Sensitivities in complex-time flows: Phase transitions, Hamiltonian structure, and differential geometry.

Area of Science:

• Complex analysis
• Dynamical systems theory
• Geometric analysis

Background:

• Separatrices divide phase space in dynamical systems with multiple equilibria, influencing flow behavior.
• Understanding separatrix properties is crucial for analyzing system dynamics and stability.

Purpose of the Study:

• To introduce complex time as a novel approach for studying Riemann surface solutions of holomorphic and meromorphic flows.
• To investigate the geometric properties of separatrices and their relation to complex-valued Hamiltonian systems.
• To apply this framework to analyze the complex-time Newton flow of Riemann's ξ-function.

Main Methods:

• Introduction of complex time to analyze Riemann surface solutions.
• Explicit solution of the sensitivity differential equation for these flows.
• Identification of a related Hamiltonian structure and associated geometry.
• Application to polynomial approximations of Riemann's ξ-function's Riemann surface solution.

Main Results:

• A method for studying separatrix properties using complex time and associated geometry.
• A connection established between complex-valued Hamiltonian systems and the geometry of Riemann surface solutions.
• Analysis of Riemann's ξ-function's complex-time Newton flow using polynomial approximations.

Conclusions:

• Complex time provides a powerful framework for understanding separatrix properties in dynamical systems.
• The geometric properties derived from this approach may offer insights into the global separatrix structure.
• This method could potentially reveal information about the root locations of Riemann's ξ-function and its derivative.