Figure 1 shows the architecture of the coupled oscillator model for dynamic image segmentation [10] . The mechanism of dynamic image segmentation is shown in Figure 2.

The coupled model comprises a global oscillator and N one-dimensional oscillators, where N denotes the number of pixels in an input image. With the exception of the global oscillator, the one-dimensional oscillators are arranged on the grid so that each corresponds to a pixel. They are connected to the eight neighboring oscillators with similar pixel values. The global oscillator connects all the other oscillators and acts as a relay between them. Oscillators with similar pixel values in the eight-neighborhood connection are coupled together. Figure 2 shows second and third (seventh and eighth) oscillators have neighboring connections. Although the responses of the directly coupled oscillators are synchronized, those of the uncoupled ones are out of phase, being connected to the global oscillator with a specific coupling strength. By associating the output value of the ith oscillator with the ith pixel value at each discrete time, segmented images are output as a time series. The discrete-time coupled models are described as follows:

Here, , with denoting the set of integers, and the internal state variable. Functions f and h are described by

and

where denotes the sigmoid function described by

Function f is based on the Rulkov map, where h denotes the effect of the global oscillator on each of the other oscillators, and k and d in (2) are tunable system parameters. in (1) represents the connection between each oscillator and its eight neighboring oscillators, described by

where denotes a group of pixels to which the ith pixel connects, is the number of elements in, and and a in (1) represent the coupling coefficients for the eight neighboring oscillators and the global oscillator, respectively. Finally, is defined by

where is the pixel value of the ith pixel, and is an arbitrary threshold. From the function, the oscillator is inhibited when the pixel value is less than.

The dynamics of an N-coupled system are described by the P-dimensional discrete-time dynamical system () as

or, equivalently, by an iterated map defined by

where denotes the set of real numbers. The nonlinear function describes the dynamical system of the P-coupled system and is given by

where. The three-coupled model shown in Figure 3 is analyzed in detail in Section 4.

The coupling of oscillators defined by (5) is uniformly based on the pixel value of the input image. We replaced this fixed coupling with an adaptive coupling based on the clustering method proposed in [8] . This adaptive coupling was represented as

where is the pixel value of the ith pixel at iteration t, represents the initial value of the ith pixel, denotes a group of pixels with around the ith pixel at iteration t, is the number of elements in, and the new variables and enable each oscillator to adaptively connect to its neighbors.

In our bifurcation analysis, the point satisfying

becomes a fixed point of in (9). If is a fixed point of, the characteristic equation for fixed point is

where is the identity matrix and denotes the derivative of. We consider to be hyperbolic if none of the absolute eigenvalues of are at unity. Note that an m-periodic point can be investigated by replacing with, i.e., the mth iteration of, in (15). In the following discussion, we consider only the properties of a fixed point of, though a similar argument can be applied to a periodic point of.

Next, we considered the topological classification of a hyperbolic fixed point. Let be a hyperbolic fixed point and be the intersection of and the direct sum of the generalized eigenspaces of corresponding to eigenvalue such that, and let. Then, the topological type of a hyperbolic fixed point is determined by and the orientation-preserving or reversing property of. Bifurcation occurs when the topological type of a fixed point is changed by varying a system parameter. The generic co-dimension-one bifurcations are the tangent, period-doubling, and Neimark-Sacker bifurcations. In addition, a D-type branching appears in a system that possesses some symmetric properties as a degenerate case of the tangent bifurcation. These bifurcations are observed when hyperbolicity is destroyed, which corresponds to the critical distribution of the characteristic multiplier such that for tangent bifurcation and D- type branching, for period-doubling bifurcation, and for the Neimark-Sacker bifurcation, where.

Bifurcation sets of a fixed point were computed by solving the simultaneous Equations (15) and (16). For the numerical determination [17] , we used Newton’s method. The Jacobian matrix of the set of equations was derived from the first and second derivatives of map F.

To investigate the bifurcation phenomena in (2), we used finite-time Lyapunov exponents in which local expansion rates are defined by

Here, is the derivative of function f in (2).

To investigate the relationship between the coupling coefficients and the phase difference of oscillators (in- phase or out-of-phase), we used the order parameter defined by

where and are arbitrary time steps after sufficient time has passed, T is the time step interval from to, and is described by

which represents the phase difference between and. The value of in (18) becomes one when the ith and jth oscillators oscillate in-phase; otherwise, the order parameter converges to a value other than one.

Figure 4 presents the bifurcation sets observed in a single oscillator defined by (2) in the (k, d)-plane. In the white region, periodic and non-periodic points can be seen. Figure 5(a) shows the one-dimensional bifurcation diagram, while Figure 5(b) shows the local expansion rate calculated by (17) at for different values of parameter d. Periodic and non-periodic points appear periodically in Figure 5, with the non-periodic points considered to be chaotic because the local expansion rates are greater than zero. Figure 6(a) and Figure 6(b) show waveforms of the stable 56-periodic point and the chaotic behavior at and, respectively. Based on these results, we set the system parameters of our coupled model to and.

Figure 7 plots the bifurcation sets of the three-coupled model in the (a, e)-plane. At the right-hand region of curve, a high order of periodic and non-periodic points is observed. In the non-shaded left-hand region of curve, non-periodic points with small amplitudes can be seen. Bifurcation analysis suggested that oscillatory responses appropriate to segmentation of a large image was dependent on the coefficients of coupling. Based on these results, we set parameters a and e to 0.001 and 1.0, respectively.

Figure 8 shows the relationship between the order parameters calculated by (18) and the coupling coefficients

with, , and. Figure 8(a) and Figure 8(b) show and calculated at different values of parameter and. As shown in Figure 8(a), remained at one in the range of approximately from zero to one, and was otherwise less than one. Figure 8(b) shows that was less than one in the range of from −0.2 to 1.2. These results demonstrate that the first and second oscillators were in-phase in the range approximately from zero to one of, whereas the first and third oscillators were out-of-phase under the given parameters. Figure 8(c) and Figure 8(d) show the relationship between the order parameters and parameter a at. Figure 8(c) shows that, whereas parameter a changed, remained at one, suggesting that the first and second oscillators were in-phase, and had no relation to parameter a. Conversely, the phase between the first and third oscillators periodically changed in response to parameter a, as shown in Figure 8(d). These results suggested that the phase difference between the oscillators corresponding to disconnected

regions could be controlled by adjusting parameter a.