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Attractiveness of periodic orbits in parametrically forced systems with time-increasing friction
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10.1063/1.4757650
/content/aip/journal/jmp/53/10/10.1063/1.4757650
http://aip.metastore.ingenta.com/content/aip/journal/jmp/53/10/10.1063/1.4757650

Figures

Image of FIG. 1.
FIG. 1.

Relative areas A(ω, γ) of the basins of attraction versus log γ for the values of γ listed in Table III.

Image of FIG. 2.
FIG. 2.

Relative areas A(ω, γ) of the basins of attraction versus log γ: a magnification of Figure 1 for the periodic orbits with ω = 1/4, 1, 1/6, 1/3, 3/8.

Image of FIG. 3.
FIG. 3.

Relative measures of the basins of attraction versus Δ for γ0 = 0.015.

Image of FIG. 4.
FIG. 4.

Relative areas of the basins of attraction versus Δ for γ0 = 0.005.

Image of FIG. 5.
FIG. 5.

Relative areas of the basins of attraction versus Δ for γ0 = 0.0005.

Image of FIG. 6.
FIG. 6.

Basins of attraction determined numerically for the 1:2 resonance for constant γ (gray; red online) and time-varying γ (black plus most of the gray/red region). Note that the gray/red region has priority over black and over white, so parts of the basin of attraction for the 1:2 resonance, γ varying, are obscured. Initial conditions in the white region either go to the origin or to the 1:4 resonance.

Image of FIG. 7.
FIG. 7.

Basins of attraction determined numerically for the 1:2 resonance for constant γ (gray; red online) and time-varying γ (black plus most of the gray/red region). Note that the gray/red region has priority over black and over white, so parts of the basin of attraction for the 1:2 resonance, γ varying, are obscured. Initial conditions in the white region either go to the origin or to the 1:4 resonance. This figure shows a magnified portion of Figure 6.

Image of FIG. 8.
FIG. 8.

Relative areas of the basins of attraction versus Δ for γ(t) given by (3.2), with γ0 = 0.006.

Image of FIG. 9.
FIG. 9.

Relative areas of the basins of attraction versus Δ for γ(t) given by (5.1), with γ0 = 0.006.

Tables

Generic image for table
Table I.

Values of the constants C 0(p/q) for p = 1 and q = 2, 4, 6, 8, 10 (leading primary resonances) for the cubic oscillator (2.1); the threshold values are of the form γ(ω, ɛ) = C 0(ω)ɛ + O2).

Generic image for table
Table II.

Values of the constants C 0(p/q) for p = 1 and q = 1, 3, 5, 7, 9 (leading secondary resonances) for the cubic oscillator (2.1); the threshold values are of the form γ(ω, ɛ) = C 0(ω)ɛ2 + O3).

Generic image for table
Table III.

Numerical results for the relative areas A(ω, γ), %, of the parts of the basins of attraction contained inside the square for ɛ = 0.1 and some values of γ. The attractors are identified by the corresponding frequency (0 is the origin). The number of random initial conditions taken in is 1 000 000 up to γ = 0.0001, 500 000 for γ = 0.00005, 150 000 for γ = 0.00001, and 50 000 for γ = 0.000005.

Generic image for table
Table IV.

Numerical results for the relative areas A(ω, 0.015; Δ) of the parts of the basins of attraction contained inside for ɛ = 0.1 and γ(t) given by (3.2) with γ0 = 0.015 and T 0 = Δ/γ0, for various values of Δ and ω = 0, 1/2 (ω = 0 is the origin). In each case, 1 000 000 random initial conditions have been taken in .

Generic image for table
Table V.

Numerical results for the relative areas A(ω, 0.005; Δ) of the parts of the basins of attraction contained inside the square for ɛ = 0.1 and γ(t) given by (3.2) with γ0 = 0.005 and T 0 = Δ/γ0, for various values of Δ and ω = 0, 1/2, 1/4 (ω = 0 is the origin). 500 000 random initial conditions have been taken in .

Generic image for table
Table VI.

Numerical results for the relative areas A(ω, 0.0005; Δ) of the parts of the basins of attraction contained in for ɛ = 0.1 and γ(t) given by (3.2) with γ0 = 0.0005 and T 0 = Δ/γ0, for various values of Δ and ω = 0, 1/2, 1/4, 1, 1/6, 1/3 (ω = 0 is the origin). 250 000 random initial conditions have been taken in .

Generic image for table
Table VII.

Values of the constants e, ɛ, and γ for some cases of physical interest for the spin-orbit model (4.1).

Generic image for table
Table VIII.

Values of the constants C 0(p/q) for some primary resonances of the the spin-orbit model (4.1); the threshold values are of the form γ(ω, ɛ) = C 0(ω)ɛ. Only positive ω have been explicitly considered.

Generic image for table
Table IX.

Numerical results for the relative areas of the parts of the basins of attraction contained inside the square for ɛ = 0.5. (ω = 0 denotes the origin). 1 000 000 random initial conditions have been taken in .

Generic image for table
Table X.

Numerical results for the relative areas of the parts of the basins of attraction contained inside the square for ɛ = 0.01. (ω = 0 denotes the origin). 500 000 random initial conditions have been taken in .

Generic image for table
Table XI.

Numerical results for the relative areas A(ω, 0.006, 0.1; Δ) of the parts of the basins of attraction contained inside for ɛ = 0.1 and γ(t) given by (3.2) with γ0 = 0.006 and T 0 = Δ/γ0, for various values of Δ and ω = 0, 1/2, 1/4. (ω = 0 denotes the origin). 500 000 random initial conditions have been taken in .

Generic image for table
Table XII.

Numerical results for the relative areas A(ω, 0.006, 0.1; Δ) of the parts of the basins of attraction contained inside for ɛ = 0.1 and γ(t) given by (5.1) with γ0 = 0.006 and T 0 = Δ/γ0, for various values of Δ and ω = 0, 1/2, 1/4 (ω = 0 denotes the origin). 500 000 random initial conditions have been taken in .

Generic image for table
Table XIII.

Values of ω T (angular velocity), M (satellite mass), M 0 (primary mass), R (satellite radius) and ρ (mean distance between satellite and primary) for the systems considered in Sec. IV. CGS units are used.

Generic image for table
Table XIV.

Values of T (orbital period) and γ for the systems considered in Sec. IV, with 3k 2Q = 0.1 for the systems with Jupiter and Saturn as primary. In the third column γ is computed by using T as time unit, whereas the fourth column gives the value of the damping coefficient expressed in years−1.

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/content/aip/journal/jmp/53/10/10.1063/1.4757650
2012-10-16
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Attractiveness of periodic orbits in parametrically forced systems with time-increasing friction
http://aip.metastore.ingenta.com/content/aip/journal/jmp/53/10/10.1063/1.4757650
10.1063/1.4757650
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