arXiv subjects

Sang-Yoon Kim

Publications and source records attributed to Sang-Yoon Kim.

52 records · Page 3Linked to original sources

arXiv1999

Nonlinear Dynamics of A Damped Magnetic Oscillator

We consider a damped magnetic oscillator, consisting of a permanent magnet in a periodically oscillating magnetic field. A detailed investigation of the dynamics of this dissipative magnetic system is made by varying the field amplitude $A$. As $A$ is increased, the damped magnetic oscillator, albeit simple looking, exhibits rich dynamical behaviors such as symmetry-breaking pitchfork bifurcations, period-doubling transitions to chaos, symmetry-restoring attractor-merging crises, and saddle-node bifurcations giving rise to new periodic attractors. Besides these familiar behaviors, a cascade of ``resurrections'' (i.e., an infinite sequence of alternating restabilizations and destabilizations) of the stationary points also occurs. It is found that the stationary points restabilize (destabilize) through alternating subcritical (supercritical) period-doubling and pitchfork bifurcations. We also discuss the critical behaviors in the period-doubling cascades.

chao-dyn↗

arXiv1999

Bicritical Behavior of Period Doublings in Unidirectionally-Coupled Maps

We study the scaling behavior of period doublings in two unidirectionally-coupled one-dimensional maps near a bicritical point where two critical lines of period-doubling transition to chaos in both subsystems meet. Note that the bicritical point corresponds to a border of chaos in both subsystems. For this bicritical case, the second response subsystem exhibits a new type of non-Feigenbaum critical behavior, while the first drive subsystem is in the Feigenbaum critical state. Using two different methods, we make the renormalization group analysis of the bicritical behavior and find the corresponding fixed point of the renormalization transformation with two relevant eigenvalues. The scaling factors obtained by the renormalization group analysis agree well with those obtained by a direct numerical method.

chao-dyn↗

arXiv1998

Critical Behavior of Period Doublings in Coupled Inverted Pendulums

We study the critical behaviors of period doublings in N (N=2,3,4,...) coupled inverted pendulums by varying the driving amplitude $A$ and the coupling strength $c$. It is found that the critical behaviors depend on the range of coupling interaction. In the extreme long-range case of global coupling, in which each inverted pendulum is coupled to all the other ones with equal strength, the zero-coupling critical point and an infinity of critical line segments constitute the same critical set in the $A-c$ plane, independently of $N$. However, for any other nonglobal-coupling cases of shorter-range couplings, the structure of the critical set becomes different from that for the global-coupling case, because of a significant change in the stability diagram of periodic orbits born via period doublings. The critical scaling behaviors on the critical set are also found to be the same as those for the abstract system of the coupled one-dimensional maps.

chao-dyn↗

arXiv1998

Intermittency in coupled maps

Using a renormalization method, we study the critical behavior for intermittency in two coupled one-dimensional (1D) maps. We find two fixed maps of the renormalization transformation. They all have common relevant eigenvalues associated with scaling of the control parameter of the uncoupled 1D map. However, the relevant ``coupling eigenvalues'' associated with coupling perturbations vary depending on the fixed maps. It is also found that the two fixed maps are associated with the critical behavior in the vicinity of a critical line segment. One fixed map with no relevant coupling eigenvalues governs the critical behavior at interior points of the critical line segment, while the other one with relevant coupling eigenvalues governs the critical behavior at both ends. The results of the two coupled 1D maps are also extended to many globally-coupled 1D maps, in which each 1D map is coupled to all the other ones with equal strength.

chao-dyn↗

arXiv1998

Renormalization analysis of intermittency in two coupled maps

The critical behavior for intermittency is studied in two coupled one-dimensional (1D) maps. We find two fixed maps of an approximate renormalization operator in the space of coupled maps. Each fixed map has a common relavant eigenvaule associated with the scaling of the control parameter of the uncoupled one-dimensional map. However, the relevant ``coupling eigenvalue'' associated with coupling perturbation varies depending on the fixed maps. These renormalization results are also confirmed for a linearly-coupled case.

