Infinite towers in the graph of a dynamical system
IInfinite towers in the graph of a dynamical system
Roberto De Leo ∗ and James A. Yorke † January 6, 2021
Abstract
Chaotic attractors, chaotic saddles and periodic orbits are examples of chain-recurrent sets.Using arbitrary small controls, a trajectory starting from any point in a chain-recurrent set canbe steered to any other in that set. The qualitative behavior of a dynamical system can beencapsulated in a graph. Its nodes are chain-recurrent sets. There is an edge from A to node B if, using arbitrary small controls, a trajectory starting from any point of A can be steered toany point of B . We discuss physical systems that have infinitely many disjoint coexisting nodes.Such infinite collections can occur for many carefully chosen parameter values. The logistic mapis such a system, as we showed in [14]. To illustrate these very common phenomena, we comparethe Lorenz system and the logistic map and we show how extremely similar their bifurcationdiagrams are in some parameter ranges. In Seventies, Charles Conley introduced the idea of describing the qualitative behavior of a dynamicalsystem by the graph that we describe below. In [14], we showed that the graph of the logistic map µx (1 − x ) is surprisingly complicated for certain values of µ . We argue that the most complicatedlogistic map graphs appear within the graphs of much more general and complicated systems. Toillustrate this fact, we compare the logistic map with the Lorenz system. That requires that weestablish the corresponding facts about the Lorenz system. We alert the reader that there is asimilarity between some of the pictures in this paper and in [14].It might seem to the reader that the Lorenz system and the logistic map might appear to becompletely unrelated. That is exactly why we have selected the Lorenz system, when we could havechosen any of a wide variety of physical systems. On the other hand, we have chosen the logisticmap because of the rich rigorous literature that is available on it.Bifurcation diagrams for the logistic map typically show how the attractor changes as the pa-rameter changes. In addition to an attractor, the logistic map has several other disjoint invariantsets, and there are parameter values for which there are infinitely many of them. The invariant setswe speak of are “chain-recurrent”, as we describe below. An example of dynamical system with a simple graph.
Consider the map z (cid:55)→ z onthe complex plane, to which we add the point at ∞ . The plane plus ∞ should be thought of as atopological sphere. For many important cases, we can “compactify” a space by adding a point at ∞ and often, as for this map, ∞ is a fixed point and the map is still continuous.We can use this map as an example of how to represent a dynamical system by a graph. Thismap has three invariant sets that will be nodes of the graph. Both { } and {∞} are attractors andare nodes, and the third node is the unit circle, a repelling chaotic invariant set. Notice that notall invariant sets are nodes. Explaining what a node is will take some care. Even for such a simplemap, the dynamics within a node can be quite complicated. For instance, in the z example, thedynamics on the unit circle z = e iϕ , ϕ ∈ [0 , π ], is given by the doubling map ϕ (cid:55)→ ϕ (also known ∗ Department of Mathematics, Howard University, Washington DC 20059, [email protected] † Institute for Physical Science and Technology and the Departments of Mathematics and Physics, University ofMaryland College Park, MD 20742, [email protected] a r X i v : . [ n li n . C D ] J a n igure 1: Examples of chain-recurrence. (LEFT)
Example of nodes in a continuous dynamicalsystem. The set D is the disc bounded by the outer periodic orbit. Three nodes are visible in thepicture: the outer periodic orbit N , the inner periodic orbit N and a fixed point N . The edgesof this graph go from the repellors N and N to the attractor N . (RIGHT) An example of an ε -chain from x and back to itself. The dashed circles represent circles of radius ε .as shift or Bernoulli map). This map is one of the best known examples of a chaotic map. Noticethat there are infinitely many periodic orbits on the unit circle but none of them is a node. The setof nodes of a general dynamical systems can be quite a bit more complicated than the set of threenodes in this case.This paper is about a type of control theory. For each point p , it identifies the downstream point q such that either the trajectory from p goes to q or an arbitrarily small amount of control can beadded such that the controlled trajectory goes from p to q . We now extend the stream analogy. If p is downstream from q and q is downstream from p , then we say p and q are in the same pond. A node is a pond. In other words, a node N is the set of points so that, if p is in N , then q is in N ifand only if p and q are in the same pond. A trajectory starting form any point in the node can beforced to stay in the node by using arbitrarily small perturbations that we call controls. We makethis precise as follows. Chain-recurrence.
