^{1,a)}, Jonathan Deane

^{1,b)}and Guido Gentile

^{2,c)}

### Abstract

We consider dissipative one-dimensional systems subject to a periodic force. As a model system, particularly suited for numerical analysis, we investigate the driven cubic oscillator in the presence of friction, and study numerically how time-varying friction affects the dynamics. We find that, if the damping coefficient increases in time up to a final constant value, then the basins of attraction of the leading resonances are larger than they would have been if the coefficient had been fixed at that value since the beginning. From a quantitative point of view, the scenario depends both on the final value and the growth rate of the damping coefficient. The relevance of the results for the spin-orbit model is argued and discussed in some detail.

We are grateful to Giovanni Gallavotti for very fruitful discussions and critical remarks, especially on the spin-orbit model and the formation and evolution of celestial bodies. We are also indebted to James Wright for providing us with the results of his simulations based on Adams-Bashforth-Moulton method, as a further support to our own numerical results.

I. INTRODUCTION

II. THE DRIVEN CUBIC OSCILLATOR WITH CONSTANT FRICTION

III. THE DRIVEN CUBIC OSCILLATOR WITH INCREASING FRICTION

IV. THE SPIN-ORBIT MODEL

V. REMARKS AND COMMENTS

A. Different values of the perturbation parameter

B. Different functions

VI. CONCLUSIONS AND OPEN PROBLEMS

### Key Topics

- Friction
- 44.0
- Attractors
- 35.0
- Oscillators
- 13.0
- Numerical modeling
- 7.0
- Phase space methods
- 7.0

## Figures

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

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

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.

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.

Relative measures of the basins of attraction versus Δ for γ_{0} = 0.015.

Relative measures of the basins of attraction versus Δ for γ_{0} = 0.015.

Relative areas of the basins of attraction versus Δ for γ_{0} = 0.005.

Relative areas of the basins of attraction versus Δ for γ_{0} = 0.005.

Relative areas of the basins of attraction versus Δ for γ_{0} = 0.0005.

Relative areas of the basins of attraction versus Δ for γ_{0} = 0.0005.

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.

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.

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.

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.

Relative areas of the basins of attraction versus Δ for γ(*t*) given by (3.2), with γ_{0} = 0.006.

Relative areas of the basins of attraction versus Δ for γ(*t*) given by (3.2), with γ_{0} = 0.006.

Relative areas of the basins of attraction versus Δ for γ(*t*) given by (5.1), with γ_{0} = 0.006.

Relative areas of the basins of attraction versus Δ for γ(*t*) given by (5.1), with γ_{0} = 0.006.

## Tables

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 + *O*(ɛ^{2}).

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 + *O*(ɛ^{2}).

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)ɛ^{2} + *O*(ɛ^{3}).

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)ɛ^{2} + *O*(ɛ^{3}).

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.

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.

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 γ_{0} = 0.015 and *T* _{0} = Δ/γ_{0}, 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 .

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 γ_{0} = 0.015 and *T* _{0} = Δ/γ_{0}, 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 .

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 γ_{0} = 0.005 and *T* _{0} = Δ/γ_{0}, 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 .

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 γ_{0} = 0.005 and *T* _{0} = Δ/γ_{0}, 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 .

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 γ_{0} = 0.0005 and *T* _{0} = Δ/γ_{0}, 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 .

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 γ_{0} = 0.0005 and *T* _{0} = Δ/γ_{0}, 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 .

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

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

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.

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.

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 .

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 .

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 .

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 .

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 γ_{0} = 0.006 and *T* _{0} = Δ/γ_{0}, 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 .

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 γ_{0} = 0.006 and *T* _{0} = Δ/γ_{0}, 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 .

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 γ_{0} = 0.006 and *T* _{0} = Δ/γ_{0}, 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 .

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 γ_{0} = 0.006 and *T* _{0} = Δ/γ_{0}, 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 .

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.

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.

Values of *T* (orbital period) and γ for the systems considered in Sec. IV, with 3*k* _{2}/ξ*Q* = 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}.

Values of *T* (orbital period) and γ for the systems considered in Sec. IV, with 3*k* _{2}/ξ*Q* = 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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