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A method of time-frequency analysis (TFA) based on wavelets is applied to study the phase space structure of three-dimensional asymmetric triaxial galaxy enclosed by spherical dark halo component. The investigation is carried out in the presence and absence of dark halo component. Time-frequency analysis is based on the extraction of instantaneous frequency from the phase of the continuous wavelet transform. This method is comparatively fast and reliable.
This method can differentiate periodic from quasi-periodic, chaotic sticky from chaotic non-sticky, ordered from chaotic and also, it can accurately determine the time interval of the resonance trapping and transitions too.
Apart from that, the phenomenon of transient chaos can be explained with the help of time-frequency analysis. Comparison with the method of total angular momentum (denoted as L_{tot}) proposed recently is also presented.

We know that the phase space of nonlinear dynamical systems consists of periodic, quasi-periodic and chaotic trajectories. Chaotic trajectories visit resonance islands, remain there for some time and then escape to the chaotic region during its evolution. To know the time interval of resonance trapping and resonance transition and to visualize the phenomenon of transient chaos are some important questions which compel us to study more about the application of different chaos indicators. Over the last few years, several chaos indicators have been introduced to study those aspects. Moreover, for dynamical system of two degrees of freedom, there are several chaos indicators such as the Poincare Surface of Section (PSS), Largest Lyapunov Characteristic Exponent (LLCE), Smaller Alignment Index (SALI). Fast Liapunov Indicators (FLI), the Generalized Alignment Index (GALI) and the Correlation Dimension (CD) (see [

We have organized the paper as follows:

In Section 2, we have given a brief description of asymmetric triaxial galaxy enclosed by spherical dark halo component (3D). In Section 3, a brief description of TFA based on the phase of CWT and its implementation in Matlab are given. Results and discussion based on the application of TFA to the three-dimensional galactic model are shown in Section 4. The conclusion is given in Section 5.

The potential for triaxial galaxy enclosed by spherical dark halo (see [

where

and

Equation (2) denotes a triaxial galaxy with a bulge and a small asymmetry introduced by the term

The Hamiltonian of the potential given by Equation (1) can be expressed as

where

where

Unit of length = 1 kpc;

Unit of mass = 2.325 ´ 10^{7} M_{e};

Unit of time = 0.97748 ´ 10^{8} yr;

Unit of velocity = 10 km×s^{-}^{1};

Unit of energy (per unit mass) = 100 km^{2}×s^{-}^{2};

G = 1 (gravitational constant).

While integrating the equations of motion in (4) for the computation of all the orbits, we use the fixed value of

Total angular momentum for a star of mass

where

Note:

Time-frequency analysis based on phase of continuous wavelet transform is described in this Section. At first we define continuous wavelet transform, instantaneous frequency and the mother wavelet. The continuous wavelet transform is defined in terms of

The function

Here,

Note: The parameter

serves the purpose. Here due to the part

ing to frequency. Due to this unique feature i.e. capability of adaptation of time window according to frequency range gives better localization in frequency and time.

Let us consider an analytic signal

Now it’s unique polar representation is

where

and

where R and Im denote the real and imaginary part of the signal. Also a unique representation of

Instantaneous frequency is defined as

The Ridge of the wavelet transform of

The algorithm for computing ridges from the phase of continuous wavelet transform is already explained in [

In this Section, we analyze the data of

Initial condition | Mass length (Mh) | Scale length (Ch) | Energy const. (h_{3}) | Ridge-plot (in sec) | Type of orbit | |
---|---|---|---|---|---|---|

(−9.55, 0, 0.1, 0, 5.8029, 0) | 10000 | 18 | 68 | 4.32 | 14.55 | Periodic |

(5.5, 0, 0.6, 0, 20.4251, 0) | 10000 | 18 | 68 | 4.19 | 14.58 | Quasi-Periodic |

(−0.5, 0, 0.85, 0, 27.9848, 0) | 10000 | 18 | 68 | 4.30 | 15.01 | Chaotic-sticky |