By a dynamical system Φ , we mean a 1-parameter family of continuousmaps Φ t from a space X into itself. Write dist ( x, y ) for the distance between x and y . The timeparameter t can be either continuous or discrete. Given two points p, q , with p (cid:54) = q , in X and ε >
0, we say that there is a ε -chain from p to q (see Fig. 1) if there is a finite sequence of points p = x , x , . . . , x n = q on X such that, for i = 0 , . . . , n − Φ ( x i ) , x i + i ) ≤ ε. (1)To our knowledge, ε -chains were introduced in literature by R. Bowen in 1975 [8].We say that q is downstream from p if, for every ε >
0, there is a ε − chain from p to q ;equivalently, we say that p is upstream from q . We write p ∼ q if p is upstream and downstreamfrom q , and we say that p is chain recurrent if p ∼ p . We let R Φ denote the chain-recurrentset , i.e. the set of all chain recurrent points of Φ . Chain recurrence was introduced by C. Conley inhis celebrated monograph in 1978 [13] and it is a central concept for this article. Examples of chain-recurrent points . Points on a periodic orbit are chain-recurrent and, if p and q are on the same periodic orbit, then p ∼ q .Chaotic sets are defined in various ways but a usual requirement is that there is a trajectorythat comes arbitrarily close to every point infinitely often. So, if p and q are in a chaotic set, atiny perturbation of p will land on the dense trajectory and, when it comes sufficiently close to q , asecond tiny perturbation will push it onto q . Hence, p ∼ q .Consider now a dynamical system on a vector space and suppose that all trajectories convergeto 0 as time goes to infinity. Then 0 is the only chain-recurrent point. Subtle control of dynamical systems.
The idea of ε − chains in (1) can be rephrased as thefollowing question in control theory. Assume X is in a linear space. Given two points p, q, p (cid:54) = q, in X , does there exist for each ε > u i such that | u i | ≤ ε for a sequence of i ’s and2 N N N N N N N Figure 2:
An example of graph. (LEFT)
Dynamics induced on the 2-torus by the gradientvector field of the height function. In this case the Lyapunov function is the height function itself,some level set of which is shaded in white. In blue are shown the heteroclinic trajectories joiningthe critical point (which are exactly the invariant sets of this dynamical system). (RIGHT)
Thegraph of the dynamical system on the left. In this case it is a 4-levels tower. N N m N n N ∞ ......... N N +2 N − N +3 N − N + n − N − n − N + n N − n ... ... Figure 3:
Examples of graphs. (LEFT)
An infinite tower graph. (RIGHT)
The graph of thesemiflow of the Chafee-Infante PDE (see Sec. 4).a controlled trajectory x i + i = Φ ( x i ) + u i where p = x , x n = q. (2)If p ∼ q , then there are such controls and it is possible to create control u i that allow us to steer atrajectory from p to q and back to p . Furthermore, max | u i | can be made as small as desired, i.e. ,less than any specified positive number. A trajectory of a dynamical system.
Here we restrict attention to discrete time dynamicalsystems. For a map Φ , we will say that the sequence p n is a trajectory if p n is defined for all n ∈ Z ,where Z is the set of all integers, n = 0 , ± , ± , . . . , and p n +1 = Φ ( p n ) for all n ∈ Z .For some maps, the inverse is not unique. For the map z (cid:55)→ z , each point other than 0 hastwo inverses. Hence there will be infinitely many trajectories through a given p (cid:54) = 0 . Two differenttrajectories through p will have the same forward limit set but might have different backward limitsets. Assumptions on the phase space.
In this paper, aside from our infinite dimensional examples,we examine continuous dynamical systems on a compact set X . In the above example, we haveadded a point at infinity to make the set compact. In this paper we use the following definition.3 set X is compact if for each sequence of points x n ( n = 1 , , . . . , ∞ ), there is a subsequence x n j ( j = 1 , , . . . , ∞ ) that converges to some point p . Considering all convergent subsequences, theset of limit points p is the limit set of x n . Where are the limit sets.
For any point x , its forward limit set ω ( x ) is the set of its limitpoints, namely those points that are the limit a subsequence of points belonging to the forward orbitof x . Its trajectory might diverge, i.e. , its limit set is empty. Then we can say it converges to thenode ∞ . Otherwise, its limit set must be a subset of a single node. For example picture a situationwhere a trajectory in the plane lies between two invariant lines and it spirals outward toward thoselines. Then the node includes all the points on those two lines. If that node Ω is a compact set,then the distance of Φ t ( x ) from Ω goes to 0 as t → ∞ . Attractors.
We call a node N an attractor , also sometimes called a Milnor attractor , if itsbasin of attraction, i.e. , the set of points x such that ω ( x ) is contained in N , has positive measure [53].A non-trivial example of a Milnor attractor occurs at the Feigenbaum parameter value. The graph of a dynamical system and Lyapunov functions.
