Transactions on Automatic Control 57: 2153–2158.
- 11.
Valtaoja E, Terësranta H, Tornikoski M, Sillanpä A, Aller MF, et al. (2000) Radio monitoring of OJ 287 and binary black hole models for periodic outbursts. The Astrophysical Journal 531: 744–755.
- 12.
Kreilos T, Eckhardt B (2012) Periodic orbits near onset of chaos in plane Couette ow. Chaos 22: 047505.
- 13.
Neufeld Z (2012) Stirring effects in models of oceanic plankton populations. Chaos 22: 036102.
- 14.
Sun J, Bollt EM, Porter MA, Dawkins MS (2011) A mathematical model for the dynamics and synchronization of cows. Physica D 240: 1497–1509.
- 15.
Qi GX, Huang HB, Shen CK, Wang HJ, Chen L (2008) Predicting the synchronization time in coupled-map networks. Physical Review E 77: 056205.
- 16.
Grabow C, Hill SM, Grosskinsky S, Timme M (2010) Do small worlds synchronize fastest? Europhysics Letters 90: 48002.
- 17.
Nishikawa T, Motter AE (2010) Network synchronization landscape reveals compensatory structures, quantization, and the positive effect of negative interactions. Proc Natl Acad Sci USA 107: 10342–10347.
- 18.
Klebanoff A, Bollt EM (2001) Convergence analysis of Davidchack and Lai Õs algorithm for finding periodic orbits. Chaos Solitons and Fractals 12: 1305–1322.
- 19.
Grobman DM (1959) Homeomorphisms of systems of differential equations. Doklady Acad Nauk SSR 128: 880–881.
- 20.
Hartman P (1960) On local homeomorphism of Euclidean spaces. Bol Sc Mat Mexicana 5: 220–241.
- 21.
Lan Y, Mezic I (2013) Linearization in the large of nonlinear systems and Koopman operator spectrum. Physica D 242: 42–53.
- 22.
Brin M, Stuck G (2002) Introduction to Dynamical Systems. Cambridge, UK: Cambridge University Press.
- 23.
Singer D (1978) Stable orbits and bifurcation of maps of the interval. SIAM Journal of Applied Mathematics 35: 260–267.
- 24.
Myrberg PJ (1963) Iteration de reellen polynome zweiten grades iii. Ann Acad Sci Fenn 336: 1–18.
- 25.
Feigenbaum MJ (1978) Quantitative universality for a class of nonlinear tranformations. Journal of Statistical Physics 19: 25–52.
- 26.
Feigenbaum MJ (1979) The universal metric properties for nonlinear tranformations. Journal of Statistical Physics 21: 669–706.
- 27.
Schmelcher P, Diakonos FK (1997) Detecting unstable periodic orbits of chaotic dynamical systems. Physical Review Letters 78: 4733–4736.
- 28.
Schmelcher P, Diakonos FK (1998) General approach to the localization of unstable periodic orbits in chaotic dynamical systems. Physical Review E 57: 2739–2746.
