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The behavior of systems such as periodicity, fixed points, and most importantly chaos has evolved as an integral part of mathematics, especially in dynamical system. This research presents a study on chaos as a property of nonlinear science. Systems with at least two of the following properties are considered to be chaotic in a certain sense: bifurcation and period doubling, period three, transitivity and dense orbit, sensitive dependence to initial conditions, and expansivity. These are termed as the routes to chaos.

Dynamical systems are part of life. Quite often it has been studied as an abstract concept in mathematics. Chaos is one of the few concepts in mathematics which cannot usually be defined in a word or statement. Most dynamical systems are considered chaotic depending on the either the topological or metric properties of the system. Yorke et al., 1976, concluded that period three implies chaos. They discussed how a dynamical system with period three orbits gives an assurance that the system is chaotic. Several approaches and conditions are factored before the construction of any definition of chaos [

The deterministic nature of these systems does not make them predictable and this is due to chaos theory. Properties of an unknown map can be associated with that of the known via topological conjugacy; hence properties of unknown maps can always be studied in terms of the unknown. The tent map and logistic maps are two known chaotic maps. At a particular point in time, a certain type of chaos may imply or could be equivalent to another type of chaos depending on the routes of chaos the system exhibits.

The size of the scrambled set of systems is one of the most usually considered conditions in terms of defining chaos in systems [

Given dynamical system

If

The orbit of a point

The individual elements of the set

Assume

Under the circumstance the orbit of

Let

Let

This idea of sensitivity dependence is otherwise called

Numerically, sensitivity is measured by Lyapunov exponent such that a positive value implies the system is really sensitive to initial conditions.

This implies that Lyapunov exponents measure the rate of divergence of orbits away from each other.

Let

The Lyapunov exponent

Given

Likewise on the interval between 1/2 and 1,

Now

Topological entropy defines the rate of exponential growth of the number of distinct orbits (periodic orbits as

Entropy is one of the most important quantities in dynamical systems so far as numerical values are concerned. It basically measures the rate of complexity of the dynamical system as time varies largely and towards infinity. For autonomous systems in compact metric space, topological entropy is commutative for composite functions such that

Topological entropy can be defined and determined in terms of fixed points of

Li Yorke chaos can be implied from entropy if the system is autonomous and in a compact metric space [

A dynamical system

A dynamical system

Transitivity usually implies the existence of dense orbit, if

Topological transitivity guarantees that there always exists a point which results from the intersection of open sets under iterative process of map.

A point

Let

There exists an obvious and more direct link between expansivity and sensitivity dependence on initial conditions. Every expansive map exhibits sensitivity dependence to initial conditions. The converse does not hold and the two conditions are never equivalent.

Expansivity implies sensitive dependence because expansivity deals with the distance between two nearby points and how their orbits separate continuously. In expansivity, the separation is observed between two nearby points by at least the constant

Let

The map

For a given map, since

Let

Since

Given

Let

Let

Bifurcation is defined as the structural changes of a dynamical system as a result of the alteration or changes in the parameter value. This change is usually sudden and could be topological or qualitative. In such cases, we expect the dynamical system to be a function of both the dependent variable and the parameter in context. An example is

The idea of bifurcation can be grouped into two, which are global bifurcation and local bifurcation. In local bifurcation, the interest is the change that happens in the system near the fixed point. It is usually analyzed through changes in stability properties and periodic orbits. Global bifurcation occurs when larger invariant sets of the system collide with each other.

Period doubling is the splitting of a trajectory into two during iteration. This unusual scene is influenced by the parameter value most of the time. Chaos is observed as the period doubling increases. This is because the path of trajectory at certain points is mixed up and inseparable making the system chaotic.

Let

Let

the periodic orbit of

Let

Let

Let

Let

The dynamical system is positively expansive chaotic (E-chaotic) iff

Let

Let

We verify that

Since the mappings in all two examples are conjugated topologically, they share the same topological properties. Hence if any of the mappings is identified with any topological property, we can as well associate the other mapping with the same property and vice versa.

At this point we take a study through the equivalent relations in various types of chaos and the implications as well.

