Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
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 $\delta$, $\alpha$, and $M$ ($\delta$ and $\alpha$ are the parameter and orbital scaling factors of 1D maps, respectively). Here the second and third ones, $\alpha$ and $M$, called the ``coupling eigenvalues'', govern the scaling behavior associated with coupling, while the first one $\delta$ 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.

Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
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 $\delta_3$ $(=1.8505\dots)$ associated with scaling of the coupling parameter, in addition to the previously known scaling factors $\delta_1$ $(=-8.7210\dots)$ and $\delta_2$ $(=-4.4038\dots)$. These numerical results confirm the renormalization results reported by Mao and Greene [Phys. Rev. A {\bf 35}, 3911 (1987)].

Abstract:
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.

Abstract:
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.