Conley realized that chain-recurrence could be used to define a graph of a dynamical system [12, 13]. His investigationsconcerned dynamical systems that come from ordinary differential equations on compact spaces.Over the years his results have been extended to several other settings, in particular: continuousmaps [58], semi-flows [63, 30, 60], non-compact [33, 60] and even infinite-dimensional spaces [63,47, 11, 28] (here and throughout this article, we sort multiple citations in the order of their year ofpublication).The main contribution of Conley is the discovery that the dynamics outside of the nodes is always gradient-like , namely there is a continuous function L : X → R such that:1. L is constant on each node;2. L assumes different values on different nodes;3. L ( Φ t x ) < L ( x ) for all t > x not in a node [54].In particular, nodes are equilibria for L . Note also that properties 1, 2 and 3 make L a Lyapunovfunction ( e.g. see [83, 58]).The graph of a dynamical system consists of nodes and edges between the nodes. The forwardand backward limit sets of a trajectory are each contained inside a single node. That limit set canalso be the entire node. There is an edge from node N to node N if and only if there is a trajectorywhose backward limit set is in N and its forward limit set is in N ( e.g. , see Fig. 2 (right) andFig. 3). That edge can be denoted by N → N , which reads that N is above N . In particular, N → N implies that L ( N ) > L ( N ), so that it is impossible that also N → N .Each node N has a closest point to the critical point c = 1 /
2. Let ρ ( N ) denote the distancebetween c and that closest point. We show in [14] that N → N is equivalent to saying ρ ( N ) >ρ ( N ).The classification of more complex nodes was an important milestone even in the setting of 1-dimensional dynamics (a list of specific references is given in Sec. 3). In this last case, though, itseems that the dynamical system community put the emphasis in the classification of the nodes andsomehow overlooked the description of the rest of the dynamics, that is, which pairs of nodes N , N have an edge N → N .We call tower a finite or infinite sequence of nodes N i such that:1. there is a first node, denoted by N ;2. there is a final node, which is the unique attractor; all other nodes are unstable;3. for any two nodes N i and N j , with j > i , we have that N i → N j .In particular, for each node N i , where i >
0, there is a previous node N i − and, unless N i is theattractor, a next node N i +1 .Our main result on the logistic map is the following: Tower Theorem.
The set of all nodes of the logistic map is a tower.4igure 4:
The Bifurcation Diagram of the Logistic Map.
To the left of the Feigenbaum-Myrberg parameter value µ F M (cid:39) . i.e. , intervals in parameter space that begin witha periodic attractor which evolves through period-doubling into small intervals of chaos. This pictureis created by plotting trajectories. More frequently visited regions are darker. Points on attractingperiodic orbits of period less than 26 are indicated by black dots. Notice, in particular, that manyof these points are near where x = 0 .
5. In colors are highlighted, besides µ F M , the largest period-6window, the intersection parameter value µ Int = 1 . − √ / + (19 + 3 √ / ] (cid:39) . µ Int , x
Int ) (see figure). Each of the high-density lines is the image of the x = 0 . (cid:96) nµ for some n . In the bottom right box it is shown a detail of the cascade about x = 0 . Bifurcation diagram and sample graphs of the logistic map.
This picture showsthe bifurcation diagram of the logistic map in the range of parameter values [2 . , µ , the attracting set is painted in shades of gray, depending on the density of the attractor, repellingperiodic orbits in green and repelling Cantor sets in red. Below the µ axis we show seven samples ofthe graphs illustrating some of the possible variability. In these graphs, each colored disk is a node.Each black disk represents an attractor, each green disk represents a repelling periodic orbit andred represents a chaotic Cantor set repellor. For simplicity we always omit the top node, which isthe point 0. Graph T4 represents the infinite tower at the first Feigenbaum point. It has infinitelymany unstable periodic orbit nodes. 6igure 6: Towers of nodes shown below the period-3 window of the logistic map bifur-cation diagram.
This figure is a blow-up from Fig. 5, and uses the color-coding from that figure.Graph T8 has two levels of nodes that are Cantor sets repellors and the second is painted in blue.In the bifurcation diagram, the chain recurrent sets have the same coloring as their nodes. GraphT5 represents the infinite tower at the first Feigenbaum point of the main cascade of the period-3window. 7igure 7:
A periodic window in the bifurcation diagram of the Poincar´e map of theLorenz system . This figure is placed here for comparison with the very similar Fig. 6 for thelogistic map. More information about the Lorenz system and its Poincar´e map is given in thetext. This window runs from r (cid:39) .
520 to r (cid:39) . y outside of the range shown here and there is another attractor.Several nodes that are unstable Cantor sets are shown in red and blue.For specific parameters, the logistic map has infinitely many nodes. In this case, we refer to thegraph as an infinite tower . We believe that infinite towers are not restricted to one-dimensionalmaps: Tower Conjecture.
