Communications on Applied Nonlinear Analysis 

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Periodic Behaviour of General Systems 

 

L. Praveen Kumar1, Vajha Srinivasa Kumar2 
1Research Scholar, Department of Mathematics, JNTUH College of Engineering, JNTU, Kukatpalle, Hyderabad - 

500085 Telangana State, India.  

2Sr. Assistant Professor, Department of Mathematics, JNTUH College of Engineering, Hyderabad, JNTU, 

Kukatpalle, Hyderabad - 500085, Telangana State, India.  

1Email: praveenkumarl55@gmail.com 

2Email: vajhasrinu@gmail.com 

 

Article History: 

Received: 15-02-2024 

Revised: 09-04-2024 

Accepted: 24-04-2024 

 

Abstract: 

A continuous function on the product of compact metric spaces to itself and back 

to the same space is known as a genera system. Where each element's orbit is an 

infinite sequence and where the first two elements are the same as given and next 

to it depends on the two elements prior to it form stronger conditions for the orbit. 

We could define an m-step dynamical system by extending the definition of a 

compact metric space to its m-times product. Because the system's current state 

frequently depends directly on the conditions of previous terms, this system 

appears more realistic. The basic theorems regarding periodic points and their 

related points, such as fixed point, limit points, recurrent points, will be proved in 

this paper. We also define the topological transitivity and its properties. In the end 

we find periodic points with periods one and two for affine maps and periodicity of 

tent map in dynamical system and generalised dynamical system. 

Keywords: generalized dynamical systems, periods and periodic points. 

 
 

1. Introduction  

Topological dynamics is an intriguing field of mathematics. In that periodicity is interesting as it is 

related to real life situations like the path of orbit of planets etc. A number of mathematicians have 

explained the periodic behaviour on various dynamical systems, such as linear operators[1] and 

operators on Hilbert spaces[2] and periods and periodic points on linear cellular automata in[3]. The 

generalized systems that define X×X →X unlike X→X as described in dynamical systems. This new 

concept was defined[4] in the year 2008. In this instance, the author views X as a complete metric 

space. Dumitru[5] defined the topological version of generalized iterated functions based on this 

novel idea. By creating new concepts that work for generalized systems[6] explored chaos and 

shadowing properties in generalized dynamical systems in 2023[7] defined the Generalized function 

systems on metric spaces. The basic ideas of periodic points, such as fixed points, periodic points 

that repeat, non-wandering points, and the relationships between them, will be covered in this work. 

The powerful character of a point and the fact that it depends on the two or m elements (as in m step 

dynamical system) before it in that element's orbit present the biggest challenge in solving this 

system. Most of the statements that are true for dynamical systems are not valid in the present system 

mailto:vajhasrinu@gmail.com


Communications on Applied Nonlinear Analysis 

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like the fundamental elements in a periodic orbit are periodic in dynamical systems but not true in 

our case. 

2. Preliminaries 

In this paper we refer (𝑋, 𝑑) as compact metric space throughout. The continuous map 𝑓: 𝑋 × 𝑋 → 𝑋 

is Generalised system. In this paper we analyse the preliminary data and extending domain to, 

𝑓: 𝑋 × 𝑋 × … . 𝑋(𝑚 𝑡𝑖𝑚𝑒𝑠) → 𝑋 𝑜𝑟 𝑓: 𝑋𝑚 ⟶ 𝑋     (2.1) 

Defining as 𝑚 step generalized system considering the same preliminaries. This assumption makes 

us calling the regular dynamical system as 1-step dynamical system and generalized system as 2-step 

dynamical system. 

