University of Birmingham The chaotic behavior on the unit circle

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Dynamical System
Dynamical systems are systems that change over time and they are very useful in modeling many different kinds of phenomena. Examples of dynamical systems include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish in a lake. The time may be continuous ( = [0, ∞)) or discrete ( = ℕ). When the system depends on a continuous time, we call it a continuous dynamical system, otherwise it is a discrete dynamical system. A continuous dynamical is represented by ordinary differential equations (̇= ( ) for one dimensional case) while a discrete dynamical system consists of a space and a function that acting on the space into itself. In this paper, we will consider the discrete dynamical system and denote it as ( , ).
The dynamical study of ( , ) is all about the attempt to understand the behavior of evolution of every point ∈ under the iteration of function . Therefore, it describes how one state develops into another state ( ) and so forth. Chaotic dynamics is the behavior of dynamical systems that are highly sensitive to small changes in initial conditions, so that small alterations can give rise to strikingly great consequences. Therefore, when the system is chaotic, the iteration of under is not approximated without perfect information of .

Chaotic Dynamical Systems
Chaotic behavior is one of the interesting topics in the study of dynamical systems. Chaos theory is very interesting because surprisingly chaos can be found within almost trivial system. The tent map is an example of a system with simple equation but has a very complex behavior [6]. Conversely, a complex system can also exhibit a non-chaotic system. For an example there is complex biological system, which has been described as "anti-chaotic" [15]. Therefore, it is very important to define chaotic dynamical system clearly and various attempts have been made to give the notion of chaos a mathematically precise meaning, but chaos is not easy to define and there is no universally agreed definition of chaos. Li and Yorke [11] have firstly defined chaos in mathematical terms and the chaos in their sense is called Li-Yorke chaos. However, there are a number of different definitions of chaos and an incomplete list of other popular definitions includes distributional chaos [13], topological chaos [16], -chaos [10], P-chaos [2], Block and Coppel chaos [3] and many more. One of the most frequently used is Devaney chaos [6], which isolates two essential ingredients of a chaotic function.

Definition 1[6]:
A dynamical system ( , ) is Devaney chaotic whenever it satisfies two conditions; i) Topologically transitive, that is for any two open nonempty sets and , there is some ∈ ℕ such that ( ) ∩ is nonempty; ii) Has a dense set of periodic points, i.e., that every open set contains a periodic point.
Another well-known definition of chaos is topologically chaos i.e.
Definition 2 [6]: A dynamical system ( , ) is topologically chaotic whenever the system has positive entropy.
Entropy is a tool to measure dynamical behavior of a system. There are some equivalent versions of its definition. Some examples and the original definitions can be found in [1,18]. It has been shown that any topologically chaotic dynamical system is Li-Yorke chaotic [4] and any Devaney chaotic dynamical system is also Li-Yorke chaotic [9]. The following are well-known results on the unit interval, and the unit circle, 1 about positive entropy and topological chaos. However, there are some common remarkable results on the interval and on the circle, as follows; Theorem 3 [4,12]: If is a continuous map on the unit circle, then the following statements are equivalent: 1. the entropy of is positive, 2. there exists a closed invariant subinterval ⊆ 1 such that |_ is Devaney chaotic, 3. has a periodic point of period 2 for an odd and integer . In addition, if is a function on the interval, the statements 1,2 and 3 are also equivalent.
The next remarkable result only holds on the unit interval.
The surprising equivalence on the interval is because transitivity implies dense periodic points, but the converse is not necessarily true [7]. The result cannot be generalized to higher dimensions or the unit circle because the proof of this result uses the ordering in ℝ in an essential way. Furthermore, an irrational rotation on a circle is a transitive map but does not have any periodic points, which is a counterexample.

Some Other Chaotic Characterizations
In this paper, we also consider some other strong chaotic concepts that relate closely to two main ingredients of Devaney chaos i.e. locally everywhere onto and a strong dense periodicity property, which are defined as follows; Definition 5 [8]: Let : → be a continuous map on a compact metric space . The function is said to be locally everywhere onto or simply l.e.o or exact, if for every open subset ⊂ there exists a positive integer such that ( ) = . It is obvious from the definition that l.e.o implies transitivity. Since transitivity is equal to Devaney chaos on the interval, then we directly have the following Theorem 6: On the interval, locally everywhere onto implies Devaney chaos.
On the unit circle, transitivity is not equal to Devaney chaos, so the above implication is not straightforward. We are going to clarify this in the next section.
Definition 7 [7]: Let : → be a continuous map on a compact metric space . The function is said to poses a strong dense periodicity property whenever the set is dense for all integer and is a collection of periodic points of prime period larger than .
As far as we are concerned, the property of strong dense periodic points were firstly introduced in 2015 [7] with following result on the interval. Theorem 8 [7]: If is Devaney chaotic on a compact metric space with no isolated points, then the set of periodic points with prime period at least is dense for each .
Conversely, if is a continuous function from a closed interval to itself, for which the set of points with prime period at least is dense for each , then there is a decomposition of the interval into closed subintervals on which either or 2 is Devaney Chaotic.