Infinite towers occur within the graphs of generic chaotic dynamical sys-tems in any dimension.In other words, many chaotic processes have a much more complicated structure than theoreti-cians previously expected. We present numerical arguments in support of our conjecture.The article is structured as follows.In Sec. 2 we discuss our numerical results on the bifurcation diagram of the Lorenz map. Inparticular, we plot the attractor together with some of the repelling chain-recurrent sets and arguethat there are parameter ranges where the diagram looks exactly as the one of the logistic map. OurTower Conjecture is a direct consequence of these observations.Motivated by these results, in Sec. 3 we review some fundamental results on the logistic map anddescribe the most important features of its bifurcation diagram.Finally, in Sec. 4 we describe the graphs of some partial differential equations and differentialdelay equations. All the published results we know of describe the graphs of these systems as beingfinite and hence simpler than the most complicated cases of the logistic map.Section 4 describes the graphs of some partial differential equations and differential delay equa-tions. All the published results we know of describe the graphs of these systems as being finite andhence simpler than the most complicated cases of the logistic map.8 view from above. A view from below.Figure 8:
The Lorenz Butterfly . This picture shows the attractor of the Lorenz system for r = 28.The color of the trajectory being plotted slowly varies to help visualize the flow. If z is thought ofas the vertical coordinate, then the left picture is viewed from above and the right one from theside. In the left, blue represents the z = r − z < r −
1. On the right, the attractor is colored with a blue tint when z > r − ad infinitum . In the 1960s, Edward Lorenz introduced and investigated the ODE system x (cid:48) = − σx + σyy (cid:48) = − xz + rx − yz (cid:48) = xy − bz , (3)that is now named after him [43], for a specific set of parameters: σ = 10, r = 28 and b = 8 / r between 13.9and 24.06, but this chaotic behavior happens only for a measure-zero set of points. At r (cid:39) . r (cid:39) . Attractors of the Lorenz system . This picture shows the attractors of the Lorenzsystem for r = 208 (left) and r = 209 . − ≤ x ≤ − ≤ y ≤
100 in the plane z = r −
1. If z is thought of as the vertical coordinate, both pictures are viewedfrom below. The Poincar´e map for the Lorenz system is built out of the intersections of the Lorenzorbits crossing this rectangle downwards. The yellow rectangle shown for r = 209 . A small region from the Lorenz system Poincar´e return map P r for r = 209 . The region shown corresponds to the small yellow region on the right-hand side of Fig. 9. In thatfigure, a periodic orbit is shown piercing the yellow rectangle in three points. Those points areshown here as the centers of three circles. Almost all points in the colored region are in the basinof attraction of the periodic orbit. Yellow indicates rapid convergence to the periodic orbit. Redindicates slow convergence. Red points are close to points that are attracted to the Cantor set onthe blue line. The blue curve is the unstable manifold of a Cantor set that lies within it. Points inthe white region are attracted to the other off-screen attractor. The blue curve includes a chain-recurrent Cantor set of saddle points and the unstable manifolds of all of the periodic orbits in theCantor set. 11igure 11:
Attractors of the Poincar´e Map of the Lorenz system.
The pictures above showthe attractor of the Poincar´e map of the Lorenz system in the ( x, y ) plane for five different valuesof r . Whenever the trajectory hits the z = r − z is decreasingand otherwise in red. There are places where the color switches from blue to red, due to the vectorfield being tangent to the plane. In all pictures x ranges from −
40 to 40 and y ranges from −
100 to100. Each panel is the plot of a single trajectory, hence low density regions of the attractor may notbe represented or it may only be represented by a few isolated points. There are two steady stateson this plane (Eq. 4). They are indicated with black dots.12ased on these works, Sparrow investigated numerically the Lorenz system [73] for a wider rangeof values of r . His Fig. 5.12 on p. 99 shows intervals of r values ( i.e. , windows) where the chaoticattractor is replaced by periodic attractors. He reports that below r = 30 . r = 28 (see [20] for a review of the analytical study of the Lorenz system and its crucialrole in the development of chaos theory). This is the 14th of the list of “mathematical problems forthe next millennium” made by Smale in 1998 [70]. It is noteworthy to mention that the proof ofTucker is computer assisted (see [74, 80] for interesting reviews of Tucker’s result). See also rigorousresults related to the Lorenz attractor [62, 61, 25, 82]. Fig 8 shows two views of the attractor for r = 28.More recently, Kobayashi and Saiki [39, 40] investigated how periodic windows arise as theparameter r increases from the Lorenz value and argue that the first bifurcation diagram’s windowsare contained in the interval 30 ≤ r ≤ r >
1, the Lorenzsystem has three fixed points: the origin and the twin points C ± = (cid:16) ± (cid:112) b ( r − , ± (cid:112) b ( r − , r − (cid:17) . (4)The origin is a saddle while C ± have a pair of complex eigenvalues. On the plane π r defined by z = r −
1, integral trajectories passing through points p close to C ± will return and cut again thesame plane in some other point q and so on. As long as the trajectory passes through π r , it will cutthe plane one time directed upwards and the next time directed downwards. Poincar´e Return Maps.