The orbit of any 𝑥 ∈ 𝑋 is the infinite sequence 𝑂(𝑥) = {𝑥𝑛}0
∞ where 𝑓: 𝑋𝑚 ⟶ 𝑋 is a continuous 

map and, 

 𝑥0 = 𝑥1 = ⋯ = 𝑥𝑚 = 𝑥 and 𝑓(𝑥𝑛−𝑚, 𝑥𝑛−𝑚+1, … . , 𝑥𝑛−1) = 𝑥𝑛, 𝑛 ≥ 𝑚 (2.2) 

For 𝑚 = 2 it becomes 𝑥0 = 𝑥1 = 𝑥 and 𝑥𝑛 = 𝑓(𝑥𝑛−1, 𝑥𝑛−2), 𝑛 ≥ 2 

Here 𝑥𝑛 denotes 𝑛𝑡ℎ term in the orbit 𝑂(𝑥) = {𝑥𝑛}0
∞. 

For the convenience we consider 𝑚 = 2 and define the preliminaries.  

We say that 𝑥 ∈ 𝑋 is fixed if 𝑓(𝑥, 𝑥) = 𝑥.     (2.3) 

We say that 𝑥 ∈ 𝑋 is periodic of periodic of period 𝑛 if 𝑥𝑘𝑛+𝑖 = 𝑥𝑖 for all 𝑘 ∈ ℕ and 0 ≤ 𝑖 < 𝑛. We 

may call periodic point of period one is fixed point.               (2.4) 

We say that 𝑥 ∈ 𝑋 is strongly periodic if every element of the orbit is periodic. Unlike in                1-

step dynamical systems in general systems every element in periodic orbit need not be periodic. 

         (2.5) 

The recurrent point is 𝑥 ∈ 𝑋 and 𝑥 ∈ 𝜔(𝑥, 𝑓) that is for each 𝜖 > 0 there is 𝑛 ∈ ℕ such that 

𝑑(𝑥𝑛, 𝑥) < 𝜖.          (2.6) 

The non-wandering is 𝑥 ∈ 𝑋 and for each > 0, 𝛿 > 0 there is 𝑛 ∈ ℕ, 𝑧 ∈ 𝐵𝛿(𝑥) implies 𝑑(𝑧𝑛, 𝑥) <

𝜖.           (2.7) 

We Also define strong non wandering if 𝑥 is non-wandering and every element of the orbit is also 

non wandering.         (2.8) 

A subset 𝐵 ⊆ 𝑋 is said to be invariant if {𝑥𝑛}0
∞ ⊆ 𝐵 for every 𝑥 ∈ 𝐵.   (2.9) 

We denote the following notions for generalized system. 

𝑓𝑖𝑥(𝑓) = Fixed points in the space 𝑋. 

𝑃𝑒𝑟(𝑓)=Periodic points in 𝑋. 

ℛ(𝑓) = Recurrent points in 𝑋. 

Ω(𝑓) =Non-Wandering points in 𝑋.  

 

 



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3. Results 

Since the results do not hold true for generalized systems, we must demonstrate all of the results that 

hold true for 1-step dynamical systems. 

Theorem 3.1. For any generalized system defined in Eq.(1)  

𝑓𝑖𝑥(𝑓) ⊆ 𝑃𝑒𝑟(𝑓) ⊆ ℛ(𝑓) ⊆ Ω(𝑓) 

Proof:  

1. The first two proofs are trivial. For any 𝑥 ∈ 𝑓𝑖𝑥(𝑓) it is clear that fixed point are of period one. 

2. For any 𝑥 ∈ 𝑃𝑒𝑟(𝑓) then for some 𝑛, 𝑥𝑛 = 𝑥. 

So for each 𝜖 > 0 for we get 𝑑(𝑥𝑛, 𝑥) <  𝜖 that implies 𝑥 ∈  ℛ(𝑓). 

3. For any 𝑥 ∈  ℛ(𝑓) then from Eq. (1.6), for any 𝜖 > 0 there is 𝑛 ∈ ℕ such that 𝑑(𝑥𝑛, 𝑥) < 𝜖. Let 

𝑧 ∈ 𝐵𝛿(𝑥) for 𝛿 > 0.  