Corollary 9:
If is a continuous function from a closed interval to itself, for which the set of points with prime period at least is dense for each , then there is a closed subinterval on which is Devaney Chaotic.
This notion was firstly introduced in 2015 [7] and as far as we know, there is no other stronger dense periodicity property introduced to describe chaos. The identity map has dense periodic points but is not transitive. Having only fixed points become an obstacle for the system to be chaotic. Motivated by this example, it is important to highlight the stronger dense periodicity property, dense for all . They show that for a system without any isolated point, this property is equivalent to Devaney chaos. On the interval they shown that if is a continuous function from a closed interval to itself, has dense for all , then there is a decomposition of the interval into closed subintervals on which either or 2 is Devaney Chaotic.

Representation of the Circle Maps
Analogous to the interval map, we then call a function : 1 → 1 a circle map. When dealing with a circle map, we can think the unit circle 1 as the interval [0, 2 ) with 0 and 2 identified. Therefore a continuous map on 1 will be associated with an interval map on [0, 2 ) which satisfies some properties. 2 ) where its graph is given in Figure 1. Proof. Since ( 1 , 2 ) = ( 1 , 2 ) and 0 ∉ ( 1 2 ), then for every ∈ ( 1 , 2 ), ( ) ≠ 0. Therefore is continuous at . ▄ Finally we will give some remarks on the relationship between a circle map and its corresponding interval map .

Remark 12:
The following are facts about and for a continuous circle map ; 1. If has dense periodic points then so does . This representation of the circle map is useful for our further discussion on the dynamics of the circle maps later.

The Chaotic Behavior of Circle Maps
Even though transitivity implies Devaney chaos on the interval, the same is not true on the circle. However Silverman [14] proved that transitivity almost implies Devaney chaos, as follows: Theorem 13 [14]: Let : 1 → 1 be a transitive continuous circle map. If is not one-to-one, then is Devaney chaotic.
Using this result, we will show the significant implication of locally everywhere onto on the circle map. ( ) = ( ) = 1 . Therefore there exists 1 ∈ and 2 ∈ such that ( 2 ) = ( 1 ). Hence is not one-to-one and by Theorem 13, is Devaney chaotic. ▄ Nevertheless, dense periodic points in the definition of Devaney chaos may be weaken to the existence of two periodic points.
Theorem 15: Suppose that : 1 → 1 is a continuous circle map with periodic points of periods < . If is transitive, then is Devaney chaotic.

Conclusion
It is interesting to highlight the implication of another chaos characterization, locally everywhere onto which is stronger than transitivity. Locally everywhere onto implies Devaney chaos on two spaces, the interval and the unit circle. This is a surprise result since transitivity is equal to Devaney chaos on the interval but not on the unit circle. We end by highlighting the difference between the implication of dense for all on the interval and on the unit circle. On the interval, dense for all implies that the whole system can be decomposed into subsystems where every subsystem is Devaney chaotic. Therefore, the system with no invariant proper subinterval is Devaney chaotic whenever it satisfies this strong dense periodicity property. In fact, this strong property implies that the system has positive entropy. That also happens on the unit circle.
dense for all can guarantee that the system on the unit circle has positive entropy i.e. is behaving topologically chaotic. Unlike what happens on the interval, this stronger density property cannot guarantee that the whole system is chaotic in the sense of Devaney even if it does not have any invariant proper subset. This is because the only invariant subinterval under is the whole interval [0, 2 ) which contains 0 and therefore discontinuity of at some points is possibly occurs. Hence its prevent us to use the same argument to show that the whole system is Devaney chaotic. However, it is an interesting fact that this strong property can guarantee that a Devaney chaotic subsystem exists which means the stronger property is more significant than the property of dense periodic points since dense periodic points does not implies any sort of chaos.