Poincar´e discussed trajectories that crossed some special plane orline. When he investigated the planar restricted three-body problem, he found it useful to recordonly half the crossings, those for which a particular coordinate was increasing. He encountered notangencies to the line. We usually take Poincar´e’s approach but, in Fig. 11, we record both crossingsin two colors, as was done in [64].We define the Poincar´e map P r at a point p to be the point q at which the trajectory startingfrom p cuts the plane π r with z decreasing. In Fig. 9 we show two examples of attractors for theLorenz system, a chaotic one (left) and a periodic one (right). The pictures also show, in gray, therectangle − ≤ x ≤ − ≤ y ≤
100 in the corresponding planes π r . By bifurcation diagramof the Lorenz system we mean the bifurcation diagram of the family of maps P r .In Fig. 12 we show a few projections of the bifurcation diagram: on the ( y, r ) plane (top), on the( x, r ) plane (middle) and on some intermediate plane (bottom). In particular, the bottom picturesuggests that the bifurcation diagram is the union of two disjoint components, one the image of theother. This fact is also suggested by the ( x, y ) sections of the diagram for several values of r shownin Fig. 11.The bifurcations pattern of the Lorenz system evolves backwards with respect to the one of thelogistic map. At r = 235 . r decreases, a bifurcation cascade completely analogous to the one of thelogistic map. The largest window of the diagram (see Fig. 12 and 13), centered at about r = 150 . Bifurcation Diagram . These are the projections onto the ( r, y ) plane (Top panel),( r, x ) plane (Middle panel) and ( y − x, r ) plane (Bottom panel) of the bifurcation diagram for thePoincar´e Return map of the Lorenz equations (3) using the plane π r defined by z = r −
1. A dotis plotted in the ( r, y ) (resp. ( r, x )) plane when a trajectory crosses downward past π r throughthe point ( x, y, r − r = 150. The period-3 window shown in Fig. 7 is anenlargement of the black rectangle shown in the top projection. For some parameter values, thereare two attractors. They are shown in red and blue. When there is a single attractor, it is shown inmagenta. 14igure 13: Bifurcation Diagram . These ( r, y ) projections are enlargements of the main period-4 window of the Lorenz system bifurcation diagram.
Top:
We show a zoom of the full window.
Middle:
We show the content of the region enclosed in the black rectangle in the top picture,namely the upper cascade.
Bottom:
We show the middle cascade of the period-3 window of thecascade above, enclosed in a black rectangle in the middle picture.15e also investigate the structure of the chain-recurrent set of the diagram. We focus on itslargest period-3 window, that is the one contained inside its very first cascade from the right. Therange of this window is from about 208 .
52 to 209 . The logistic map (cid:96) µ ( x ) = µx (1 − x ) (5)is among the simplest continuous maps giving rise to a (highly) non-trivial dynamics. In this article we focus exclusively on the parameter interval µ ∈ (1 , For these values, (cid:96) µ maps [0 , , our graphs only include nodes in (0 , In particular, ourgraphs omit the fixed point at x = 0, which is always the top-most node , and we ignorepoints outside of [0 , −∞ . Some history.
In a series of celebrated works starting in 1918, Julia and Fatou gave birth tothe study of the dynamics of the quadratic map in the complex plane. Surprisingly, the study ofthe quadratic map in the real line began later. The first example we know of is by Chaundy andPhillips [10] in 1936, inspired by early Mathematical Biology works such as [3].The study of iterations of real quadratic maps reappeared in a few clever abstract articles (“ab-stract” in that no applications were mentioned) around 1960 by P.J. Myrberg [55, 56, 57]. Myrbergdiscovered the infinite number of period-doubling bifurcations in the logistic map. In the same years,fundamental properties on the existence of cycles for general continuous maps of the real line intoitself were discovered by A.N. Sharkovski˘ı [66] (in Russian; see English translation in [67]).Possibly the first time that (cid:96) µ was called the “logistic map” was in 1968 in J. Maynard Smith’sbook “Mathematical ideas in Biology” [72]. Smith used it as a toy model for population dynamics,analogous to the one-century old logistic ordinary differential equation model of Verhulst [79, 81].In the 1970’s, many more works on the logistic map appeared in literature, some purely theoretical( e.g. Metropolis, Stein and Stein [52], Li and Yorke [42], Hoppensteadt and Hyman [32]) and someapplied ( e.g.
May [48], Smale and Williams [71], May and Oster [50], Guckenheimer, Oster andIpaktchi [24], Feigenbaum [17]). The celebrated article by R. May [49] brought the importance of1-dimensional dynamics to a broad scientific audience.The theoretical study of the logistic map evolved in 1980s in the study of families of more generalone-dimensional real maps such as, in order of generality, S-unimodal, unimodal and multimodal( e.g. see [16, 76, 46]).