Then 𝑑(𝑥, 𝑧) < 𝛿 implies 𝑑(𝑥𝑛, 𝑧𝑛) < 𝜖 as the function as the function 𝑓 defined on the compact 

metric space. 

So, 𝑑(𝑧𝑛, 𝑥) ≤ 𝑑(𝑧𝑛, 𝑥𝑛) + 𝑑(𝑥𝑛, 𝑥) < 𝜖 + 𝜖 = 2𝜖.  

Then 𝑥 ∈  Ω(𝑓). 

Lemma 3.1. If an orbit of an element converges then its converges to fixed point. 

Proof: Let 𝑥 ∈ 𝑋 be any point as orbit of 𝑥 and 𝑂(𝑥) = {𝑥𝑛}0
∞ converges to for some 𝑝. 

That is for any 𝜖 > 0 there is 𝑚 ∈ ℕ such that 𝑑(𝑥𝑛, 𝑝) < 𝜖, 𝑛 ≥ 𝑚. 

By Eq (1.1), we can write 𝑑(𝑓(𝑥𝑛−2, 𝑥𝑛−1), 𝑝) < 𝜖 

⟹ 𝑑(𝑓(𝑝, 𝑝), 𝑝) < 𝜖 

We can say 𝑓(𝑝, 𝑝) = 𝑝. That is 𝑝 is a fixed point. 

Theorem 3.2. 𝑓𝑖𝑥(𝑓) closed in 𝑋. 

Proof: Let 𝑝 ∈ 𝑓𝑖𝑥(𝑓)̅̅ ̅̅ ̅̅ ̅̅ ̅ is the closure of the set. There exist an orbit 𝑂(𝑥) = {𝑥𝑛}0
∞ converges to 𝑝.  

𝑝 is a fixed point By the lemma 3.1. Which means set of all fixed point is a closed set as every limit 

point 𝑝 is in that set. 

Theorem 3.3. The set of Strong non-wandering points is closed set in 𝑋. 

Proof: As we defined 𝑆 is the set of Strong non-wandering points in X. 

Let 𝑝 is a limit point of 𝑆. 

Then There exist an orbit 𝑂(𝑥) = {𝑥𝑛}0
∞ converges to 𝑝. 

So for every 𝛿 > 0 we get 𝐵𝛿(𝑝) which contains large number of non wandering points 𝑥𝑛. For any 

𝑦 ∈ 𝐵𝛿(𝑝) which is non-wandering then there exist 𝑧 ∈ 𝐵𝛿(𝑝) and for some natural number 

𝑛,𝑑(𝑧𝑛, 𝑦) < 𝛿. This is true for every ∈ 𝐵𝛿(𝑝).  

So 𝑝 is a strong non-wandering point. 



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The strong version of the theorem is taken into consideration because every element of the non-

wandering orbit is non-wandering. 

Lemma 3.2. Set of fixed points, strong periodic points and strong non-wanderings points is invariant 

in 𝑋. 

Proof: The proof is direct application of the definition of invariant set and consideration strong form. 

The Lemma 3.2 is true for set of periodic points and non-wandering points in m-step dynamical 

system. 

4. Transitivity 

Function 𝑓: 𝑋 × 𝑋 → 𝑋 is topological transitive if for every pair of non empty open sets 𝑈, 𝑉 ⊆ 𝑋 

then for some 𝑥 ∈ 𝑈 then for some ∈ ℕ, 𝑥𝑛 ∈ 𝑉.  

𝑥 ∈ 𝑋 is a transitive point if it has a dense orbit. 

Every element of a dense orbit is also a dense orbit in a strong dense orbit. 

A non empty, closed, invariant subset 𝑌 of 𝑋 is minimal set if it is not contained in any proper 

invariant closed set, Which is equivalent to orbit of every element of 𝑌 is dense. 

Theorem 4.1. 

