Öz
It is known that a chaotic function in the Devaney sense has three main properties: namely, topological transitivity, density of periodic points, and sensitive dependence on initial conditions. The product method is a main tool for constructing new discrete dynamical systems as well as topological space. In the theory of discrete dynamical systems, it has been investigated whether the properties of two chaotic functions carry over to their product function, and it has been shown that product of two Devaney chaotic functions may not always be chaotic in the Devaney sense. In this work, we consider the reverse problem: if the product system is chaotic, what can be said about the chaotic behavior of its components? So, we construct chaotic functions in the product form φ×γ, where the second component is chaotic while first is not. Additionally, we show that φ×γ is chaotic in the Li-Yorke sense if γ is Li-Yorke chaotic, even though φ is not.
Anahtar Kelimeler
Devaney chaos Product chaos Chaos conditions Logistic map Li-Yorke chaos
Kaynakça
- [1] Değirmenci N, Koçak Ş. Chaos in product maps. Turk J Math 2010; 34 (4): 593-600.
- [2] Kwietniak D, Misiurewicz M. Exact Devaney chaos and entropy. Qual Theor Dyn Syst 2005; 6: 169-179.
- [3] Devaney RL. An introduction to chaotic dynamical systems. New York, NY, USA: Addison Wesley, 1989.
- [4] Armstrong MA. Basic Topology. Springer-Verlag, New York, 1983.
- [5] Mangang KB. Li-Yorke chaos in product dynamical systems. Adv Dyn Syst Appl 2017; 12 (1): 81-88.
- [6] Banks J, Brooks J, Cairns G, Davis G, Stacey P. On Devaney's definition of chaos. Am Math Mon 1992; 99 (4): 332-334.
Öz
It is known that a chaotic function in the Devaney sense has three main properties: namely, topological transitivity, density of periodic points, and sensitive dependence on initial conditions. The product method is a main tool for constructing new discrete dynamical systems as well as topological space. In the theory of discrete dynamical systems, it has been investigated whether the properties of two chaotic functions carry over to their product function, and it has been shown that product of two Devaney chaotic functions may not always be chaotic in the Devaney sense. In this work, we consider the reverse problem: if the product system is chaotic, what can be said about the chaotic behavior of its components? So, we construct chaotic functions in the product form φ×γ, where the second component is chaotic while first is not. Additionally, we show that φ×γ is chaotic in the Li-Yorke sense if γ is Li-Yorke chaotic, even though φ is not.
Anahtar Kelimeler
Devaney chaos Product chaos Chaos conditions Logistic map Li-Yorke chaos
Kaynakça
- [1] Değirmenci N, Koçak Ş. Chaos in product maps. Turk J Math 2010; 34 (4): 593-600.
- [2] Kwietniak D, Misiurewicz M. Exact Devaney chaos and entropy. Qual Theor Dyn Syst 2005; 6: 169-179.
- [3] Devaney RL. An introduction to chaotic dynamical systems. New York, NY, USA: Addison Wesley, 1989.
- [4] Armstrong MA. Basic Topology. Springer-Verlag, New York, 1983.
- [5] Mangang KB. Li-Yorke chaos in product dynamical systems. Adv Dyn Syst Appl 2017; 12 (1): 81-88.
- [6] Banks J, Brooks J, Cairns G, Davis G, Stacey P. On Devaney's definition of chaos. Am Math Mon 1992; 99 (4): 332-334.
Ayrıntılar
| Birincil Dil | Türkçe |
|---|---|
| Konular | Adi Diferansiyel Denklemler, Fark Denklemleri ve Dinamik Sistemler, Topoloji |
| Bölüm | Makaleler |
| Yazarlar | |
| Yayımlanma Tarihi | 25 Ağustos 2025 |
| Gönderilme Tarihi | 18 Mart 2025 |
| Kabul Tarihi | 31 Mayıs 2025 |
| Yayımlandığı Sayı | Yıl 2025 Cilt: 13 Sayı: 2 |