Universal two-dimensional map with various bifurcation scenarios

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Model map

was introduced by A.P. Kuznetsov and S.P. Kuznetsov as an example manifesting all basic variants of a limit cycle stability loss and associated scenarios of transition to chaos. The map contains three parameters, S, J and e. S and J represent trace and determinant of Jacobian matrix of linearization at the fixed point x=y=0. This point may be thought as intersection of a closed orbit (limit cycle) of a flow system with a surface of cross-section of Poincare. Parameter e is not so essential for qualitative behavior and will be regarded as a constant: e=0.35.

There are three typical situations of the limit cycle stability loss:

one of the multipliers (Floquet eigenvalues) m becomes equal – 1, while the secont remains less than 1 in modulus (period-doubling bifurcation);

one of the multipliers m becomes equal 1, while the second is less than 1 in modulus (saddle-node bifurcation that may lead e.g. to intermittency);

two complex conjugate multipliers become equal to 1 in modulus: |m|=1, i.e. arrive at the unit circle (Neimark - Sacker bifurcation, which gives rise to a birth of quasiperiodic motion or, at resonance frequency ratios, to a birth of a new limit cycle).

By linearization at the fixed point at the origin the map becomes

(In fact, our model map was constructed from this relation by adding the nonlinear quadratic terms.) Multipliers of the fixed point appear as eigenvalues of the Jacobian matrix

from the quadratic equation

The mentioned above bifurcation conditions correspond to three sides of the triangle representing the stability domain on the plane (S, J):

The left picture shows a general view of the chart of regimes for the map on the parameter plane (S, J), and the right one presents a magnified fragment of the chart.

Regions of periodic dynamics are shown by colors, and periods are indicated by numbers. Black regions correspond to non-periodic regimes, including quasiperiodic behavior and chaos. In the diagram one can see a green stability triangle (period 1). At its left side the period-doubling bifurcation takes place with subsequent period-doubling cascade and Feigenbaum transition to chaos. The right side corresponds to a hard transition via the saddle-node bifurcation, and the top side to Neimark - Sacker bifurcation and birth of the quasiperiodicity. Inside the domain of quasiperiodicity synchronization tongues take place corresponding to rational ratios of the components of the motion. Vertices are associated with bifurcation situations of codimension 2, where the multipliers become equal (+1,+1), (+1,-1), (-1,-1).

The magnified fragment of the chart shows details of structure of the synchronization tongue 1:4. Notice a difference with the case of tongues in a standard circle map: the period-doubling bifurcation curves are not placed inside the tongue, but stick its edges, at some codimension-2 points. It appears that in a course of the period-doubling bifurcation cascade they tend to the critical point classified as C-type point: S=-0.548966, J=1.547188.

About stability triangle see S.P. Kuznetsov. Dynamical Chaos (in Russian) , p. 213. Construction, description and properties of the universal map see in the paper of A.P. Kuznetsov, A.Yu. Kuznetsova, and I.R. Sataev "On critical behavior of a map with the Neimark - Sacker bifurcation at destruction of the phase synchronization at the limit point of Feigenbaum cascade" (in Russian), 2003.

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