The author: Doctor of Sciences, Professor Kuznetsov A. P.

**17 lectures**

education

problems and lessons

**1.1 About classification of dynamic systems. **Flow systems and mappings. The
dissipative and conservative systems, character of evolution of a cloud of points in
a phase space. Attractors. Dimensionality of a phase space, growth of variety and complexity
of attractors with increasing number of dimensions. The elementary examples of discrete
mappings: a chain of resistances and a biological population. A method of sections of
Poincare. About interrelation of streams and mappings. Invertible and noninvertible
mappings. Autonomous and nonautonomous dynamic systems. Poincare section for nonautonomous
systems.

**1.2 Examples of dynamic systems.** Why should we consider dynamic systems in nonlinear
dynamics? The kinetic equations and the law of corresponding states. Brusselator. Oscillatory
chemical reactions. Oregonator. Rosler system as a an artificial model of a chemical
kinetics. Fermi acceleration and problems with collisions. Ulam model. Zaslavski and
Chirikov system (springing ball on a vibrating table), approximate discrete mapping
for the case of a small vibration amplitude, its dissipativity and invertable. System
with viscous friction under impuls forcing, Henon-type maps and Henon map, their dissipativity
and invertibility. The logistic mapping, cubic mapping, their noninvertibility. A periodically
forced relaxation generator. Circle map, representation of its dynamics on the iterative
diagram. When this mapping is noninvertible?

**2.1 Bifurcations of the one-dimensional flow systems.** Links between the theory
of bifurcations and theory of catastrophes in case of the one-dimensional systems. Classification
of bifurcations by a codimension. A saddle - node bifurcation. Transcritical bifurcation.
A "fork" bifurcation. Interconnections of various bifurcation diagrams at
the two-parameter analysis. Subcritical and supercritical types of bifurcations.

**2.2 Bifurcations of the one-dimensional flows in two-dimensional systems.** Nontriviality
of a problem (after Prigogine). A conservative oscillator near the points of catastrophes.
Dependence of frequency on parameters and a phenomenon of a softening of modes. Physical
examples. Power laws near the points of catastrophes. A case of the dissipative oscillator.
Transmutations of focuses into nodes near the points of catastrophes or why the "saddle-focus"
bifurcation is not possible. Deceleration of time and universality of bifurcations.
Phase portraits of bifurcations of the one-dimensional flows in two-dimensional systems.

**2.3 Andronov-Hopf bifurcation.** About a possibility of bifurcations of the dynamic
systems which do not have analogs in the theory of catastrophes. Andronov-Hopf bifurcation
and its universality. Evolution of points in a three-dimensional phase space at a threshold
of Andronov-Hopf bifurcation and concept of central manifold. The "linear part"
of the problem and its reduction to a canonical form in a framework of the slow amplitudes
formalism. A role of nonlinearity and the universal equations for slow amplitudes. Configuration
of the born limit cycle in canonical variables and in an initial phase space. A threshold
of Andronov-Hopf bifurcation in two-dimensional systems. A threshold of Andronov-Hopf
bifurcation in three-dimensional systems.

**2.4 Other bifurcations of bidimentional streams.** A bifurcation of birth of a
limit cycle from an inspissation of phase trajectories and its explanations in terms
of slow amplitudes. Nonlocal bifurcations of a codimension one: occurrence and decay
of the saddle bundle, birth of a limit cycle from a loop of a separatrix, birth of a
limit cycle from a common separatrix of a saddle and a node. Bifurcations of a codimension
two. "Critical point" of a line of Andronov-Hopf bifurcation (transmutation
of a line of a subcritical bifurcation into supercritical) and its connection with a
bifurcation of birth of a limit cycle from an inspissation of phase trajectories. Cusp
of bifurcation lines of birth of a limit cycle from an inspissation of trajectories
and analog of a "fork" bifurcation for cycles. A bifurcation "a common
point of Andronov-Hopf bifurcations and a saddle - node", its relationship with
a birth of a limit cycle from a loop of a separatrix.

**2.5 From two-dimensional flows - to one-dimensional maps.** A method of sections
of Poincare for two-dimensional flows. Various examples of phase portraits and the relevant
iterative diagrams. A problem of reconstruction of the one-dimensional mapping. Invertibility
of mappings associated with two-dimensional flows.

**3.1 The elementary properties of the one-dimensional mappings.** The fixed points.
Stable and unstable fixed points. Character of convergence (divergence) near the fixed
point. A multiplicator and its geometrical interpretation. Stability criterion for the
fixed point. Examples of iterative diagrams for different values of a multiplicator.
Cycles. Relationship between a problem of searching a N-cycle and a problem of searching
the fixed point. A multiplicator of a cycle and its expression as product of derivatives.
Examples of evaluation of a multiplicator of a cycle. Concept of superstable cycles.
About variety of cycles by an example of the logistic mapping.

