Frequency locking, Quasi periodicity and Chaos due to special relativistic effects
FFrequency locking, Quasiperiodicity and Chaos due tospecial relativistic effects
Derek C. Gomes and G. Ambika
Department of Physics, Indian Institute of Science Education and Research (IISER) Tirupati ,Tirupati - 517507 , India
Abstract.
We study quasiperiodic and frequency locked states that can occur ina sinusoidally driven linear harmonic oscillator in the special relativistic regime.We show how the shift in natural frequency of the oscillator with increasing rel-ativistic effects leads to frequency locking or quasi periodicity and the chaoticstates that arise due to the increasing nonlinearity. We find the same system canhave multi-stable states in the presence of small damping. We also report an en-hancement of chaos in the relativistic H´enon-Heiles system.
Keywords: quasiperiodicity, frequency locking, relativistic harmonic oscillator,H´enon-Heiles system
Most of the studies in nonlinear dynamical systems deal with nonrelativistic regime andchaos exists in such systems due to their inherent nonlinearity. The study of chaos inrelativistic systems is an interesting area of research both for its own nature as well asfor its applications in many experimental contexts where particle oscillations occur ateffectively very high velocities [1,2]. The addition of relativistic corrections to lineardynamical systems can induce nonlinearity in them and hence has been reported re-cently to exhibit chaotic dynamics [3,4,5,6]. The classical and quantum dynamics ofkicked relativistic particle in a box and driven oscillator are also studied recently [7].The 2-dimensional relativistic anisotropic, harmonic oscillator is shown to be chaotic[3]. Also the dynamics of a quartic oscillator at relativistic energies, exhibits bifurca-tions and chaos and is found to have a transition to periodic regular motion for largeforcing [8]. The nonlinear dynamics of the constant-period oscillator under externalperiodic forcing, displays nonlinear resonances and chaos when the driving force issufficiently strong [9].While it is thus established that chaos must appear in most integrable classical sys-tems due to special relativistic corrections to the dynamics, the details of the dynamicalstates and route to chaos in them are still not fully understood. In our study of the one-dimensional forced harmonic oscillator we show how the natural frequency varies withincreasing relativistic effects. We then find that the system exhibits frequency lockedand quasiperiodic states as the natural frequency changes. We also indicate a novelroute to chaos in this system as the effects of relativistic corrections are tuned. Whendamping is present, we report that the system exhibits multi stable states due to nonlin-ear effects induced by relativistic corrections. Also, to the best of our knowledge, our a r X i v : . [ n li n . C D ] M a r Derek C. Gomes and G. Ambika study for the first time on the relativistic version of the H´enon-Heiles system, reportsan enhancement of chaotic regions in the relativistic regime.
The harmonic oscillator is a popular prototype, widely used in all branches of Physicsto understand periodic oscillations as well as to approximate a variety of vibrations inreal systems. One of the reasons for this is the extreme simplicity with which it canbe used both in classical and quantum regimes giving analytic solutions. This meansharmonic oscillator is integrable in classical mechanics and analytically solvable inquantum mechanics and this holds true in many dimensions even with damping andforcing. However, in the special relativistic regime, even the 1-d harmonic oscillator isnot integrable and we see the relativistic harmonic oscillator under forcing and dampingcan give rise to nonlinear dynamical states due to the nonlinearity introduced by specialrelativistic effects.The Hamiltonian for the relativistic forced harmonic oscillator (with rest mass unity)is H = (cid:112) p c + c + kx + xFcos ω t (1)which leads to the equations of motion dxdt = p (cid:113) + p c (2) d pdt = − kx − Fcos ω t (3)It is clear that the above system (all variables in suitable units) is nonlinear , withthe spring constant ( k ) , the forcing amplitude ( F ) and the driving frequency ( ω ). Asreported by Kim and Lee [6], we control the relativistic effects in the system by treatingthe speed of light, c , as an additional parameter. As we will see in the next section,the relativistic corrections occur as functions of p / c (where p is the momentum) anddecreasing the value of c for similar values of p effectively increases the impact ofrelativistic effects in the system. One of the main changes in the system in Eq. (1) from its non-relativistic counterpartis that its natural frequency is no longer constant with amplitude, just like a nonlinearoscillator. In order to get a better understanding of how