A skeleton structure of self-replicating dynamics

Nishiura, Yasumasa; Ueyama, Daishin

doi:10.1016/s0167-2789(99)00010-x

Cited by 164 publications

(196 citation statements)

References 11 publications

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“…A related phenomenon is that of self-replication, where bifurcations of the underlying stable solutions can lead to space-filling pattern propagation. We provide evidence that this phenomenon is an example of the Nishiura-Ueyama self-replication mechanism [29].…”

Section: Multipeaked Solutions and Self-replicationmentioning

confidence: 67%

“…A general theory for this phenomenon was proposed by Nishiura and Ueyama [29], which appears to be applicable here. Their idea relies on the disappearance of all multipeaked solutions via saddle-node (or fold) bifurcation, more or less simultaneously.…”

Section: Self-replicationmentioning

confidence: 82%

“…There is a considerable body of literature of self-replicating patterns in reactiondiffusion systems (e.g., [29,32,34]). A general theory for this phenomenon was proposed by Nishiura and Ueyama [29], which appears to be applicable here.…”

Section: Self-replicationmentioning

confidence: 99%

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Spatially Localized Structures in Diblock Copolymer Mixtures

Glasner¹

2010

SIAM J. Appl. Math.

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Abstract. Above the critical temperature for the order-disorder transition, diblock copolymer melts have been observed to exhibit localized structures that exist within the homogeneous mixture. This paper uses an Ohta-Kawasaki-type density functional to explore this regime. Spatially localized peak-shaped equilibria are studied in one, two, and three dimensions, corresponding to amphiphilic bilayers, cylindrical micelles, and spherical micelles, respectively. A combination of rigorous estimates, asymptotic analysis, and numerical computation is used to characterize solutions and the regime where they exist. The interaction of superpositions of these solutions is studied by a perturbation analysis and shows how steady multipeak configurations can be achieved. Evidence is found for a secondary bifurcation slightly below the spinodal instability threshold, beyond which self-replication phenomena are observed. Dynamics in two dimensions are also illustrated, suggesting other mechanisms for instability and growth.Key words. diblock copolymer, micelle, density functional theory, self-replication AMS subject classifications. 82D60, 35K55DOI. 10.1137/080743913 1. Introduction. Diblock copolymers are linear chain molecules composed of two distinct subchains. At low temperatures, repulsion between subchains can result in phase segregation, but macroscopic segregation cannot occur because of covalent bonding between subchains. This incomplete segregation can give rise to a variety of stable microstructures (see, e.g., [1,11,14,24]), which are typically characterized as spatially periodic patterns whose domain boundaries closely resemble minimal surfaces, such as spheres, cylinders, and gyroids [39].Outside of the regime where mixtures are unstable to spinodal decomposition, the situation for copolymers is more complicated than simple binary systems, such as those described by the Cahn-Hilliard theory [5]. In addition to a stable homogeneous state, there are other observed behaviors. Physical experiments have identified inhomogeneous states above the critical temperature, such as sparse arrangements of micellar structures [35,36,43] and anisotropic lamellar grains [15]. This paper seeks to investigate this regime and its dynamics using density functional theory and the corresponding gradient flow dynamics.Theoretical work on the disordered regime has focused on spherical micelles. Dormidontova and Lodge [9] use Semenov's strong segregation theory [37] to predict the concentration of micelle assemblies. Their theory predicts that the number of micelles increases rapidly below a "critical micelle temperature" (CMT), which is greater than the temperature at which a homogeneous mixture is spinodally unstable. Wang, Wang, and Yang [42] use self-consistent field theory calculations to find the density profile of individual spherical micelles and evaluate their excess free energy. They find that micelles exist (within the mean field approximation) for volume frac-

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Section: Multipeaked Solutions and Self-replicationmentioning

confidence: 67%

Section: Self-replicationmentioning

confidence: 82%

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Spatially Localized Structures in Diblock Copolymer Mixtures

Glasner¹

2010

SIAM J. Appl. Math.

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“…Transition from standing wave or traveling wave to selfreplicating pattern was clarified in [2]. The essence is that all the ordered patterns disappear due to the saddle-node structures almost simultaneously, but the strong (fading) memory remains in the phase space if the parameter is close to the location of the saddle-node bifurcation.…”

Section: Self-replicationmentioning

confidence: 99%

“…This aftereffect and the connections among saddlenode branches are key ingredients to understand the self-replication dynamics. For details, see [2], 2.3. Self-destruction.…”

Section: Self-replicationmentioning

confidence: 99%

Global bifurcational approach to the onset of spatio-temporal chaos in reaction diffusion systems

Nishiura¹

2001

Methods and Applications of Analysis

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Abstract. A new criterion for the onset of spatio-temporal chaos arising in the Gray-Scott model is presented. This is based on the interrelationship of global bifurcating branches of stationary or periodic solutions with respect to the removal rate contained in the model, especially their locations of saddle-node points and the Hopf bifurcation point of a constant state play a key role. At the onset point there exists a generalized heteroclinic cycle on the whole line and spatio-temporal chaos emerges by unfolding this cycle.

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Effects of shear heating, slip‐induced dilatancy and fluid flow on diversity of 1‐D dynamic earthquake slip

Suzuki

Yamashita

2014

JGR Solid Earth

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We theoretically study effects of thermal pressurization, slip-induced dilatancy and fluid flow on slip evolution assuming 1-D fault model. We generalize the analysis made in our former papers by introducing an upper limit for the inelastic porosity evolution. The expression for nondimensional parameter T a , which is related to the upper limit, is derived in the present paper. We find that the parameter T a together with two nondimensional parameters S u and S u ′ derived in our former papers completely determine the qualitative nature of system behavior once the initial condition is given. For example, changes in S u and T a generate two qualitatively different slip behaviors, both of which can be models for ordinary earthquake. The slip is accelerated with time after experiencing an initial deceleration in one of them, while the slip ceases after monotonic deceleration in the other. The initial slip deceleration observed in the former case may be interpreted as a dynamic event preceding the main shock in seismological observations. We mathematically derive the expression for the ranges of the nondimensional parameters in which the above two slip behaviors appear. Slow earthquakes are also modeled in the same framework, and nonzero values of the nondimensional parameter S u ′ together with relatively large values of S u and T a are found to contribute to the generation of such slow earthquakes. The mathematical framework constructed here can be applied to understanding of some other natural phenomena such as a kind of reaction-diffusion system.

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A skeleton structure of self-replicating dynamics

Cited by 164 publications

References 11 publications

Spatially Localized Structures in Diblock Copolymer Mixtures

Spatially Localized Structures in Diblock Copolymer Mixtures

Global bifurcational approach to the onset of spatio-temporal chaos in reaction diffusion systems

Effects of shear heating, slip‐induced dilatancy and fluid flow on diversity of 1‐D dynamic earthquake slip

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