Bifurcation and dynamics of a normal form map

Reza KHOSHSIAR GHAZIANI

Article Details

Bifurcation and dynamics of a normal form map

Authors:

Reza KHOSHSIAR GHAZIANI

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2015 Vol.39 Issue 1, pp.23-36

Document Type:

Article

Publication Date:

3/2015

Document Language:

Turkish

Document ID (Org No.):

2000747220

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098

Keywords:

Abstract:

 

This paper investigates the dynamics and stability properties of a so-called planar truncated normal form map. This kind of map is widely used in the applied context, especially in normal form coefficients of n-dimensional maps. We determine analytically the border collision bifurcation curves that characterize the dynamic behaviors of the system. We first analyze stability of the fixed points and the existence of local bifurcations. Our analysis shows the presence of a rich variety of local bifurcations, namely stable fixed points, periodic cycles, quasiperiodic cycles that are constraints to stable attractors called invariant closed curves, and chaos, where dynamics of the system change erratically. Our study is based on the numerical continuation method under variation of 1 and 2 parameters and computation of different bifurcation curves of the system and its iterations. For the all codimension 1 and 2 bifurcation points, we compute the corresponding normal form coefficients to reveal the criticality of the corresponding bifurcations as well as to identify different bifurcation curves that emerge around the corresponding bifurcation point. We further perform numerical simulations to characterize qualitatively different dynamical behaviors within each regime of parameter space.

 

This paper investigates the dynamics and stability properties of a so-called planar truncated normal form map. This kind of map is widely used in the applied context, especially in normal form coefficients of n-dimensional maps. We determine analytically the border collision bifurcation curves that characterize the dynamic behaviors of the system. We first analyze stability of the fixed points and the existence of local bifurcations. Our analysis shows the presence of a rich variety of local bifurcations, namely stable fixed points, periodic cycles, quasiperiodic cycles that are constraints to stable attractors called invariant closed curves, and chaos, where dynamics of the system change erratically. Our study is based on the numerical continuation method under variation of 1 and 2 parameters and computation of different bifurcation curves of the system and its iterations. For the all codimension 1 and 2 bifurcation points, we compute the corresponding normal form coefficients to reveal the criticality of the corresponding bifurcations as well as to identify different bifurcation curves that emerge around the corresponding bifurcation point. We further perform numerical simulations to characterize qualitatively different dynamical behaviors within each regime of parameter space.

Article Details

Bifurcation and dynamics of a normal form map

Authors:

Reza KHOSHSIAR GHAZIANI

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2015 Vol.39 Issue 1, pp.23-36

Document Type:

Article

Publication Date:

3/2015

Document Language:

Turkish

Document ID (Org No.):

2000747220

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098

Normal form coefficients

stable fixed point

chaotic behavior

invariant curve

Issues

Article Details

Bifurcation and dynamics of a normal form map

Authors: Reza KHOSHSIAR GHAZIANI

Publisher: TUBITAK

Journal: Turkish Journal of Mathematics 2015 Vol.39 Issue 1, pp.23-36

Document Type: Article

Publication Date: 3/2015

Document Language: Turkish

Document ID (Org No.): 2000747220

Frequency: Bimonthly

eISSN: 1303-6149

pISSN: 1300-0098

Normal form coefficients

stable fixed point

chaotic behavior

invariant curve

Issues

Authors:

Reza KHOSHSIAR GHAZIANI

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2015 Vol.39 Issue 1, pp.23-36

Document Type:

Article

Publication Date:

3/2015

Document Language:

Turkish

Document ID (Org No.):

2000747220

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098