At its core, recursive dynamics refers to iterative processes where energy or information is transferred across different dimensions, whether spatial or temporal. Each interaction influences the subsequent iterations, creating a feedback loop. In this context, the influence function represents a physical quantity, such as gravitational potential or strain, in the system. This influence function evolves through recursive interactions with neighboring dimensions, which allows energy to propagate not just within one dimension but across multiple dimensions simultaneously.
We begin with the key equation governing this system, which describes how the influence function evolves over time:
∂I_d/∂t = -φ_d ∇²I_d + π_d ∇²I_d - S_d I_d
Here, φ_d is the compression term that describes the influence from the dimension below, π_d is the expansion term representing the influence from the dimension above, and S_d is the stabilization term ensuring the system remains stable by preventing unbounded growth of energy. The spatial Laplacian, ∇²I_d, models the influence propagation across space.
The recursive nature of this system is captured in the following energy conservation equation:
φ_d I_(d-1) + π_d I_(d+1) + S_d I_d = 0
This ensures that the energy transferred between neighboring dimensions is balanced by the stabilization term, preventing runaway energy growth and maintaining a self-consistent system.
The stabilization term, S_d, is analogous to the cosmological constant (Λ) in standard cosmology. Just as Λ counteracts gravitational collapse and stabilizes the expansion of the universe, S_d works to balance the recursive energy flow, ensuring long-term stability in the system. This parallel suggests that recursive feedback and energy transfer in this framework may offer new insights into cosmological dynamics, potentially shedding light on phenomena like dark energy and the large-scale structure of spacetime.
Fractal recursive feedback plays a key role in shaping the evolution of the influence function. The influence function evolves in a self-similar manner, where each recursive step is scaled by a factor b and attenuated by a decay factor γ. This fractal recursion ensures that the process repeats itself at progressively smaller scales, introducing a fractal-like structure into the dynamics of the system. This self-similar behavior could have implications for understanding the structure of spacetime itself, as recursive dynamics might reflect the underlying geometric properties of the universe.
One of the significant implications of this framework is its potential connection to cosmological phenomena. The recursive nature of the dynamics could help explain the fluctuations observed in the cosmic microwave background (CMB), with recursive energy propagation across dimensions generating self-similar patterns that mirror the large-scale structure of the universe. The balance between compression and expansion terms, along with the stabilization mechanism, provides a way to model the behavior of energy across scales, which could be crucial for understanding the dynamics of the early universe.