Devaney chaos implies Wiggins chaos, Knudsen chaos, and Lyapunov chaos. Sensitivity to initial conditions is a common route in Devaney, Wiggin, and Knudsen chaos while positive Lyapunov exponents which are equivalent to sensitive dependence are used in Lyapunov chaos. Now, Devaney chaos combines both topological transitivity and the existence of dense orbits. Since Wiggins, Lyapunov, and Knudsen chaos depend on either of the two topological conditions, then Devaney chaos implies them all. A system that is Devaney chaotic has to be Wiggins chaotic, Lyapunov chaotic, and Knudsen chaotic.

Positive expansive implies Wiggins chaos, Knudsen chaos, and Lyapunov chaos. Wiggins definition, Knudsen chaos, and Lyapunov’s definitions satisfies either the condition of transitivity or dense orbits.

In relation to Wiggins chaos, their distinct conditions are positive expansiveness and sensitive dependence to initial conditions. Expansivity implies sensitive dependence; hence positive expansive chaos implies Wiggins chaos.

In the case of Knudsen chaos, beyond the existence of dense orbit, the distinct condition is also sensitivity. Similarly, since expansivity implies sensitive dependence, expansive chaos implies Knudsen chaos.

Also for positive Lyapunov exponents, the distinct condition is expansivity and positive Lyapunov exponents. Every expansive map has a positive Lyapunov exponent but the converse is not true. As said earlier, if just two orbits separate apart at a point in the iteration, the Lyapunov will be positive but the map might not necessarily be expansive.

Wiggins chaos and Knudsen chaos imply each other and are quite similar or equivalent. They share a common property of transitivity. They also share an equivalent property of sensitivity and positive Lyapunov exponents. The difference in the two definitions is the space on which it is defined. Wiggins considers a continuous map on a metric whiles Lyapunov deals with differentiable maps.

Knudsen chaos shares an equivalent property of sensitivity dependence to initial condition and positive Lyapunov exponents with Wiggins definition and Lyapunov’s definition. In some systems, transitivity is equivalent to dense orbits though not always. In such systems, all three types of chaos are the same apart from the space on which each is defined.

Devaney chaos and Li York chaos are interrelated via topological entropy. Devaney chaos implies positive topological entropy and the converse is not true. Positive topological entropy implies Li Yorke chaos and here too the converse does not hold. According to the law of transitivity in analysis Deveney chaos implies Li Yorke chaos. On the interval map Deveney chaos is the strongest whereas Li Yorke’s chaos is the weakest.

From above, Devaney chaos implies Wiggins chaos, Lyapunov chaos, and Knudsen chaos. Positive expansive chaos implies Wiggins chaos, Lyapunov chaos, and Knudsen chaos. Possibly then, there should be a link between the two. The two definitions share two common properties of topological transitivity and dense periodic orbit. Positive expansivity is a stronger property than sensitive dependence to initial conditions. Since positive expansivity implies sensitive dependence, the positive expansive chaos implies Deveney chaos.

There should always be a way to calculate chaos or numerically determine chaos. Entropy usually is quite difficult to obtain numerically compared to Lyapunov exponent. Since every expansive map has positive Lyapunov exponent, then by confirming the expansiveness of a map, chaos is possible if not guaranteed.

Lastly, that gives a more stronger form of chaos in that the existence of sensitive dependence alone in itself would not and cannot guarantee chaos in dynamical systems. Hence other routes become relevant in the definition of the types of chaos.

Chaos as a nonlinear science has become part of our daily lives. There is an implication or equivalence relation between the kinds of chaos based on the particular routes to chaos. Topological conjugacy is one of the means of associating topological properties of a map with known properties and a map with unknown topological properties. Expansive chaotic system implies Lyapunov chaos. Devaney chaos implies Wiggins chaos, Knudsen chaos, and Lyapunov chaos. Wiggins chaos and Knudsen chaos imply each other and are quite similar or equivalent. Devaney chaos and Li York chaos are interrelated via topological entropy.

The authors declare that there are no conflicts of interest regarding the publication of this paper.