Is this project an undergraduate, graduate, or faculty project?
Undergraduate
Project Type
individual
Campus
Daytona Beach
Authors' Class Standing
Martyna Wojcik, Senior
Lead Presenter's Name
Martyna Wojcik
Lead Presenter's College
DB College of Arts and Sciences
Faculty Mentor Name
Dr. Keshav Acharya
Abstract
Bridging Discrete and Continuous Systems: Fibonacci Sequences and Exponential Growth from Ordinary Differential Equations It has been observed that nature often exhibits specific patterns of growth and structure in biological systems and spiral formations. The Fibonacci sequence, defined as a discrete recursive sequence where each term is generated as the sum of the two preceding terms, “has been applied extensively to understand some natural phenomena” (Pakdemirli, 2023). In contrast, exponential growth describes a continuous process in which the rate of change of a quantity is proportional to its current value. Such behavior is modeled using differential equations that “produce solutions which are continuous throughout the domain of interest” (Pakdemirli, 2023), expressed by dP/dt=rP, where r is the growth rate (constant), and P represents the quantity that is growing (dependent on time). The purpose of this project is to compare these two frameworks and determine how closely Fibonacci dynamics approximate exponential growth derived from ordinary differential equations. The analysis involves numerical computation and graphical visualization to show that Fibonacci growth exhibits exponential-like characteristics, with the ratio of successive terms converging to the golden ratio. The results demonstrate that discrete recursive systems can be approximated by continuous differential equation models. This work is potentially applicable in engineering and applied sciences, including control systems, embedded systems, population dynamics, and financial modeling, where discrete-time implementations are used to model and approximate continuous-time dynamics. References: Pakdemirli, M. (2023). Fibonacci Differential Equation and Associated Spiral Curves. CODEE Journal, 16(1), 1–7. https://doi.org/10.5642/codee.rqgl4401 Supriatna, A. K., Carnia, E., & Ndii, M. Z. (2019). Fibonacci numbers: A population dynamics perspective. Heliyon, 5(1), e01130. https://doi.org/10.1016/j.heliyon.2019.e01130
Did this research project receive funding support (Spark, SURF, Research Abroad, Student Internal Grants, Collaborative, Climbing, or Ignite Grants) from the Office of Undergraduate Research?
No
Included in
Number Theory Commons, Numerical Analysis and Computation Commons, Ordinary Differential Equations and Applied Dynamics Commons
Bridging Discrete and Continuous Systems: Fibonacci Sequences and Exponential Growth from ODEs
Bridging Discrete and Continuous Systems: Fibonacci Sequences and Exponential Growth from Ordinary Differential Equations It has been observed that nature often exhibits specific patterns of growth and structure in biological systems and spiral formations. The Fibonacci sequence, defined as a discrete recursive sequence where each term is generated as the sum of the two preceding terms, “has been applied extensively to understand some natural phenomena” (Pakdemirli, 2023). In contrast, exponential growth describes a continuous process in which the rate of change of a quantity is proportional to its current value. Such behavior is modeled using differential equations that “produce solutions which are continuous throughout the domain of interest” (Pakdemirli, 2023), expressed by dP/dt=rP, where r is the growth rate (constant), and P represents the quantity that is growing (dependent on time). The purpose of this project is to compare these two frameworks and determine how closely Fibonacci dynamics approximate exponential growth derived from ordinary differential equations. The analysis involves numerical computation and graphical visualization to show that Fibonacci growth exhibits exponential-like characteristics, with the ratio of successive terms converging to the golden ratio. The results demonstrate that discrete recursive systems can be approximated by continuous differential equation models. This work is potentially applicable in engineering and applied sciences, including control systems, embedded systems, population dynamics, and financial modeling, where discrete-time implementations are used to model and approximate continuous-time dynamics. References: Pakdemirli, M. (2023). Fibonacci Differential Equation and Associated Spiral Curves. CODEE Journal, 16(1), 1–7. https://doi.org/10.5642/codee.rqgl4401 Supriatna, A. K., Carnia, E., & Ndii, M. Z. (2019). Fibonacci numbers: A population dynamics perspective. Heliyon, 5(1), e01130. https://doi.org/10.1016/j.heliyon.2019.e01130