- Introductions
- Basic mathematical concepts
- Mathematical tools for analyzing autonomous linear and non linear ordinary differential equations.
- Introduction to qualitative analysis.
- Criteria and theorems for the asymptotic stability of equilibrium.
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Nonlinear systems
- Equations of nonlinear systems in continuous and discrete time.
- Linearization and Hartman-Grobman theorem.
- Nonlinear oscillations (limit cycles).
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Bifurcations
- Saddle-node bifurcation (continuous time).
- Transcritical bifurcation (continuous time)
- Pitchfork bifurcation (continuous time).
- Hopf bifurcations (continuous time).
- Flip bifurcation (discrete time).
- Period doubling bifurcations (discrete and continuous time).
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Deterministic chaos
- Bifurcation cascades and routes to chaos.
- Chaotic attractors.
- The Lorenz system (continuous time).
- The Rossler systems (continuous time).
- The logistic map (discrete time).
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Networked and distributed systems
- Complex Networks.
- Dynamic systems on networks.
- Synchronization.
- Fractals.
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Applications
- Complex systems and cooperation in human societies.