Dynamics asks what iteration or time evolution does over many steps, even when an explicit solution is unavailable. Teschl studies flows, stability and discrete maps using analysis and differential equations. Bruin develops invariant measures, recurrence and ergodic behavior. Choose the smooth or discrete route for local stability and phase portraits; choose the measure-preserving route when the question concerns typical long-term behavior with respect to a specified measure.
What to work on
Describe a flow, its orbits and invariant sets.
Distinguish stability, attraction and asymptotic stability.
Use a Lyapunov function and state what its derivative proves.
Determine local stability of fixed points of a discrete map.
A starting text
Ordinary Differential Equations and Dynamical Systems — Gerald Teschl. Read 6.1-6.3 on flows, orbits and invariant sets and 6.5-6.6 on stability and Lyapunov methods; use 9.1 for linear stability. For a discrete route read 10.1-10.2 and 10.4 on logistic maps, periodic points and local behavior.
Choosing what to read
Find fixed points and compute the derivative or linearization, then state when that local test is inconclusive. Compare a contracting flow with a rotation so that stability and attraction remain separate. For a discrete map, follow a few orbits but use an argument to justify any limiting claim. Before an ergodic theorem, identify the invariant measure, the observable and the mode of convergence. Revisit measure theory if almost-everywhere statements are being inferred from a picture.
Study invariant sets and Lyapunov methods alongside explicit examples.
How it reads
Analytic route through ODE flows, local stability and discrete maps.
What it expects
Real analysis and linear algebra; elementary differential equations.
What to keep in mind
Local derivative tests at unit-modulus multipliers need further analysis; this route does not itself establish ergodic theory.
Practice
Public numbered problems verified within the author PDF, some with hints; a public worked solution collection was not verified. In chapter 6 start with Problems 6.8-6.10 on limits, periodicity and nonwandering points. Then use problems embedded in 6.5-6.6 and chapter 10 alongside the corresponding readings.
Complete public PDF.
Edition, reading range and source
Read 6.1-6.3 on flows, orbits and invariant sets and 6.5-6.6 on stability and Lyapunov methods; use 9.1 for linear stability. For a discrete route read 10.1-10.2 and 10.4 on logistic maps, periodic points and local behavior.
Flows, stability and discrete-dynamics problems. In chapter 6 start with Problems 6.8-6.10 on limits, periodicity and nonwandering points. Then use problems embedded in 6.5-6.6 and chapter 10 alongside the corresponding readings.. Public numbered problems verified within the author PDF, some with hints; a public worked solution collection was not verified.
Try these before moving on
Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.
For x'=-x and y'=-2y, write the flow and use V=x squared+y squared to prove global asymptotic stability of the origin. Explain why V=x squared+y squared cannot prove attraction for x'=-y, y'=x.
A hint
Differentiate V along each vector field. For stability give a delta that controls all future times; for attraction show decay from every initial point.
Work through the solution
The first system has flow ϕt(x0,y0)=(e−tx0,e−2ty0) for all real t. Along any trajectory, V′=−2x2−4y2≤−2V. Since (e2tV)′≤0, V(t)≤e−2tV(0) for t≥0. Thus ∥ϕt(x0,y0)∥≤e−t∥(x0,y0)∥. For every ε>0, choose δ=ε; an initial norm below δ stays below ε for every future time, proving stability. The same bound tends to zero for every initial point, proving global attraction and hence global asymptotic stability.
For x′=−y, y′=x, one instead has V′=2x(−y)+2y(x)=0. Every nonzero initial point stays at its positive initial distance from the origin (indeed the explicit flow is rotation). It cannot tend to zero. This origin is stable but not attractive; the failure to prove attraction reflects actual non-attraction, not merely a weak choice of estimate.
Real Analysis — Review the quantifiers distinguishing stability from attraction.
Already comfortable with this? Skip flow and stability review if you can prove the first check with the definitions of stability and attraction. Skip the discrete selection if you can give the derivative argument and recognize its inconclusive boundary cases. Ergodic theory needs a separate measure-based route.
Connects rigorous ODE theory to stability, dynamical systems and chaos. A bridge to graduate dynamics; the author permits personal-use downloads, not unrestricted redistribution.
Preparation: Real analysis; linear algebra; calculus.
A measure-theoretic path into ergodic theory. Best after measure and integration; the author warns that the notes have not been proofread and should be used carefully.
The complete published book is subscription or purchase content. A preview or sample is not the complete book; institutional access depends on your library licence.
Studies multiplicative cocycles, invariant measures and Lyapunov exponents.
Preparation: Ergodic theory; linear algebra; probability.
Use: Advance from introductory dynamics to nonuniform growth.
Purchase or library subscription required; previews are not the complete book.
Develops ergodicity, mixing, entropy and hyperbolic examples. Use after measure theory, with concrete invariant-measure computations alongside abstract results.
I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai · Textbook · Graduate · English · Purchase / institutional access
Purchase or library subscription required; previews are not the complete book.
A substantial classical treatment of measure-preserving dynamics and entropy. Read selectively with a modern course when notation or exposition differs.
Herbert B. Keller · Long lecture notes · Graduate · English · PDF
Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.
An applied computational route to bifurcation, complementing analytic dynamics. Numerical results require attention to approximation and stability.
Manfred Einsiedler and Thomas Ward · Textbook · Graduate · English · Purchase / institutional access
Purchase or library subscription required; previews are not the complete book.
Develops measure-preserving dynamics with number-theoretic motivations. A bridge into homogeneous dynamics and equidistribution rather than a first ODE course.
Carl Ludwig Siegel · Long lecture notes · Research · English · PDF
Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.
A specialized course in celestial-mechanics dynamics and singularity analysis. This is a research branch, not a first qualitative ODE course.
Preparation: ODE, mechanics, complex analysis and dynamical systems.
Patrick J. Rabier · Long lecture notes · Research · English · PDF
Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.
A specialist course on how solutions change with a parameter. Functional-analytic preparation is needed beyond an elementary phase-plane course.