this change occurs and thefactors that influence this, we expand Eq. (2) upto first order, to get dxdt = p ( − p / c ) (4)neglecting higher order terms since c >> p . In this limit the relativistic effects act likea perturbation as in Eq. (4). In this context, perturbation approaches can be used as requency locking, Quasiperiodicity and Chaos due to special relativistic effects 3 reported in the case of the constant period oscillator [9] using canonical perturbationtheory. We use a different(but equivalent) approach developed by Lindstedt [10] to cal-culate the effect of the perturbation on the natural frequency. Then the expression forthe frequency, ω , rel (upto first order) is : ω , rel = ω [ − c { p ( ) + ω ( x ( ) + F ω − ω ) } − ω F c ( ω − ω ) ] (5)Here ( ω ) is the natural frequency in the non-relativistic limit ( c → ∞ ). We find that theresultant frequency decreases due to the perturbation. Although we consider the case ofa weak relativistic perturbation for the analytical calculation in Eq.(5) , the approach issuggestive of the parameters that play a role in the shift in frequency. We present thiseffect more explicitly by computing the power spectra from the numerically obtainedtime series of the position variable of the system. (a) (b)(c) (d) Fig. 1:
Power spectra of x(t) for varying c . a. c = b. c = c. c =
50, the other pa-rameters are fixed as k = F = ω = x ( ) = − p ( ) = d. Poincar´e plot of thephase space showing resonances corresponding to the parameters in the power spectra, colourcoded as the red points correspond to ( a. ), the blue points to ( b. ) and the green points (enlargedfor visualization) to ( c. ) In Fig. 1 we show the power spectra for increasing relativistic effects in the system.It is clear that the natural frequency decreases, while, as expected, the driving frequency
Derek C. Gomes and G. Ambika (taken as ω =
4) remains the same. We note that the greater the relativistic correction(i.e, smaller the parameter c ), the smaller is the natural frequency corresponding to thelargest peak. The dynamical states corresponding to the three values of c used, are clearfrom the Poincar´e plots in Fig 1d . Fig. 2:
Shift in the natural frequency of the relativistic harmonic oscillator as a function of c By tuning c as a parameter, we compute the shift in natural frequency from thepower spectra and plot this shift with c in Fig. 2. The solid curve in the figure showsthe numerical fit to the variation of shift in frequency( ω − ω , rel ) ≈ ω ( + ( c / a ) ) − ,where ω =
1. The fitting parameter, a = . The fact that the natural frequency continuously decreases makes it possible for thesystem to get into rational and irrational ratios with the driving frequency and thereforethe dynamics also leads to frequency locked and quasiperiodic states. The frequencylocked states corresponding to rational ratios and the values of c at which they occurare clear from the plateaus in the Devil’s Staircase plot in Fig. 3. The most prominentplateau is for w , rel = .
8, with the driving frequency at 4 giving a ratio of 5. As thenatural frequency continuously decreases, the ratios 6 , 7 , 8 etc. also appear. Also it isequally likely that as the ratios of frequencies vary due to changes in c , they enter intoirrational ratios and hence can give rise to quasiperiodic dynamics in the system. requency locking, Quasiperiodicity and Chaos due to special relativistic effects 5 Fig. 3:
Devil’s staircase plot of natural frequency as c is varied ,keeping forcing amplitude con-stant .The other parameters are the same as in Fig.1 The structure of the trajectories corresponding to both types of dynamical statesdiscussed above are clear from the Poincar´e plots that show trajectories in phase spacesampled at the driving frequency. The resonances corresponding to the rational ratiosof frequencies are shown in Fig. 4. As the natural frequency decreases with decrease in c , the ratio of frequencies and hence the number of resonances also increase. We note (a) (b) Fig. 4:
Poincar´e plots showing frequency locked states with their islands a. c = b. c =
200 , N = 6 that the odd numbered (corresponding to odd ratio) resonances are more prominent thantheir even-numbered counterparts. The occurrence of only odd-numbered resonances
Derek C. Gomes and G. Ambika (a) (b)
Fig. 5:
Poincar´e plots showing quasiperiodic states. In a. c =