The bifurcation diagram.
The logistic map has exactly one attractor for each parametervalue (e.g. see [16], Thm. 4.1, or [46]). Bifurcation plots for µ to the left of the so-called Myrberg-Feigenbaum or Feigenbaum parameter value µ F (cid:39) . µ =1, µ = 3, µ = 1 + √ (cid:39) . , . . . , whose speed increases exponentially until the Myrberg-Feigenbaum parameter value µ F (cid:39) . µ n such that for µ n < µ < µ n +1 , the attractor is a periodiccycle of 2 n distinct points. Feigenbaum’s fundamental discovery was that the speed of the bifurcationcascade lim n →∞ µ n − µ n − µ n +1 − µ n (cid:39) . Example of Cantor set.
First steps in the construction of the standard Cantor subsetof [0 , In [23] Guckenheimer proved that, for every value of µ in [0 , (cid:96) µ (or, to be precise,any S-unimodal map) has exactly one attractor and that this attractor must be precisely of thefollowing three kinds. First kind: a periodic orbit.Second kind: a finite union of intervals.
In this case, the attractor is a collection of intervals J , . . . , J n such that (cid:96) µ ( J i ) = J i +1 except that (cid:96) µ ( J n ) = J . Furthermore the map is chaotic. Mostoften, there is a single interval J and (cid:96) µ ( J ) = J . In particular, for most µ between µ Int and 4(see Fig. 4 for µ Int ), the attractor is an interval and the dynamics on it is chaotic. However, thereare windows, i.e. , intervals in parameter space, where the attractor is not an interval. Now, alsoin Fig. 4, each window has a bifurcation diagram that is tiny but extremely similar to the entirediagram (see Subsec. 3.2). Such windows occur not only after µ Int but more generally after µ F M . Third kind: a Cantor set attractor.
We call this attractor almost periodic [53], alsosometimes called “odometer”. This kind of attractor is a Cantor set and it is not chaotic. It occursprecisely when the graph has infinitely many nodes. Each node, other than the attractor, is eitheran unstable periodic orbit or a repelling chaotic Cantor set.For each point x in the attracting Cantor set and each ε >
0, there is a periodic point x ε suchthat the n th iterate of the map on x and the n th iterate of the map on x ε stay within ε of eachother for all time. At µ F M , every period orbit has period 2 n for some n and none of them belongsto the Cantor set. They converge to the Cantor set as n → ∞ .Recall that any Cantor subset C in an interval I is an uncountable set that contains no intervals.Also, it has no isolated points in the sense that each point in the set is a limit of other points in theset.In all three cases, “almost every” x ∈ (0 ,
1) belongs to the basin of attraction. By almost everywe mean that the points that are not attracted can be covered by a finite or countably infinitecollection of intervals with arbitrarily small total length.Notice that, in case of an attracting Cantor set, the basin has empty interior. Each openneighborhood of such an attractor contains infinitely many nodes of the graph.We can write the parameter space as (1 ,
4] = A P ∪ A Chaos ∪ A AP , where the union is disjoint, A P is the set of parameters for which the attractor is a periodic orbit, A Chaos is the set of parametersfor which the attractor is chaotic and A AP the set of those for which it is a Cantor set (see [35]).The set A P is open, which simply follows from the stability of attracting cycles under smallperturbations, and dense, which instead requires heavy machinery and was proved rigorously onlyin 1997, independently by Lyubich [44] and Graczyk and Swiatek [21]. Heuristically, the density ofthis set follows from the fact that it is to be expected that, for almost all chaotic parameters, the17rbit of the critical point c will be dense in the attractor and so, in particular, its orbit will getarbitrarily close to itself. This way, arbitrary small changes in µ should be able to make the orbit of c become periodic [59], and a periodic orbit containing the critical point is always super-attractingand lies within a regular window.Since A P is dense and it contains (1 , µ F M ), its complement A Chaos ∪ A AP is a Cantor subset of[ µ F M , A Chaos has positive measure while it was proved onlyin 2002 by Lyubich [45] that A AP has measure zero.Notice that all results above hold not just for the logistic map but also for every non-trivialreal analytical family and any generic smooth family of unimodal maps [15, 69]. However, it isknown that there are non-generic smooth families showing “robust chaos” [4, 2, 77], namely withoutwindows. The edges of the graph.
Our approach to the logistic map aims at finding the nodes, thechain-recurrent sets, and the edges of the graph.Most of the traditional literature on logistic maps look at the non-wandering sets. One non-wandering set can contain many nodes. Most of the ideas are similar but one must be careful.The non-wandering set of unimodal maps was first described by Jonker and Rand in 1980 [35] (seealso [75, 31, 7, 16, 6, 68]).Furthermore, no one seems to have examined the edges of the graph, therefore we do in [14]. Weproved the following.