The Following are equivalent. (These also hold true for 1-step dynamical systems, but additional 

proof is required in m-step dynamical systems case.) 

1. Topological transitivity. 

2. The set 𝑉∗ = {𝑥 ∈ 𝑋: 𝑥𝑛 ∈ 𝑉𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑛 ∈ ℕ} is dense for every non empty open subset 𝑉 of 

𝑋. 

3. If 𝐾 is any invariant subset of 𝑋 then either 𝐾 is dense or 𝐾 is nowhere dense. 

Proof: 1⇒2  

Assume 𝑓 is a topological transitive. Then for every non empty pair of open subsets 𝑊, 𝑉 ⊆ 𝑋, there 

exist 𝑥 ∈ 𝑊 and 𝑥𝑛  ∈ 𝑉 for some 𝑛. So𝑉∗(≠ ∅) ⊆ 𝑋. And it is true for every pair of 𝑊 and fixed 

𝑉. Then for every 𝑥 ∈ 𝑋 we get a non empty open set which intersects the given 𝑉∗. We can 

conclude 𝑉∗ is dense in 𝑋. 

2⇒ 3  

Assume 𝐾 be non empty invariant subset of 𝑋.We have 𝐾 is invariant so compliment of 𝐾 in 𝑋 

doesn’t contain any element in the orbit of element of 𝐾.  

Define (𝐾𝑐)∗ = {𝑥 ∈ 𝑋: 𝑥𝑛 ∈ 𝐾𝑐 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑛} is 𝐾𝑐 only.  

If 𝐾𝑐 has an interior point then from consequence of (2) it is dense and 𝐾 has empty interior. In other 

case 𝐾𝑐 has empty interior. In both cases 𝐾 or 𝐾𝑐 has empty interior. Which means 𝐾 or 𝐾𝑐 is dense. 

Which is same as either 𝐾 is dense or 𝐾𝑐 is dense. 

Theorem 4.2: 𝑓 is topological transitive then set of transitive points are residual. 



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Proof: 𝑥 is a transitive point then 𝑂(𝑥) is dense in 𝑋. Let {𝑈𝑖} be the arbitary collection of open sets 

whose union is 𝑋.  

Define 𝐷(𝑈𝑖) = {𝑥 ∈  𝑋: 𝑂(𝑥) ∩ 𝑈𝑖 ≠ ∅} which is non empty as 𝑓 is topological transitive and is 

open subset of 𝑋 as 𝑓 is continuous [6]. 𝐷(𝑈𝑖) is open dense set for each 𝑖.  

As 𝑋 is Compact so complete and we have ∩ 𝑈𝑖 ≠ ∅ for each 𝑖 and 𝑥 ∈ ( ∩ 𝑈𝑖) ≠ ∅. Which means 

𝑥 is the residual point. 

Theorem 4.3: (𝑋, 𝑑) is a compact metric space and 𝑓: 𝑋𝑚 ⟶ 𝑋 is continuous. If 𝑂(𝑥)̅̅ ̅̅ ̅̅  is minimal 

for some 𝑥 ∈ 𝑋 then for any non empty open subset 𝑈 of 𝑋 the cardinality of the set 𝑍 =

{𝑛: (𝑥𝑛)𝑝  ∈ 𝑈 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑛 ∈ ℕ, 𝑝 ∈  ℕ} is finite. 

Proof: Assume 𝑌 = 𝑂(𝑥)̅̅ ̅̅ ̅̅  is minimal. Assume 𝑈 be open subset of 𝑌. 

Given 𝑌 is minimal then orbit of every element is dense in 𝑌. So for every 𝑦 ∈ 𝑌 there is some 𝑛 

such that 𝑦𝑛 ∈ 𝑈. We Define 𝑂−𝑛(𝑈) = {𝑦 ∈ 𝑋: 𝑦𝑛 ∈ 𝑈 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑛}. Which is open as 𝑓 is 

continuous.  