chao-dyn↗

arXiv1998

Bifurcations and Transitions to Chaos in An Inverted Pendulum

We consider a parametrically forced pendulum with a vertically oscillating suspension point. It is well known that, as the amplitude of the vertical oscillation is increased, its inverted state (corresponding to the vertically-up configuration) undergoes a cascade of ``resurrections,'' i.e., it becomes stabilized after its instability, destabilize again, and so forth ad infinitum. We make a detailed numerical investigation of the bifurcations associated with such resurrections of the inverted pendulum by varying the amplitude and frequency of the vertical oscillation. It is found that the inverted state stabilizes via alternating ``reverse'' subcritical pitchfork and period-doubling bifurcations, while it destabilizes via alternating ``normal'' supercritical period-doubling and pitchfork bifrucations. An infinite sequence of period-doubling bifurcations, leading to chaos, follows each destabilization of the inverted state. The critical behaviors in the period-doubling cascades are also discussed.

chao-dyn↗

arXiv1997

Symmetry and Dynamics of A Magnetic Oscillator

We consider a permanent magnetic dipole in an oscillating magnetic field. This magnetic oscillator has two dynamical symmetries. With increasing the amplitude $A$ of the magnetic field, dynamical behaviors associated with the symmetries are investigated. For small $A$, there exist symmetric states with respect to one of the two symmetries. However, such symmetric states lose their symmetries via symmetry-breaking pitchfork bifurcations and then the symmetry-broken states exhibit period-doubling transitions to chaos. Consequently, small chaotic attractors with broken symmetries appear. However, as $A$ is further increased they merge into a large symmetric chaotic attractor via symmetry-restoring attractor-merging crisis.

chao-dyn↗

arXiv1997

Experimental Study of Bifurcations in A Parametrically Forced Pendulum

An experimental study of bifurcations associated with stability of stationary points (SP's) in a parametrically forced magnetic pendulum and a comparison of its results with numerical results are presented. The critical values for which the SP's lose or gain their stability are experimentally measured by varying the two parameters $Ω$ (the normalized natural frequency) and $A$ (the normalized driving amplitude). It is observed that, when the amplitude $A$ exceeds a critical value, the normal SP with $θ=0$ ($θ$ is the angle between the permanent magnet and the magnetic field) becomes unstable either by a period-doubling bifurcation or by a symmetry-breaking pitchfork bifurcation, depending on the values of $Ω$. However, in contrast with the normal SP the inverted SP with $θ=π$ is observed to become stable as $A$ is increased above a critical value by a pitchfork bifurcation, but it also destabilizes for a higher critical value of $A$ by a period-doubling bifurcation. All of these experimental results agree well with numerical results obtained using the Floquet theory.

chao-dyn↗

arXiv1996

Period p-tuplings in coupled maps

We study the critical behavior (CB) of all period $p$-tuplings $(p \!=\!2,3,4,\dots)$ in $N$ $(N \!=\! 2,3,4,\dots)$ symmetrically coupled one-dimensional maps. We first investigate the CB for the $N=2$ case of two coupled maps, using a renormalization method. Three (five) kinds of fixed points of the renormalization transformation and their relevant ``coupling eigenvalues'' associated with coupling perturbations are found in the case of even (odd) $p$. We next study the CB for the linear- and nonlinear-coupling cases (a coupling is called linear or nonlinear according to its leading term), and confirm the renormalization results. Both the structure of the critical set (set of the critical points) and the CB vary according as the coupling is linear or nonlinear. Finally, the results of the two coupled maps are extended to many coupled maps with $N \geq 3$, in which the CB depends on the range of coupling.

chao-dyn↗

arXiv1996

Period Doublings in Coupled Parametrically Forced Damped Pendulums

We study period doublings in $N$ $(N=2,3,4, \dots)$ coupled parametrically forced damped pendulums by varying $A$ (the amplitude of the external driving force) and $c$ (the strength of coupling). With increasing $A$, the stationary point undergoes multiple period-doubling transitions to chaos. We first investigate the two-coupled case with $N=2$. For each period-doubling transition to chaos, the critical set consists of an infinity of critical line segments and the zero-coupling critical point lying on the line $A=A^*_i$ in the $A-c$ plane, where $A^*_i$ is the $i$th transition point for the uncoupled case. We find three kinds of critical behaviors, depending on the position on the critical set. They are the same as those for the coupled one-dimensional maps. Finally, the results of the $N=2$ case are extended to many-coupled cases with $N \geq 3$, in which the critical behaviors depend on the range of coupling.

