OSCILATE documentation

Welcome to the documentation for OSCILATE, a Python project for solving systems of coupled nonlinear second-order ODEs.

OSCILATE: Oscillators’ nonlinear analysis through SymboliC ImpLementATion of the mEthod of multiple scales

Contents:

• README
  • Project Home
  • Nonlinear systems considered
  • Overview
    • Solver
    • Examples
    • Outputs
  • Citation
  • Installation guide
  • Disclaimer
  • License
• The Method of Multiple Scales
  • Dynamical systems of interest
  • Application of the MMS
    • Asymptotic series and time scales
    • Scaling
    • The multiple scales system
    • Frequency of interest
    • Iteratively solving the MMS system
      • Leading order solution
      • Higher order solutions
      • Secular analysis
    • Modulation equations
      • Polar coordinates
      • Autonomous phase coordinates
      • Reconstituted equations
  • Evaluation at steady state
    • Steady state solutions
      • Steady state conditions
      • MMS modulation equations at steady state
      • MMS solutions at steady state
    • Stability analysis
  • Stability analysis
    • Stability information
      • Jacobian matrix
      • Perturbation of the steady state solution
      • Stability condition
      • Bifurcations
      • Bifurcation curves
    • Stability analysis in cartesian coordinates
      • Limitations of polar coordinates
      • Cartesian coordinates
      • Cartesian modulation equations
      • Jacobian matrix
• Modules
  • MMS Module
    • dyn_sys
      • Dynamical_system
      • Forcing
    • mms
      • Chain_rule_d2fdt2()
      • Chain_rule_dfdt()
      • Coord_MMS
      • Forcing_MMS
      • Multiple_scales_system
      • Sol_MMS
      • Substitutions_MMS
      • cartesian_to_polar()
      • rescale()
      • scale_parameters()
    • steady_state
      • Coord_SS
      • Forcing_SS
      • Sol_SS
      • Stab_SS
      • Steady_state
      • Substitutions_SS
    • visualisation
      • numpise_ARC()
      • numpise_FRC()
      • numpise_F_ARC()
      • numpise_omega_FRC()
      • numpise_omega_bbc()
      • numpise_omega_bif()
      • numpise_phase()
      • plot_ARC()
      • plot_FRC()
  • sympy_functions Module
    • check_solvability()
    • get_block_diagonal_indices()
    • get_exponent()
    • is_block_diagonal()
    • polynomial_terms()
    • solve_poly2()
    • sub_deep()
    • sympy_to_numpy()
• Main module architecture
• Installation guide
  • Install possibilities
    • Install from PyPI (recommended)
    • Install from a GitHub release
    • Install from the repository (latest version)
  • Dependencies
  • Optional: use a virtual environment (recommended)
  • Test the install
• Application examples
  • Example 1: Duffing oscillator
    • System description
    • Code description
  • Example 2: Coupled Duffings in 1:3 internal resonance
    • System description
    • Code description
  • Example 3: Coupled Duffings in 1:1 internal resonance
    • System description
    • Code description
  • Example 4: Coupled quadratic oscillators in 1:2 internal resonance
    • System description
    • Code description
  • Example 5: Parametrically excited Duffing oscillator
    • System description
    • Code description
    • Validation
  • Example 6: Van der Pol oscillator
    • System description
    • Code description
  • Example 7: Parametrically excited Rayleigh oscillator
    • System description
    • Code description
  • Example 8: Superharmonic response of a Duffing oscillator
    • System description
    • Code description
  • Example 9: Subharmonic response of a Duffing oscillator
    • System description
    • Code description
  • Example 10: Subharmonic response of parametrically excited coupled centrifugal pendulums in 1:1 internal resonance
    • System description
    • Code description
• SymPy basics
  • Symbols and functions
  • Working with expressions
  • Display
• Citation
• License
• Legal notices