MSc Physics | Computational Physics (PhD aspirant) Nepal

Physics educator and computational researcher focusing on the explicit numerical modeling of non-linear dynamical systems. My work evaluates the long-term phase-space stability of Hamiltonian, Lagrangian, and dissipative chaotic models.

I treat my code repositories as formal numerical laboratories. My methodology prioritizes mathematical rigor, strict physical conservation proofs, and custom algorithmic execution over pre-built library wrappers. I am systematically building a technical and mathematical foundation for a PhD in Computational Physics and Nonlinear Dynamics.

A computational framework designed to model, visualize, and analyze deterministic chaos within the three-dimensional, non-linear atmospheric flow model derived by Edward Lorenz.

$$ \frac{dx}{dt} = \sigma(y - x), \quad \frac{dy}{dt} = x(\rho - z) - y, \quad \frac{dz}{dt} = xy - \beta z $$

• Numerical Engine: Implemented a standard fourth-order Runge-Kutta (RK4) integrator with adaptive time-stepping to preserve trajectory integrity across high-gradient vector fields.
• Bifurcation Analysis: Mapped phase-space trajectories across a spectrum of Rayleigh numbers ($\rho$). Safely isolated the subcritical Hopf bifurcation ($\rho \approx 24.74$) where steady-state fixed points lose stability, giving rise to the iconic "butterfly" strange attractor.
• Chaos Quantification: Evaluated sensitive dependence on initial conditions (SDIC) by tracking twin trajectories with an initial perturbation of $\delta \sim 10^{-8}$, calculating the maximal Lyapunov exponent ($\lambda_{max} > 0$) to verify true deterministic chaos rather than numerical noise.
• Topology Mapping: Visualized 3D phase-space projections, plotting the dual-wing geometry to study trajectory ergodicity and fractal structure bounds.

An examination of multi-degree-of-freedom systems and the explicit onset of deterministic chaos through analytical mechanics.

• Euler-Lagrange Derivation: Formulated the coupled, highly non-linear transcendental equations of motion using generalized coordinates ($\theta_1, \theta_2$).
• Phase-Space Mapping: Isolated the transition boundaries where regular, low-energy invariant tori break apart into high-energy chaotic seas.
• Poincaré Sections: Generated state-space slices to visually map the breaking of system integrability and track coordinate divergence.

A high-precision study of gravitational stability, transition to chaos, and long-term orbital conservation.

• Symplectic Engineering: Developed a custom Velocity-Verlet (Leapfrog) engine in Python to eliminate numerical energy dissipation, maintaining a relative Hamiltonian error of $\sim 10^{-9}$.
• Chaos Quantification: Identified a critical velocity perturbation threshold ($\delta \approx 0.5$) for stellar ejection using Lyapunov divergence mapping.
• Astronomical Validation: Verified engine stability through a 12-year simulation of the Sun-Earth-Jupiter system using Astronomical Units and Solar Masses ($G = 4\pi^2$).
• Visualization: Authored a cinematic rendering suite to visualize "Butterfly Effect" divergence and total system collapse.

A structural migration of a 2023 procedural Fortran 90 dissertation into a high-performance, vectorized Python framework.

• Numerical Validation: Verified custom RK4 algorithms against SciPy, achieving a maximum residual error of $< 10^{-7}$.
• Legacy Cross-Validation: Achieved 100% trace overlap between modern Python outputs and legacy Fortran data.
• Performance Auditing: Benchmarked Python interpreter loops versus compiled Fortran for 1,000,000-step execution cycles.

An investigation of ballistics kinematics evaluating the transit from idealized vacuum baselines to dissipative media.

$$ m \frac{d\mathbf{v}}{dt} = m \mathbf{g} - b\mathbf{v} - c|\mathbf{v}|\mathbf{v} $$

• State-Space Vectorization: Implements array-based evaluations in NumPy to resolve non-linear quadratic drag terms where exact analytical solutions are unavailable.
• Error Benchmarking: Compares custom discrete step allocations against adaptive-step algorithms to map absolute spatial residuals.

My technical stack maps directly to strict methodological requirements designed for reproducible execution on standard consumer hardware (16 GB RAM, CPU-only):

• Numerical Integration: Symplectic algorithms (Leapfrog/Verlet) and adaptive high-order solvers (RK45, DOP853).
• State-Space Modeling: Vectorized array operations using NumPy and SciPy for multi-variable ODEs.
• Chaotic Quantification: Implementation of phase-space metrics, including Lyapunov exponents and Poincaré mapping.
• Physics Validation: Test-driven workflows designed to verify structural invariants (e.g., energy and momentum conservation) during runtime.
• Environment Reproducibility: Strict adherence to deterministic random seed controls and continuous versioning for repository continuity.

Consolidated Knowledge I have verified my capacity to construct vectorized ODE solvers for many-body and chaotic systems. I can successfully track structural conservation laws directly within computation loops, quantify non-linear trajectories using established chaos indicators, and enforce mathematical accuracy over millions of integration steps.

Targeted PhD-Level Methodologies (Future Research Roadmap) To transition from standalone numerical simulations to high-performance, doctoral-level research, I have identified the following computational methodologies as the core technical objectives of my upcoming academic trajectory:

1. Compiled Acceleration: Migrating computationally intensive components from interpreted Python to compiled frameworks (Numba, native C++ extensions) to bypass interpreter overhead and execute high-speed multi-parameter sweeps.
2. Dimensional Scaling: Progressing from systems governed by Ordinary Differential Equations (ODEs) to high-dimensional Partial Differential Equations (PDEs) and large many-body lattice matrices.
3. Parallelization: Implementing multi-core CPU parallelization architectures to manage large-scale computational execution and high-density state-space mapping efficiently.

"Physics is the only profession in which prophecy is not only allowed but required." — This profile tracks my progress in mastering those prophecies through code.