Transitions between periodic orbits and control of chaos assisted by an external force in Hamiltonian systems

We investigate the possibility of inducing transitions between periodic orbits in two-dimensional Hamiltonian systems by means of a time-localized external perturbation. We show that the amplitude of the perturbation can be approximately calculated in the limit of a delta-type force in terms of the initial and final periodic orbits. For a specific Hamiltonian, we show several numerical examples where the external perturbation, varied from delta-type to gaussian, allows transitions between specifically chosen members of families of periodic orbits. The same mechanism is then applied to move aperiodic chaotic orbits into periodic ones, presenting a new way to control chaotic behavior in Hamiltonian systems.

Authorship

R.M. Angelo

UniversidadeUniversidadeUniversidade, Instituto de Física, 'Gleb Wataghin', Campinas, SP, BrazilUniversidadeUniversidadeUniversidadeBrazilCampinas, SP, BrazilUniversidadeUniversidadeUniversidade, Instituto de Física, 'Gleb Wataghin', Campinas, SP, Brazil

K. Furuya

M.A.M. de Aguiar

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables

Figures (5)
Tables (2)

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Figure 1.
(a1)Initial p.o. of the HS family with Eⁱ = 0.0048 and target p.o. of the V3a family with E = 0.0975. (a2) The transition from the initial p.o. of the HS family to an orbit of the V3U family with energy E^t = 0.0943, with A = -0.43 and delta kick (total time elapsed: t=100); (b1) Initial p.o. of the family HS and the target p.o. of the family V4U with E = 0.2349 , (b2) H ® V4U with lower final energy (total time elapsed: t=100); (c1)Initial p.o. of the family HS and the target p.o. of the V4S family with E = 0.1058 , (c2) HS ® V4S (total time elapsed: t=100); (d1) Initial p.o. of the HS family and the target p.o. of the V4U family with E = 0.2349 , (d2) HS ® V4U (total time elapsed: t=47).

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Figure 3.
Transitions with the gaussian force in the x-direction. The initial periodic orbit V4S family has Eⁱ = 0.1058 and V3S has Eⁱ = 0.0962 . (a1)target p.o. from HU family with E = 0.1370, (a2) V4S ® HU (total time elapsed: t=40); (b1)target p.o. of the IU family with E = 0.1895, (b2) V4S ® IU (total time elapsed: t=40); (c1) Initial p.o. from the V3S family with Eⁱ = 0.0962 and target p.o. from the IU family with E = 0.1370 , (c2) V3S ® IU (total time elapsed: t=70); (d1) Initial p.o. from the V4S family with Eⁱ = 0.1058 and target p.o. from the same family with higher energy (d2) V4S ® V4S (total time elapsed: t=100). See table II for the parameter values.

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Figure 4.
The transition from a non-periodic chaotic trajectory (C) with energy Eⁱ = 0.5039 to periodic orbits. (a) the chaotic trajectory evolved up to t=100; (b) transition from the chaotic trajectory to a torus close to the p.o. of the IS family with energy E^t = 0.0447, A = 1.0000 and e = 0.01. (c) transition from the chaotic trajectory to the p.o. of the IU family with energy E^t = 0.1858, A = 1.4918 and e = 0.01; In all cases the total evolution time t=100.

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Figure 5.
The time evolution of the energy during a short-time kick with e = 0.57, much larger than the integration step dt = 0.01, corresponding to the case of Fig.2(a).

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TABLE I : H®V and H®H transitions

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TABLE II : V®H and V®V transitions

Figure 1. (a1)Initial p.o. of the HS family with Eⁱ = 0.0048 and target p.o. of the V3a family with E = 0.0975. (a2) The transition from the initial p.o. of the HS family to an orbit of the V3U family with energy E^t = 0.0943, with A = -0.43 and delta kick (total time elapsed: t=100); (b1) Initial p.o. of the family HS and the target p.o. of the family V4U with E = 0.2349 , (b2) H ® V4U with lower final energy (total time elapsed: t=100); (c1)Initial p.o. of the family HS and the target p.o. of the V4S family with E = 0.1058 , (c2) HS ® V4S (total time elapsed: t=100); (d1) Initial p.o. of the HS family and the target p.o. of the V4U family with E = 0.2349 , (d2) HS ® V4U (total time elapsed: t=47).

Figure 3. Transitions with the gaussian force in the x-direction. The initial periodic orbit V4S family has Eⁱ = 0.1058 and V3S has Eⁱ = 0.0962 . (a1)target p.o. from HU family with E = 0.1370, (a2) V4S ® HU (total time elapsed: t=40); (b1)target p.o. of the IU family with E = 0.1895, (b2) V4S ® IU (total time elapsed: t=40); (c1) Initial p.o. from the V3S family with Eⁱ = 0.0962 and target p.o. from the IU family with E = 0.1370 , (c2) V3S ® IU (total time elapsed: t=70); (d1) Initial p.o. from the V4S family with Eⁱ = 0.1058 and target p.o. from the same family with higher energy (d2) V4S ® V4S (total time elapsed: t=100). See table II for the parameter values.

Figure 4. The transition from a non-periodic chaotic trajectory (C) with energy Eⁱ = 0.5039 to periodic orbits. (a) the chaotic trajectory evolved up to t=100; (b) transition from the chaotic trajectory to a torus close to the p.o. of the IS family with energy E^t = 0.0447, A = 1.0000 and e = 0.01. (c) transition from the chaotic trajectory to the p.o. of the IU family with energy E^t = 0.1858, A = 1.4918 and e = 0.01; In all cases the total evolution time t=100.

Figure 5. The time evolution of the energy during a short-time kick with e = 0.57, much larger than the integration step dt = 0.01, corresponding to the case of Fig.2(a).

TABLE I : H®V and H®H transitions

In. Family	A	e	Fi. Family	\|A_calc\|	D^ft	Figure
HS	-0.43	0.01	V3U	-0.4	0.0089	1.(a)
HS	-0.25	0.01	V4U	-0.2	0.0047	1.(b)
HS	0.7	1.0	V4S	-	0.0028	1.(c)
HS	-1.283	1.0	V4U	-	0.2335	1.(d)

IS	0.6	0.57	V4S	0.36	0.0065	2.(a)
IS	0.5283	0.18	HS	0.44	0.0095	2.(b)
IS	0.428	0.80	IS	0.55	0.0063	2.(c)

TABLE II : V®H and V®V transitions

In. Family	A	e	Fi. Family	\|A_calc\|	D^ft	Figure
V4S	-0.57	0.1	HU	0.25	0.0077	3.(a)
V4S	-0.706	0.4	IU	0.41	0.0101	3.(b)
V3S	0.4880	0.02	IU	-	0.0193	3.(c)
V4S	-0.533	0.1	V4S	-	0.0178	3.(d)

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Transitions between periodic orbits and control of chaos assisted by an external force in Hamiltonian systems

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