For every µ ∈ (1 , To illustrate this idea, we now argue that the top node always consists of the point 0, except for µ = 4. For any point x > x n with n < n → −∞ . For µ ∈ (1 , p of (cid:96) µ , and the trajectory x n converges to p as n → + ∞ . That means there is a edge from 0,which is a node, to the attractor node. Of course, for µ = 1, the attractor is 0.For µ ∈ [3 , x > J for the interval [0 , x ]. When we apply themap (cid:96) µ to J , we obtain a longer interval and, as we repeatedly apply the map, we eventually obtainan interval that includes [0 , ]. Notice that is critical point of the logistic map. Hence with onemore iterate, the image interval is [0 , (cid:96) µ ( )] . For any point p in any node, there is a point ˆ x near 0such that there is a N for which (cid:96) Nµ (ˆ x ) = p . Since each node is an invariant set, the forward limitset of ˆ x is in the node. As discussed above, there is a backward trajectory from ˆ x that limits on 0as n → −∞ . Hence the graph has an edge going from 0 to the node containing p . That is true forevery p in every node. Hence, for each node other than 0, there is an edge from 0 to that node. For µ = 4 there is a single node, that is the whole interval [0 , Because of this fact, to simplify the pictures of graphs, in Figs. 5,6 we do not includethe “zero node”, the node that consists of , which is always on top of each tower. A period- k window (of parameter values). Figure 6 shows a “period-3 window”, an intervalof parameters in which there are three intervals J i , i = 1 , ,
3, in x space which are permuted by themap. Each of the intervals J i changes continuously, starting at the parameter value µ = 1 + √ (cid:39) . µ at which the attractor fills the interval, namely µ (cid:39) . − k orbit begins an analogous window or period − k window with a final µ at which the attractor fills the intervals J i . Between the first Feigenbaum parameter value and µ = 4, there are infinitely many windows andevery µ in that range is either in a window or arbitrarily close to one. In Fig. 5 several are visible.The biggest is the period-3 window, which is also shown in Fig. 6.For each parameter value inside a period- k window, there are k intervals J j mentioned aboveand there is a chain-recurrent set C of points whose forward trajectories do not fall into any of thethe J j . These sets are shown in red for the larger windows in Fig. 5. The set C is always a Cantorset. 18 indows within windows within windows . . . . In Fig. 6 we see the period-3 window.The bifurcation diagram of the attractor is plotted in black and gray. It consists of three piecesthat each look like the entire bifurcation diagram. Each piece lies within one of the J j for each µ . As such, there are windows within this bifurcation diagram, infinitely many windows within theprimary window. Each windows has secondary windows within it.For µ that has a period − k window, there are k intervals inside which the attractor lies. Wedenote them by J , . . . , J k . There is a chain-recurrent Cantor set C of points that do not fall intothem. If µ has a period − k window within the window, then k divides k and there are k intervals J (cid:48) , . . . , J (cid:48) k that are a small-scale version of the J j above.The Cantor set C for this window lies outside the union ∪ J (cid:48) j but inside the union ∪ J j ; see Fig. 6.Such a C Cantor set is shown in blue where the primary window has k = 3 and the secondarywindow has k = 3 × . The graph has a node 0 on top of C on top of C followed by possible more Cantor set fromfurther windows within windows and finally some attractor at the bottom. Tower building blocks. If µ does not belong to any window, then the only chain-recurrent setsare the left endpoint 0 and the attractor.Just after the start of a window, the attractor is a periodic orbit with some period k (e.g. see T2in Fig. 5). As µ increases, the periodic orbit goes through a bifurcation process identical to the oneat the left of µ F M (e.g. see T3–T4 in Fig. 5 and T2–T5 in Fig. 6). This reflects in the graph in thefollowing way. Each subwindow corresponds, in the graph, to a Cantor set node. Each Cantor nodemay be immediately followed by s nodes that are unstable periodic orbits. The periods of theseorbits are, in the following order, k , 2 k , 2 k , up to 2 s − k . After those nodes, there will be either theattractor or another repelling Cantor set node.Figures 5 and 6 show examples of towers. In summary, a graph can contain any number of Cantorset nodes, including none and infinitely many. Between two consecutive Cantor set nodes, there canbe any finite number of repelling periodic orbit nodes, including no such orbits. In particular, therecan be infinitely many nodes and any combination of Cantor sets and repelling periodic orbits ispossible. All possible finite or infinite patterns (including only saddles and including only Cantorsets) occur for appropriately chosen parameter values. In addition, (1) there is an attractor, thebottom-most node; (2) for the logistic, the top node is the repelling fixed point 0 for all µ < Here we report on examples of graphs that have been determined for differential delay equations andpartial differential equations. These examples do not exhibit chaos in the regimes where the graphshave been determined, but the reader should expect great complexity in other examples that havechaos.