So ℳ = {𝑂−𝑛(𝑈): 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑛} is open cover of 𝑋. Being 𝑋 is compact we have a finite subcover 

for ℳ.Then the set {𝑈, 𝑂−1(𝑈), … 𝑂−(𝑝−1)(𝑈)} is a finite sub cover for ℳ. Then for every 𝑛 ∈  ℕ 

there exist 𝑘 < 𝑝 and 𝑥𝑛 ∈ 𝑂−𝑘(𝑈).  

Which is equivalent to (𝑥𝑛)𝑘 ∈ 𝑈, 0 ≤ 𝑘 < 𝑝 

That is the set 𝑍 = {𝑛: (𝑥𝑛)𝑝  ∈ 𝑈 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑛 ∈ ℕ, 𝑝 ∈  ℕ} is finite. 

5. Periods and Periodic Points  

In this section we find period and periodic points for the general system in the form of, 

𝑓: ℝ × ℝ → ℝ 

Defined as, 

𝑓(𝑥, 𝑦) = 𝑠𝑥 + 𝑡𝑦 + 𝑐 (Eq 5.1) 

Where 𝑠, 𝑡 𝑎𝑛𝑑 𝑐 are constants. 

Case I (If 𝒄 ≠ 𝟎) 

Orbit of any point 𝑂(𝑥) = {𝑥0 = 𝑥, 𝑥1 = 𝑥, 𝑥2 = 𝑓(𝑥, 𝑥), … }  

We examine fixed points for this map by 𝑥2 = 𝑥 ⇒  𝑓(𝑥, 𝑥) = 𝑥 

⟹ 𝑠𝑥 + 𝑡𝑥 + 𝑐 = 𝑥 

⇒ (𝑠 + 𝑡)𝑥 − 𝑥 = −𝑐 

⇒ 𝑥(𝑠 + 𝑡 − 1) = −𝑐 

⇒ 𝑥 =
𝑐

1−𝑠−𝑡
     (Eq 5.2) 

If any map defined as in (Eq 5.1) then 𝑥 =
𝑐

1−𝑠−𝑡
, 1 − 𝑠 − 𝑡 ≠ 0 is a fixed point. 

Now we find the periodic points of period 2. 

𝑂(𝑥) = {𝑥0 = 𝑥, 𝑥1 = 𝑥, 𝑥2 = 𝑠𝑥 + 𝑡𝑥 + 𝑐, 𝑥3 = 𝑠𝑥1 + 𝑡𝑥2 + 𝑐, … } 



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For period 2 points 𝑥3 = 𝑥, 

⇒ 𝑠𝑥 + 𝑡(𝑠𝑥 + 𝑡𝑥 + 𝑐) + 𝑐 = 𝑥 

⇒ 𝑥(𝑠 + 𝑠𝑡 + 𝑡2 − 1) = −𝑐(1 + 𝑡) 

⇒ 𝑥(𝑠(1 + 𝑡) + (1 + 𝑡)(𝑡 − 1)) = −𝑐(1 + 𝑡) 

⇒ 𝑖𝑓 (1 + 𝑡) ≠ 0, 𝑥(𝑠 + 𝑡 − 1) = −𝑐 

⇒ 𝑥 =
𝑐

1 − 𝑠 − 𝑡
 

Which is same as fixed point so for (1 + 𝑡) ≠ 0, then, 

The map in (Eq 5.1) doesn’t have any periodic point with period 2. 

If 𝑡 + 1 = 0 ⇒ 𝑡 = −1 then the orbit will be, 

𝑂(𝑥) = {𝑥0 = 𝑥, 𝑥1 = 𝑥, 𝑥2 = 𝑠𝑥 − 𝑥 + 𝑐, 𝑥3 = 𝑠𝑥1 − 𝑥2 + 𝑐, … } 

⇒ 𝑥3 = 𝑠𝑥 − (𝑠𝑥 − 𝑥 + 𝑐) + 𝑐 

⇒ 𝑥3 = 𝑠𝑥 − 𝑠𝑥 + 𝑥 − 𝑐 − 𝑐 

⇒ 𝑥3 = 𝑥         (Eq 5.3) 

We can conclude if (1 + 𝑡) = 0 then every point is of period 2. 

