This paper presents a new model for a two-dimensional nonlinear difference system that incorporates symmetric interactions between two sequences through a scaling parameter $ d $ and a continuous one-to-one transformation function $ f $. Explicit analytical solutions are derived, establishing a direct connection with the $ d $-Fibonacci sequence. The transformation function $ f $ plays a crucial role: It accommodates diverse nonlinear iteration patterns and provides a natural mechanism for regulating both growth dynamics and sequence interactions. Moreover, the use of a continuous one-to-one function guarantees that the analytical solutions of transformed systems can be recovered through its inverse mapping. The approach highlights a unified framework linking generalized Fibonacci-type recursions with nonlinear transformations, offering new insights into the structure and solvability of higher-order discrete systems. Several illustrative examples are provided to support the theoretical findings.
Citation: Ahmed A. Al Ghafli. Closed-form solutions of a nonlinear bidimensional difference system via generalized Fibonacci sequences[J]. AIMS Mathematics, 2025, 10(11): 26545-26567. doi: 10.3934/math.20251167
This paper presents a new model for a two-dimensional nonlinear difference system that incorporates symmetric interactions between two sequences through a scaling parameter $ d $ and a continuous one-to-one transformation function $ f $. Explicit analytical solutions are derived, establishing a direct connection with the $ d $-Fibonacci sequence. The transformation function $ f $ plays a crucial role: It accommodates diverse nonlinear iteration patterns and provides a natural mechanism for regulating both growth dynamics and sequence interactions. Moreover, the use of a continuous one-to-one function guarantees that the analytical solutions of transformed systems can be recovered through its inverse mapping. The approach highlights a unified framework linking generalized Fibonacci-type recursions with nonlinear transformations, offering new insights into the structure and solvability of higher-order discrete systems. Several illustrative examples are provided to support the theoretical findings.
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Ahmed A. Al Ghafli. Closed-form solutions of a nonlinear bidimensional difference system via generalized Fibonacci sequences[J]. AIMS Mathematics, 2025, 10(11): 26545-26567. doi: 10.3934/math.20251167
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