175 and in b. c = (up to first order) has been attributed to the symmetry of the potential in the earlierwork on the relativistic driven harmonic oscillator [6] as well as in similar relativisticsystems [9]. We observe the even-numbered resonances arise from higher-order effectsbut are confined to very small regions of phase space. The continuous trajectories shownin Fig.5 indicate occurrence of quasiperiodic states for two different values of c . We observe the occurrence of chaos induced by relativistic effects in regions confinedbetween the other types of trajectories . We plot in Fig. 6 the Poincar´e plots of thephase space of the system with the same set of initial conditions and parameters butreducing the value of c . It is clear from Fig 6 that as c decreases the chaotic regionsappear in phase space due to disappearance of the even-resonance states. Thus in Fig.6b. the chaotic trajectory occurs near the N=10 resonance while the nearby trajectoriesof N=9 and N=11 resonances are still preserved. In Fig 6c. where c is decreased evenfurther, only odd resonances are visible in between chaotic regions. To further visualizechaos in this system we plot in Fig. 6d. the power spectrum corresponding to a chaotictrajectory in Fig. 6b. requency locking, Quasiperiodicity and Chaos due to special relativistic effects 7 (a) (b)(c) (d) Fig. 6:
Poincar´e plots of the phase space trajectories indicating the appearance of chaotic regionsas c is decreased a. c = b. c = c. c = d. Power spectrum for a chaotic trajectory from b In this section we consider the damped harmonic oscillator in the relativistic regimewith the following equations dxdt = p (cid:113) + p c ; d pdt = − kx − bp − Fcoswt (6)where the additional term is − bp , with b as the damping parameter. We find that rela-tively small damping in the system, with relativistic effects leads to multi-stable states,very different from the damped driven nonrelativistic oscillator. This is seen in Fig. 7where the phase space structure for the same set of 20 different initial conditions areshown for the undamped nonrelativistic, damped nonrelativistic and damped relativisticcases. In the damped relativistic case, we find the system settles to four different statesor attractors. To confirm multi-stability, we present the basin structure corresponding tothese four attractors. Derek C. Gomes and G. Ambika (a) (b)(c) (d)
Fig. 7:
Phase space trajectories starting from the same set of 20 initial conditions for a. undampednonrelativistic b. , damped nonrelativistic c. damped relativistic oscillator. Here b = .
01 and c = d. we plot the basin structure for the system where the four different colors correspondto the basins of the four different attractors in the damped relativistic case in c. (the colors black,blue, red and green represent the smallest to largest attractor respectively) As another system of interest, we study the effects of special relativity in an intrinsicallynonlinear and nonintegrable system, the H´enon-Heiles system. This system models thestellar motion about a galactic center and is known to have a rich dynamics [11]. Thepotential of the system models the galactic potential centred around the galactic centerin the x − y plane given by V ( x , y ) = ( x + y ) + λ ( x y − y ) (7)The relativistic Hamiltonian is H = (cid:112) p c + c − c + V ( x , y ) (8)where rest mass is unity and we have subtracted the rest energy( c ) from the Hamil-tonian as it is usually not considered in measuring energy in the non-relativistic case.In general, this has no significance in the equations of motion and as the total energy requency locking, Quasiperiodicity and Chaos due to special relativistic effects 9 E is treated as a parameter in studying the H´enon-Heiles system [11] and for makingcomparison with well-studied non-relativistic case, it is important to subtract this con-tribution. We note that in the limit as c → ∞ the Hamiltonian in Eq. (7) reduces to thenonrelativistic one. Our main result in this study is that adding relativistic correctionsleads to enhanced chaos in the system. This is clear from the phase space structuresshown in the y − p y plane of phase space in Fig. 8 for both non-relativistic and the cor-responding relativistic regimes for three different values of E (we take λ = p y corresponds to momentum conjugate to y ). (a) (b)(c) (d)(e) (f) Fig. 8:
Phase space structure in the y − p y plane for the relativistic H´enon-Heiles system (right)compared with that of the non-relativistic system(left) a. E = .
10, non-relativistic b. E = . c. E = . d. E = . e. E = / f. E = /
6, relativistic, where relativistic cases correspond to c = .
30 Derek C. Gomes and G. Ambika
We report the different dynamical states induced by special relativistic effects in a har-monic oscillator with sinusoidal forcing. We present perturbation methods and numeri-cal computations to show how the relativistic corrections shift the natural frequency andconsequently generate frequency locked and quasiperiodic states. We also show how re-ducing the value of c can lead to chaos with the disappearance of even resonances. In thepresence of small damping, the forced harmonic oscillator exhibits multi-stable stateswith an interesting basin structure. In an inherent nonlinear system like H´enon-Heilessystem, we report the enhancement of chaos as the relativistic effects are tuned. References
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