Toolbox Chaos procedure
Equilibria and Eigenvalues
Compute supported ODE equilibria, inspect their Jacobian eigenvalues in the complex plane, and distinguish local linear stability from global dynamics.
Question and result
Calculation performed
Use equilibrium and eigenvalue information as a local analysis layer alongside trajectories and parameter studies.
Required configuration
- Select a supported continuous ODE flow. The panel lists the model families available for equilibrium and eigenvalue analysis.
- Understand that eigenvalues describe the Jacobian linearization at a particular equilibrium and parameter set.
Panel and figure
Displayed quantities
Configuration and calculation
Control sequence
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Select the ODE and parameters
Choose the system and enter its parameter values. The static equilibrium calculation uses equations and parameters; initial condition, integration method, and duration belong to trajectory simulations.
Record the parameter vector before calculating; changing a parameter can move, create, merge, or remove equilibria.
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Calculate the supported equilibria
Press Calcular equilibrios y autovalores. The panel displays the supported equilibrium set and corresponding eigenvalues in its information area and complex-plane figure.
An unavailable-calculation message identifies the panel compatibility for the selected model type. Determine equilibrium existence through an analysis valid for that model.
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Inspect all points
Choose Todos to plot all returned eigenvalues in the complex plane and read the coordinate list in the information area.
Check multiplicity, complex-conjugate pairs, and real parts near zero, which can make finite-precision classification delicate.
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Inspect one equilibrium
Select an individual equilibrium to view its point, eigenvalues, local type, and classification.
Relate signs of real parts to local contraction or expansion only within the assumptions of linearization; nonhyperbolic cases require additional analysis.
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Connect local and global views
Run trajectories from small, explicitly declared perturbations only when the research question requires local numerical behavior.
Use phase portraits, time series, and parameter sweeps to study global behavior. Basin structure and distant invariant sets require their corresponding analysis.
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Preserve the figure and displayed values
Save the complex-plane figure and record the equilibrium coordinates and eigenvalue values exactly as shown in the information area.
The information area rounds equilibrium coordinates with .4g formatting and eigenvalues with .6g formatting. Retain the parameter vector and figure with the record; full-precision analysis remains in the source calculation.
State whether the view contains all displayed equilibria or one selected point.
Reporting language
- Supported: ‘At these parameters, the panel returned equilibrium E and the Jacobian eigenvalues had these real parts.’
- Supported: ‘At the values displayed by the GUI, the panel labeled this equilibrium with the reported local classification.’
- Additional evidence required for global chaos, attractor uniqueness, or hidden-attractor localization: the appropriate global, basin, and mathematical analyses.
Record
Computed data
- Displayed equilibrium coordinates and Jacobian eigenvalues for one parameter vector, formatted at the precision shown by the interface.
- A complex-plane plot and a cautious local classification for each selected equilibrium.
Scientific reading
Interpretation criteria
Negative real parts support local linear contraction for a hyperbolic equilibrium; positive real parts indicate expanding directions. Mixed signs indicate saddle behavior, and imaginary parts describe local oscillatory components.
Zero or nearly zero real parts, equilibrium manifolds, and nonlinear degeneracies require analysis beyond a simple eigenvalue sign rule.
Files
Experiment record
- Record equations or catalog name and the complete parameter vector before calculating.
- Retain the saved complex-plane figure and the equilibrium coordinates, eigenvalues, and local classification exactly as displayed.
- Describe displayed coordinates and eigenvalues at their stated formatting precision, and retain the parameter vector with the saved figure.
- State local classification and avoid global language unless supported by separate evidence.
Applications
Questions addressed by the calculation
- Teaching linearization and local stability.
- Choosing parameter regions for bifurcation exploration.
- Relating local saddle directions to observed trajectory geometry as a hypothesis for further study.
Scope
Evidence conditions
- The panel is limited to supported ODE-flow equilibrium contracts.
- Global basin geometry, multistability, and distant attractors are investigated with trajectories, basin sampling, and the corresponding global analysis.
- Equilibrium analysis documents local linear behavior; Hidden Attractors FO provides the separate localization and certification workflow.