Example 1: Delay-Differential Equations.
In 1986, J. Mallet-Paret [47] (see also [36, 37, 51,28, 27]) showed that the graph approach can be applied also to the infinite dimensional dynamicalsystem associated to first-order scalar delay-differential equations of the form˙ x ( t ) = f ( x ( t ) , x ( t − , (7)with the initial condition x ( t ) = ϕ ( t ) on [ − ,
0] for some continuous function ϕ . For instance, thecelebrated Wright’s equation ˙ x ( t ) = − α x ( t − − x ( t )) , α > , modeling population dynamics, is of this form, as well as equations of the form ˙ x ( t ) = − αx ( t ) − g ( x ( t − finite number of levels. Fromthe point of view of the dynamics outside of the nodes, therefore, these systems are simpler thansome logistic map. Below we describe in some detail the nodes of these graphs.19enote by x ϕ ( t ) the solution, defined up to some T >
0, of (7). This defines a dynamical system Φ t , for t ≥
0, on the space of continuous functions C ([ − , (cid:0) Φ t ϕ (cid:1) ( τ ) = x ϕ ( t + τ ) , − ≤ τ ≤ . We assume the following properties, satisfied in many applications:1. f is smooth;2. y f ( x, y ) > y (cid:54) = 0;3. f x (0 , > f x (0 ,
0) + f y (0 , > Φ of any bounded ball is bounded;6. sup ϕ ∈ C ([ − , ,t ∈ R (cid:107) Φ t ϕ (cid:107) < ∞ .Under these conditions, the solutions of Eq. (7) oscillate about zero, the map Φ is compact anddissipative [26] and the flow has a maximal compact attractor S [5] equal to the set of all initialconditions ϕ ∈ C ([ − , x ϕ is global and bounded.The decomposition found by Mallet-Paret is relative to the dynamics of the restriction of Φ to S . The unstable nodes of Φ in S are rapidly oscillating unstable periodic orbits. The more rapidlyoscillating nodes are above the more slowly oscillating nodes. Mallet-Paret has a precise definitionof rapidly oscillating, based on the number of zeros the trajectory has on every interval [ t, t + 1]. Hewas able to prove that there are orbits joining every node with all nodes that are oscillating moreslowly. Example 2: Parabolic partial differential equations (PDEs).
The setting of nonlinearparabolic PDEs proved to be an unexpectedly rich source of dynamical system graphs [28, 41].Let X be a closed segment and denote by H ( X ) the Sobolev space of square-summable func-tions on X with a weak derivative and whose first weak derivative is also square-summable. Theset H ( X ) ⊂ H ( X ) is closure, in H ( X ), of the set of smooth functions that are zero in someneighborhood of the end-points of X .We begin with the Chafee-Infante PDE on X = [0 , π ], namely u t = u xx + λ (1 − u ) u,u ( t,
0) = u ( t, π ) = 0 for all t ≥ ,u (0 , x ) = u ( x ) ∈ H ( X ) , (8)with λ ≥
0. We denote by Φ tλ : H ( X ) → H ( X ) the semiflow of the Chafee-Infante PDE. Then themap Φ λ is a C (infinite-dimensional) Morse-Smale map for every λ which is not the square of aninteger [9, 28]. In particular, each node of its graph is a fixed point, and the dynamics elsewhere isgradient-like. One of the key observations, by Henry [29], leading to the construction of a Lyapunovfunction for the system, is that the number of components of the set { x ∈ X : u ( t, x ) (cid:54) = 0 } is amonotonously decreasing function of time.The structure of the graph in this case is more interesting and shows (see Fig. 3 (right)) thefollowing bistable behavior: for ( n − < λ < n , Φ tλ has 2 n − N , N ± j ,2 ≤ j ≤ n − N ± n . The node N has edges to N ± . All N ± j , j < n , haveedges towards both N ± j +1 , as shown in the figure. Note that, in this case, no other edges arise dueto a blocking connections principle that holds for these systems (see [18] for a thorough discussionand examples). In particular, in this case the graph is not a tower.These results do not depend strictly on the analytical form of (1 − u ) u but rather hold forall C functions f ( u ) with a similar shape (see [9, 28, 41]) and hold for several other importantPDEs such as FitzHugh-Nagumo equation and the Cahn-Hilliard equation (see [41]). They werealso further generalized by Chen and Polacik [11] to the time-periodic non-autonomous version ofthe Chafee-Infante equation. 20iedler and Rocha [18], finally, further generalized the PDEs above to the autonomous semilinearvariation u t = a ( x ) u xx + f ( x, u, u x )with Neumann boundary conditions u x ( t,
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