Pavol Brunovský

# Diferenčné a diferenciálne rovnice

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Úvod

I. Diskrétne dynamické systémy a diferenčné rovnice

1. Jednorozmerný, lineárny a afinný dynamický systém

1.2. Lineárny jednorozmerný systém (LDS)

1.3. Lineárna nehomogénna diferenčná rovnica a afinný jednorozmerný LDS

1.4. Lineárny pavučinový (cobweb) model

1.5  Úlohy zloženého úrokovania

2. Jednorozmerný nelineárny dynamický systém

2.1. Pevné body, periodické body

2.2  Logistický model populačnej dynamiky

2.3. General autonomous equation

2.4. Example. Logistic equation

2.5. Example. The Solow model of economic growth

2.6. Equations with separable variables

3. Higher dimensional linear equations

3.1  Linear autonomous (homogeneous) equation

3.2  Stability of the zero solution of a LDE

4.Structure of solutions of a linear differential equation

4.1  The space of solutions

4.2  Fundamental matrix

4.3. Autonomous equation

4.4. The linear nonhomogeneous and the affine equation

4.5  Higher order linear equations

4.6  The harmonic oscillator

4.7. Example. Forced oscillations of an electric network. Resonance

5. General theory of differential equations

5.1. The need of theory

5.2. Basic theorem on existence, uniqueness and extendability of solutions

5.3. The method of Cauchy-Peano

5.4. Picard's approximations

5.6. Approximate computation of solutions

5.7 Linear equations

6. General autonomous differential equations.

6.1. Stability of stationary solutions of autonomous differential equations

6.2. Example. Continuous dynamics of the supply and demand cobweb model

6.3. Example. The IS-LM model

6.4. Trajectories of autonomous differential equations

7. Two-dimensional autonomous equations

7.1  Trajectories of two-dimensional equations

7.2. Classification of two-dimensional linear equations

7.3. Node

7.5. Focus and center

7.6. Affine equations

7.7. Nonlinear equations

7.8 The method of isoclines

7.9. Example. The Volterra-Lotka equations

7.10. Example. The predator-prey equation}

7.11. Example. Goodwin's growth cycle

8. Integrals of differential equations

8.1. The concept of integral

8.2. Example. The predator - prey equation

8.3. Conservative equations with one degree of freedom

8.4. Example. The harmonic oscillator

8.5. The pendulum equation

Pavol Brunovský, ã