chao-dyn↗

arXiv1995

Multiple Transitions to Chaos in a Damped Parametrically Forced Pendulum

We study bifurcations associated with stability of the lowest stationary point (SP) of a damped parametrically forced pendulum by varying $ω_0$ (the natural frequency of the pendulum) and $A$ (the amplitude of the external driving force). As $A$ is increased, the SP will restabilize after its instability, destabilize again, and so {\it ad infinitum} for any given $ω_0$. Its destabilizations (restabilizations) occur via alternating supercritical (subcritical) period-doubling bifurcations (PDB's) and pitchfork bifurcations, except the first destabilization at which a supercritical or subcritical bifurcation takes place depending on the value of $ω_0$. For each case of the supercritical destabilizations, an infinite sequence of PDB's follows and leads to chaos. Consequently, an infinite series of period-doubling transitions to chaos appears with increasing $A$. The critical behaviors at the transition points are also discussed.

chao-dyn↗

arXiv1994

Extension of ``Renormalization of period doubling in symmetric volume-preserving maps''

We numerically reexamine the scaling behavior of period doublings in four-dimensional volume-preserving maps in order to resolve a discrepancy between numerical results on scaling of the coupling parameter and the approximate renormalization results reported by Mao and Greene [Phys. Rev. A {\bf 35}, 3911 (1987)]. In order to see the fine structure of period doublings, we extend the simple one-term scaling law to a two-term scaling law. Thus we find a new scaling factor associated with coupling and confirm the approximate renormalization results.

chao-dyn↗

arXiv1994

Critical behavior of period doubling in coupled area-preserving maps

We study the critical behavior of period doublings in $N$ symmetrically coupled area-preserving maps for many-coupled cases with $N>3$. It is found that the critical scaling behaviors depend on the range of coupling interaction. In the extreme long-range case of global coupling, in which each area-preserving map is coupled to all the other area-preserving maps with equal strength, there exist an infinite number of bifurcation routes in the parameter plane, each of which ends at a critical point. The critical behaviors, which vary depending on the type of bifurcation routes, are the same as those for the previously-studied small $N$ cases $(N=2,3)$, independently of $N$. However, for any other non-global coupling cases of shorter range couplings, there remains only one bifurcation route ending at the zero-coupling critical point, at which the $N$ area-preserving maps become uncoupled, The critical behavior at the zero-coupling point is also the same as that for the small $N$ cases $(N=2,3)$, independently of the coupling range.

chao-dyn↗

arXiv1994

M-furcations in coupled maps

We study the scaling behavior of $M$-furcation $(M\!=\!2, 3, 4,\dots)$ sequences of $M^n$-period $(n=1,2,\dots)$ orbits in two coupled one-dimensional (1D) maps. Using a renormalization method, how the scaling behavior depends on $M$ is particularly investigated in the zero-coupling case in which the two 1D maps become uncoupled. The zero-coupling fixed map of the $M$-furcation renormalization transformation is found to have three relevant eigenvalues $δ$, $α$, and $M$ ($δ$ and $α$ are the parameter and orbital scaling factors of 1D maps, respectively). Here the second and third ones, $α$ and $M$, called the ``coupling eigenvalues'', govern the scaling behavior associated with coupling, while the first one $δ$ governs the scaling behavior of the nonlinearity parameter like the case of 1D maps. The renormalization results are also confirmed by a direct numerical method.

chao-dyn↗

arXiv1994

Period Doubling in Four-Dimensional Volume-Preserving Maps

We numerically study the scaling behavior of period doublings at the zero-coupling critical point in a four-dimensional volume-preserving map consisting of two coupled area-preserving maps. In order to see the fine structure of period doublings, we extend the simple one-term scaling law to a two-term scaling law. Thus we find a new scaling factor $δ_3$ $(=1.8505\dots)$ associated with scaling of the coupling parameter, in addition to the previously known scaling factors $δ_1$ $(=-8.7210\dots)$ and $δ_2$ $(=-4.4038\dots)$. These numerical results confirm the renormalization results reported by Mao and Greene [Phys. Rev. A {\bf 35}, 3911 (1987)].

cond-mat↗

arXiv1993

Universality of Period Doubling in Coupled Maps

We study the critical behavior of period doubling in two coupled one-dimensional maps with a single maximum of order $z$. In particurlar, the effect of the maximum-order $z$ on the critical behavior associated with coupling is investigated by a renormalization method. There exist three fixed maps of the period-doubling renormalization operator. For a fixed map associated with the critical behavior at the zero-coupling critical point, relevant eigenvalues associated with coupling perturbations vary depending on the order $z$, whereas they are independent of $z$ for the other two fixed maps. The renormalization results for the zero-coupling case are also confirmed by a direct numerical method.

cond-mat↗