(−0.9, 0, 0.85, 0, 27.6788, 0) | 10000 | 18 | 68 | 4.44 | 14.69 | Chaotic nonsticky |

(0.7, 0, 0.85, 0, 27.8624, 0) | 10000 | 18 | 68 | 4.31 | 31.57 | Transient chaos and |

Resonance trapping |

equation of motion (4) at the given initial conditions using Runge-Kutta (4/5) variable step-size Integrator. Phase-portrait and

Initial condition | Mass mass (Mh) | Scale length (Ch) | Energy const. (h_{3}) | Ridge-plot (in sec) | Type of orbit | |
---|---|---|---|---|---|---|

(−7.5, 0, −1, 0, 7.4008, 0) | 0 | 8 | 516 | 4.23 | 14.57 | Periodic |

(3, 0, 0.5, 0, 20.5996, 0) | 0 | 8 | 516 | 4.33 | 14.55 | Quasi-periodic |

(0.1, 0, 0.5, 0, 24.5701, 0) | 0 | 8 | 516 | 4.25 | 14.58 | Chaotic sticky |

(1.7, 0, −1, 0, 22.2880, 0) | 0 | 8 | 516 | 4.31 | 15.01 | Chaotic non-sticky |

(0.01, 0, 0.1, 0, 24.8801, 0) | 0 | 8 | 516 | 4.37 | 32.67 | Resonant transition |

and Transient chaos |

At first, we discuss the results obtained using the Poincare surface of section in

We have selected a sample of five representative orbits for both cases. Time interval considered for the TFA is (1, 32768) except two figures (

Note: In [

Sample of orbits considered in

Now we consider the orbits at initial conditions (−0.5, 0, 0.85, 0, 27.9848, 0) and (0.1, 0, 0.5, 0, 24.5701, 0) are shown in

Now, we consider the sample of trajectories at initial conditions (−0.9, 0, 0.85, 0, 27.6788, 0) and (1.7, 0, −1, 0, 22.2880, 0) presented in

as chaotic non-sticky.

At last, we consider sample of two orbits at initial conditions (0.7, 0, 0.85, 0, 27.8624, 0) and (0.01, 0, 0.1, 0, 24.8801, 0) shown in

phenomenon as in the previous case and hence we can say that these trajectories are chaotic. But in case of Ridge-plots scene is little different. We observe following things in the Ridge-plots:

1) Resonance trapping: In

2) Resonance transition: In

3) Transient Chaos: We know that in a time-dependent system a chaotic trajectory move from regular to chaos and vice versa (see [

As we have already discussed, the aim of the present work is to show the advantage of TFA in comparison to TFA and to explore some additional information of phase space structures of asymmetric triaxial galaxies in the presence and absence of the spherical dark halo component. Based on the discussion of Section 4, we can conclude that TFA has several advantages in comparison to TAM. We conclude following things:

1) TFA based on wavelets is comparatively fast and more reliable in comparison to TAM (C.P.U time taken for the computation of Ridge-plots for 32,768-time units is 15 seconds (maximum) whereas the time taken by TAM for 150-time units is 5 seconds (maximum)).

2) TFA can identify between periodic and quasi-periodic, chaotic sticky and non-sticky, and ordered and chaotic motion.

3) With the help of TFA, we can accurately determine and also visualize the event of trapping of a chaotic trajectory around resonance island (see

4) The phenomenon of resonance transition and transient chaos can also be explained with the help of Ridge-plot (see

5) Computational effort needed for programming of TFA based on wavelets is not easy in comparison to TAM. This is an important drawback of TFA based on wavelets. But once it is done, we can perform other computational works in comparatively negligible time.

6) We always search for an indicator which is applicable to higher-dimensional nonlinear dynamical systems. TFA is independent of the degree of freedom and requires the only solution of equations of motion which can be computed. Our present work is also an important example of the application to higher dimensional systems.

Thus, we can say that Time-frequency analysis based on wavelets can be given preference for the study of nonlinear dynamical systems for two or more degrees of freedom.