From Stability to Chaos: Fixed Points and Bifurcations in the Framework of Discrete Dynamical Systems
6th International Interdisciplinary Symposium on Chaos and Complex Systems, SCCS 2025, İstanbul, Türkiye, 8 - 10 Mayıs 2025, ss.85-95, (Tam Metin Bildiri)
- Yayın Türü: Bildiri / Tam Metin Bildiri
- Doi Numarası: 10.1007/978-3-032-09101-7_8
- Basıldığı Şehir: İstanbul
- Basıldığı Ülke: Türkiye
- Sayfa Sayıları: ss.85-95
- Anahtar Kelimeler: Bifurcation, Chaos, Discrete dynamical systems, Fixed point, Growth-rate
- Yıldız Teknik Üniversitesi Adresli: Evet
Özet
This study explores exponential and logistic growth models within the context of discrete dynamical systems, with particular emphasis on the qualitative behavior of orbits, the nature of fixed points, and the emergence of chaotic regimes. Initially, population growth is modeled through deterministic difference equations, and the transition from unbounded exponential growth to constrained logistic dynamics is discussed in light of environmental limitations. The study investigates the stability of fixed and periodic points through derivative-based criteria, illustrating how system trajectories evolve over time. As the growth rate parameter increases, the system exhibits classical bifurcation phenomena-culminating in the onset of chaos. The findings affirm that even simple nonlinear maps can exhibit highly complex and unpredictable behaviors, thereby highlighting the significance of discrete dynamical systems in mathematical modeling across diverse scientific disciplines.