Case II (If 𝒄 = 𝟎) 

Then (Eq 5.1) becomes, 

𝑓(𝑥, 𝑦) = 𝑠𝑥 + 𝑡𝑦        (Eq 5.4) 

Orbit of any point 𝑂(𝑥) = {𝑥0 = 𝑥, 𝑥1 = 𝑥, 𝑥2 = 𝑓(𝑥, 𝑥), 𝑥3 = 𝑓(𝑥1, 𝑥2) … }  

Fixed points are 𝑥2 = 𝑥 ⇒  𝑓(𝑥, 𝑥) = 𝑥 

⟹ 𝑠𝑥 + 𝑡𝑥 = 𝑥 

⇒ (𝑠 + 𝑡)𝑥 = 𝑥 

⇒ (𝑠 + 𝑡) = 1 

If 𝑠 + 𝑡 = 1 then every point is fixed point.           (Eq 5.5) 

For period 2 points we have 𝑥3 = 𝑥 ⇒ 𝑓(𝑥, 𝑠𝑥 + 𝑡𝑥) = 𝑥 

⇒ 𝑠𝑥 + 𝑡(𝑠𝑥 + 𝑡𝑥) = 𝑥 

⇒ 𝑥 ≠ 0, 𝑠 + 𝑡 ≠ 1, 𝑠 + 𝑠𝑡 + 𝑡2 = 1 

⇒ 𝑠(1 + 𝑡) + (1 + 𝑡)(𝑡 − 1) = 0 

⇒ (1 + 𝑡)(𝑠 + 𝑡 − 1) = 0 

As we have assumed 𝑠 + 𝑡 − 1 ≠ 0 ⇒ 1 + 𝑡 = 0 

⇒ 𝑡 = −1 

For 𝑠 + 𝑡 − 1 ≠ 0 and 𝑡 = −1 every point is of period 2. (Eq 5.6). 

6. Problems 

Problem 6.1 For the map 𝑓: ℝ × ℝ → ℝ defined as 

 𝑓(𝑥, 𝑦) = 2𝑥 − 3𝑦 + 4 which is continuous so it is a general system. 



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By (Eq 5.2) Fixed points for the map is 𝑥 =
4

1−2−(−3)
= 2. 

By (Eq 5.3) It doesn’t have any periodic points of period 2 as 1 + 𝑡 ≠ 0. 

Problem 6.2 For the map 𝑓: ℝ × ℝ → ℝ defined as 

 𝑓(𝑥, 𝑦) = 𝑥 − 𝑦 + 9 which is continuous so it is a general system. 

By (Eq 5.2) Fixed points for the map is 𝑥 =
9

1−1−(−1)
= 9. 

By (Eq 5.3) 1 + 𝑡 = 0 so every point is of period 2 except fixed points. 

Problem 6.3 For the map 𝑓: ℝ × ℝ → ℝ defined as 

𝑓(𝑥, 𝑦) = 2𝑥 − 𝑦 which is continuous so it is a general system. 

By (Eq 5.5) Every point is fixed point as 𝑠 + 𝑡 = 1. 

By (Eq 5.3) Every point is of period 2 except fixed points as 𝑡 = −1. 

7. Conclusion 

The major part of the fundamental dynamical system is still unknown in the generalized system and 

m-step systems, which are novel concepts. We developed relationships between periodic concepts 

such as fixed points, recurrent points, and non-wandering points in this study. Additionally, we 

looked at the theorems connected to transitivity and, finally, were provided with examples. 

References 

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