# General Template for the Analysis of a Dynamic System

## 1 Generalities

• 1.1 Introduce some background about your model
• References
• Application
• Relevance of the Model for your Project?
• Basic Assumptions of the Model
• 1.2 Describe the goals of your study

## 2 Model description

• 2.1 Write down the set of ODEs characterizing your model
• 2.2 Describe in a table the signification of each term and parameter of the model
• Use simple terms
• Give Physical Interpretation when possible
• 2.3 Define strengths and flaws of the model
• Insist on physical interpretation
• Look into relevance of model for small numbers

## 3 Stability analysis

• 3.1 Studying the steady points
• What is a steady point?
• What is the property of the system at these particular points ?
• Write down the set of ODEs to solve to find them
• Write down their expression.
• Any remarks on them ?
• 3.2 Studying the stability of the steady points
• What is the meaning of this study ?
• Why do we use the Jacobian ?
• Why do we use its Eigenvalues?
• What is the rule for stability?
• Analyse each steady point
• by writing the value of the Jacobian at this particular point
• by writing the trace and determinant of the 2D matrix
• by writing down the eigen values
• Conclude on the stability of the point considered in regards to the parameters
• 3.3 Studying the Vector Field (VF)
• What is the Vector Field?
• Depending of the value of the parameters, define different cases (different behaviours of the steady points)
• for each case, draw the VF and place the steady points with the behaviour of the flow at their vecinity
• Plot in the VF, dx/dt=0 and dy/dt=0. By using the sign the dx/dt and dy/dt, draw the general trend of the VF in each region of the VF.
• Simulate the VF for a few well chosen values of your parameters, justify choice.
• Plot Different trajectories for wel chosen initial values
• 3.4 General Remarks on the VF
• Give as thorough as possible an analysis of the VF
• In particular Focus on
• Shape of Trajectories
• Influence of initial conditions
• Influence of model parameters
• Make predictions and suggestions regarding the sensitivity analysis
• NB: Quting Poincare-Bendixson is a bonus for a 2D model....