**3.2 Bifurcations of a codimension one.** The tangential bifurcation. Motion of
the imaging point inside a "corridor" near bifurcation threshold, description
of a bifurcation with the help of the differential equation and an estimate of time
of passage of "corridor". A bifurcation of a "fork" type. Subcritical
types of the tangential bifurcation and a "fork". Bifurcation. Period doubling
bifurcation. Birth of a 2-cycle in the logistic mapping and analysis of a stability
of this cycle. Period doublings " in terms " of a twice iterated mappings.
Properties of such mappings and Schwarz derivative. A bifurcation " the hard transition
through a multiplicator -1 " as alternative to period doubling. Bifurcations of
cycles. From local to the global bifurcation analysis: period doubling cascade, a bifurcation
tree for the logistic mapping and discussion of its structure.

**3.3 Bifurcations of a codimension two.** A cusp point of the tangential bifurcations.
A point of transmutation of a line of period doubling into a line of the hard transition
(subcritical period doubling) - a flip bifurcation of a codimension two. The stable
configurations of stability domains of cycles on a plane of parameters, "crossroad
area" and "spring area" situations. Multistability and "multisheet"
structure of a parameter plane of multimodal mappings. From local to the global two-parameter
bifurcation analysis: examples of coexistence of bifurcation lines and points of various
cycles on a parameter plane. Dynamic regimes charts.

**3.4 Concept of critical phenomena. **Why period doubling bifurcations form a cascade?
Accumulation of period doublings. A critical point. Feigenbaum's law and universal constant.
Splitting laws for a bifurcation tree and second universal Feigenbaum's constant. Scaling.
Scaling along the axis of driving parameter. Scaling on a bifurcation tree. Period doublings
in two-parameter mappings. Bifurcation "scenarios" and routes in the parameter
plane. A situation of a codimension one associated with a requirement of an "extremum
onto extremum" mapping by an example of cubic map. The proof of presence of an
extremum of the fourth order for a twice iterated mappings. Tricritical points and the
arrangement of the charts of dynamic regimes in their neighbourhood. Two-parameter scaling.

**3.5 Two-dimensional mappings.** The fixed points of two-dimensional mappings.
Monodromy matrix and multiplicators. Examples of various types of dynamics on a phase
plane in a vicinity of the fixed point, its relationship with values of multiplicators.
Cases of the real and complex multiplicators. Stability of the fixed point. Boundaries
of stability domains and domains of the real and complex multiplicators on a plane "trace
- Jacobian" of a monodromy matrix. Cycles of two-dimensional mappings. Monodromy
matrix and multiplicators of cycles.

**3.6 Bifurcations, featured for the one-dimensional mappings in case of two dimensions.
**Mappings with constant Jacobian, restrictions on bifurcations for such mappings.
The tangential bifurcation in Henon maping. Period doubling bifurcation of the fixed
point of Henon mapping. Evolution of multiplicators on a complex plane when the parameter
is changed from a point of the tangential bifurcation up to a period doubling bifurcation.
2-cycle of Henon mapping and the analysis of its stability.

**3.7 Neimark bifurcation.** Model mapping in case of the arbitrary Jacobian. Dynamic
regimes chart for a model mapping. Attractors as invariant curves and cycles, evolution
of imaging point on these attractors. Synchronization tongues. A rotation number. Rational
and irrational rotation numbers. Examples of various rotation numbers for the same cycle
configuration. Relationship between the rotation number and argument of a complex multiplicator
on the line J=1. Stable and unstable manifolds of two-dimensional mappings. Configuration
of invariant manifolds for a cycle inside synchronization tongue. Boundary of synchronization
tongue as the tangential bifurcation. Fine structure of a plane of parameters in a neighbourhood
of a line of an invariant curve. Notes on physical realization of the regimes corresponding
to an invariant curve.

**3.8 Synchronization and circle mapping.** A qualitative explanation of transition
to circle mapping while arguing a problem of synchronization. Dynamic regimes chart
for a circle mapping. A rotation number for circle mapping. Examples of iterative diagrams
for various rational rotation numbers. How does the iterative diagram vary at an exit
through boundaries of synchronization tongue? The equation for searching cycle elements
and boundaries of synchronization tongue for the given rotation number. Boundaries of
the basic tongue.

**4.1 From two-dimensional mappings to three-dimensional flows.** Various types
of stability of limit cycles in three-dimensional space and their explanation with the
help of Poincare sections. Examples.

**4.2 The elementary bifurcations of three-dimensional flows. **Period doubling
bifurcation of limit cycles. A cycle of a "doubled" period as edge of a Mobius
strip. Period doubling cascades in a three-dimensional phase space. A bifurcation of
birth of a torus and its relationship with Neimark bifurcation. Two time scales, motion
at a torus, beats. Ergodic tori and quasiperiodic regimes. Resonant cycles at torus,
rational rotation numbers for three-dimensional systems and their relationship with
a rotation number in Poincare section. Motion at a torus and on a resonant cycle in
a projection onto coordinate plane in a phase space. Tori on the basis of cycles of
a doubled period, bifurcation manifolds of tori and the related cycles.

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