Yoshihito Ogasawara

Sufficient Conditions for the Existence of a Primitive Chaotic Behavior

The tent map ’ : 1⁄20; 1 ! 1⁄20; 1 ; x 7! minf2x; 2ð1 xÞg, the baker map ’ : 1⁄20; 1 ! 1⁄20; 1 ; x 7! 2x ð0 x 1=2Þ; 2x 1 ð1=2 x 1Þ, and the quadratic dynamical system ’ : 1⁄20; 1 ! 1⁄20; 1 ; x 7! 4xð1 xÞ are fundamental maps in the chaos theory, and have an intriguing property ‘‘For any infinite sequence !0; !1; !2; . . . each term !i of which is either 1⁄20; 1=2 ð1⁄4 AÞ or 1⁄21=2; 1 ð1⁄4 BÞ, there exists an initial point x0 2 !0 such that ’ðx0Þ 2 !1; ’ð’ðx0ÞÞ 2 !2; ’ð’ð’ðx0ÞÞÞ 2 !3; . . .’’. It is easy to recognize that this property guarantees the variety of their orbits. Then, if we assume that A and B mean the head and the tail of a coin, respectively, we can see that for any indeterministic time series obtained by coin tossing, all members of the series are traced deterministically. Accordingly, one may raise the general problem about the relation between determinism and indeterminism. Also, from the fact that any phenomenon expressed by the time series is totally described by such a simple law, one may raise the fundamental problem ‘‘What are the laws describing phenomena?’’. Also, assuming that A and B are alternative selections chosen successively, one may see such a situation that any history of actions, choosing selections, is recognized (like a destiny) by such a simple law, and may raise the problem ‘‘What is the choices of selections?’’. However, this property is too restricted in that the number of the selections A and B is only two. Therefore, the following generalized property (P) is naturally proposed as a primitive chaotic behavior. (P) For any infinite sequence !0; !1; !2; . . . ; there exists an initial point x0 2 !0 such that f!0 ðx0Þ 2 !1; f!1 ð f!0 ðx0ÞÞ 2 !2; . . . : Here, each !i is an element of a family fX ; 2 g of nonempty subsets of a set X, and each fX is the map from X to X. In this note, let us explore sufficient conditions for guaranteeing the existence of this chaotic behavior for the purpose of revealing the essence of its existence. As a result, we can see the emergence of dendrite known in the fields of the materials science and the fractal theory. In the beginning, we prepare the following lemma, whose proof is the same as that in the previous article. Lemma 1. If X is a countably compact space, fX ; 2 g is a family of nonempty closed subsets of X, and each fX is a continuous map from the subspace X of X onto X, the property (P) holds. Accordingly, the chaotic behavior is guaranteed by such a space X, a family fX ; 2 g, and maps fX ; 2 . However, the existence of the continuous onto maps is by no means a weak condition but an artificial one. Therefore, it is natural that we explore the sufficient conditions for this existence, and we can recall the following lemma obtained from the Hahn–Mazurkiewicz Theorem and the Tietze Extension Theorem.

The effect of curvature on the instability of a solid/liquid interface

An essential factor causing the instability (stability) of a solid/liquid interface during solidification is explored. By examining the qualitative properties of the classical solution of a nonlinear evolution equation, the bifurcation, the coalescence and the growth rate of the interface are discussed. These discussions lead to the relation between the instability (stability) and the curvature of the interface.

Flattening Properties of the Mechanism of Evaporation–Condensation under a Temperature Gradient

This paper discusses the flattening properties of the classical solution of the nonlinear evolution equation, which is derived from the model for the development of a solid surface controlled by the mechanism of evaporation–condensation under a vertical temperature gradient.

A consideration of the morphological stability of an interface

The evolution equation of an interface based on a mass transfer and that based on a heat transfer are unified into a nonlinear evolution equation for two spatial variables, and the properties of the solution of this equation are discussed without employing any linear approximations. Then, through these discussions, the intrinsic conditions which characterize the morphological stability are exhibited.

On the Existence of Stationary Solutions of a Generalized Mullins Equation

We derive a nonlinear evolution equation which describes the development of a solid surface due to the mechanism of evaporation–condensation under a temperature gradient. The existence of stationary grain boundary grooves is shown and the stationary shapes are evaluated.

Consideration of a Primitive Chaos

Since the chaos was discovered, it has been recognized that we are surrounded by diverse chaotic behaviors. The purpose of this study is to reconsider the implication of this fact through the notion of a primitive chaos. Under natural conditions, each primitive chaos leads to apparent chaotic features, such as the existence of a nonperiodic orbit, the existence of the periodic point whose prime period is n for any n ∈ N , the existence of a dense orbit, the density of periodic points, sensitive dependence on initial conditions, and topological transitivity, while infinite varieties of the primitive chaos are guaranteed by a nondegenerate Peano continuum.

Characteristic Spaces Emerging from Primitive Chaos

This paper describes the emergence of two characteristic notions, nondegenerate Peano continuum and Cantor set, by the exploration of the essence of the existence of primitive chaos from a topological viewpoint. The primitive chaos is closely related to vital problems in physics itself and leads to chaotic features under natural conditions. The nondegenerate Peano continuum represents an ordinarily observed space, and the existence of a single nondegenerate Peano continuum guarantees the existence of infinite varieties of the primitive chaos leading to the chaos. This result provides an explanation of the reason why we are surrounded by diverse chaotic behaviors. Also, the Cantor set is a general or universal notion different from the special set, the Cantor middle-third set, and the existence of a single Cantor set guarantees infinite varieties of the primitive chaos leading to the chaos. This analogy implies the potential of the Cantor set for the method of new recognizing physical phenomena.This study describes such a situation that a Cantor set emerges as a result of the exploration of sufficient conditions for the property which is generalized from fundamental chaotic maps, and the Cantor set even guarantees infinitely many varieties of the behavior with the property, as well as a typical continuum.

Property Leading to Morphological Instability

This letter describes that morphological instabilities, such as the nonsimplification of the interface and the increase in interface perturbation, which are required for the formation of matters with complex appearance, is caused by a property, that is, “the driving force of interface development that increases with the increase in curvature”, with the aid of the qualitative properties of a primitive and general evolution equation. This result has a certain universality owing to these primitiveness and generality; thus, it has the potential of playing an essential role in the realization of diverse phenomena that generate matters with intriguing appearance and in the actual manufacture of matters with complex appearance.

Addendum to “A Consideration of the Morphological Stability of an Interface”

In our previous paper, we discuss the flattening properties of a generalized interface through the harmony between phenomenal facts and a mathematical structure, and we propose an original principle of interface morphology: ‘‘The interface tends to become flat if the driving force decreases with the increase of the curvature’’. However, we see the incompleteness of its universality; that is, a question ‘‘Is the principle applied even if the contribution neglecting the curvature effect AI or AII or the curvature effect fI or fII depends not only on the height of the interface u but also on the time t or the spatial variable x1 or x2 caused by concentration distribution, impurity distribution, temperature distribution and so on?’’ naturally comes to mind. Then, letting AI, AII, fI, and fII be a function of t, x1, x2, and u, we verify the principle in the same way as that in our previous study. First, with the aid of the implicit function theorem and the mean value theorem, the notion of a weak flattening property is acquired as follows. 1) Even if AI, AII, fI, and fII actually depend on t, x, y, and u, the velocity of the peak top (the valley bottom) of the interface is lower (greater) than the function of t; x; y, and u, which corresponds to the velocity without the curvature effect. Then, with the aid of the lemmas in our previous paper on a nonlinear evolution equation and an ordinary differential equation, the notion of the hierarchy of flattening properties is acquired as follows. 2) If AI and AII are independent of x1 (or x2), the velocity of the peak top (the valley bottom) is lower (greater) than the function of t, x2 (or x1), and u, which corresponds to the velocity without the curvature effect. 3) If AI, AII, fI, and fII are independent of x1 (or x2), the velocity of the peak top (the valley bottom) is lower (greater) than the function of t, x2 (or x1), and u, which corresponds to the velocity without the curvature effect. Then, the bifurcation never takes place. 4) If AI and AII are independent of x1, x2 and u and fI and fII are independent of x1 (or x2), the velocity of the peak top (the valley bottom) is lower (greater) than the function of only t, which corresponds to the velocity of a plane interface. In particular, if AI and AII are constant, the function is constant. Then, the bifurcation never takes place. 5) If AI and AII depend only on u [that is, AIðt; x; y; uÞ 1⁄4 ~ AIðuÞ;AIIðt; x; y; uÞ 1⁄4 ~ AIIðuÞ] under the condition 9 s.t. ~ AIð Þ 1⁄4 ~ AIIð Þ, d1⁄2 ~ AIðuÞ ~ AIIðuÞ = duju1⁄4 < 0, which reinforces the flattening property, and fI and fII are independent of x1 (or x2), the interface exponentially approaches a flat plane u 1⁄4 without the bifurcation. Consequently, the difference between the role of the contributions neglecting the curvature effects (AI and AII) and that of the curvature effects ( fI and fII) for the flattening properties is revealed, and the above principle is more universally exhibited.

Addendum to ``Property Leading to Morphological Instability''

It was described that the property (P) can result in the morphological instabilities that are nonsimplification such that two peaks of the interface never coalesce into a single peak and the (exponential) increase in the perturbation of the interface, with the aid of a generalized evolution equation in two-dimensional space. (P) The driving force of interface development increases with the increase in curvature. In this addendum, we indicate that the corresponding results hold even in three-dimensional space. First, the following evolution equation in three-dimensional space can be obtained as an extension of that in two-dimensional space. zt 1⁄4 ð1þ zx þ zyÞAIðt; x; y; zÞ fIðt; x; y; z; Kðzx; zy; zxx; zxy; zyyÞÞ ð1þ zx þ zyÞAIIðt; x; y; zÞ fIIðt; x; y; z; Kðzx; zy; zxx; zxy; zyyÞÞ; AIðt; x; y; zÞ 0; AIIðt; x; y; zÞ 0; fIðt; x; y; z; KÞ > 0; fIIðt; x; y; z; KÞ > 0; fIðt; x; y; z; 0Þ 1⁄4 fIIðt; x; y; z; 0Þ 1⁄4 1; ðt; x; yÞ 2 ; ð1Þ where zðt; x; yÞ is the interface profile, is a connected open set, K is the mean curvature, Kðzx; zy; zxx; zxy; zyyÞ 1⁄4 1 2 ð1þ zyÞzxx þ ð1þ zxÞzyy 2zxzyzxy ð1þ zx þ zy2Þ ;