数理解析研究所講究録
1072 巻 1998 年 91-102

91

An Introduction to a Multi-dimensional
H\’enon-like Family
Shin Kiriki

(桐木

紳)

Department of Mathematical Sciences, Tokyo Denki University,

Hatoyama, Hiki, Saitama, 350-0349, Japan-mail: ged@r.dendai.ac.jp$E$

Masaki Nakajima (中島 正樹)
Hitachi Software Engineering Co., Ltd.,
9-12 Minato-cyo, Kanagawa-ku, Yokohama, 221-0056, Japan$E_{-ma}il$1: nmasaki@pyt.hitachi-sk.co.jp

Introduction

H\’enon map $f_{a,b}(x, y)=(a-by-X^{2},x)$ is one of the most important objects in the chaotic

dynamical systems. Through its study, many mathematicians have confirmed some essential concepts in the dynamical systems presented by Smale [4], and ncountered various$..\mathrm{e}$phenomena and interesting new problems which contain many difficulties originating fromits nonlinearity. Computer simulations are also indispensable for the progress of thisstudy. There have been many works on this map. However, we here state briefly some es-sential works which motivated our study of dynamics with a high-dimensional H\’enon-like
diffeomorphisms. This planar map was first presented by H\’enon [2] with its non-trivial
phenomenon called a strange attractor which was discovered by computational experimentas shown in the first panel of Figure 1. Nitecki and Devany showed in [1] that, whenand satisfy some conditions, it has the same structure as a horseshoe map, that is, it

$a$

$b$

has a hyperbolic structure and its dynamics is coded perfectly by a topological conjugacy
with the 2-shift. Yang moreover showed that this structure is preserved until the firsthomoclinic tangency occurs when$a$decreases and$b$takes very small positive values [6].

92

Benedicks and Carleson proved that the occurrence of the non-trivial attractors is aabundant phenomenon in the measure-theoretical sense. Mora and Viana extended theirresult to more generic one-parameter family of surface diffeomorphisms. In [5], Viana

moreover extended their result to higher dimensions, that is, it assures the existence ofcodimension-l H\’enon-like non-hyperbolic attractors near the homoclinic tangency on highdimensional manifolds.

Figure 1: (1) A 2-dimensional Heton attractor, for $a=1.4$ and $b=0.3$ . (2) and (3) The 3-dimensional

Heton attractors for $(1.7, -0.1)$ and (0.2, 0.8), respectively.

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In our recent paper [3], on a somewhat different ground from that of Viana, we extend

the H\’enon map to a high-dimensional space. One of the merits of our method is that

we can actually have a formula for this extended map which has some properties similar
to the 2-dimensional H\’enon map. Therefore, one can observe globally its dynamics andits progress of bifurcations by numerical computations. Actually, in the 3-dimensional

Euclidean space, we define a family of diffeomorphisms of H\’enon type as$F_{a,b}(x, y, z)=(a+bZ-x^{2}, x, y)$ .

As in the 2-dimensional case, some non-trivial attractors are observed easily in$\mathbb{R}^{3}$bynumerical experiments, for example for $(a, b)$ close to $(1.7, -0.1),$ $(0.2,0.8)$ , as shown inthe second, the third panel of Figure 1, respectively. However, it is not yet clear whether
these attractors are non-hyperbolic and persistently strange.

As the results of the paper [3], we present a two-parameter familyof diffeomor-$F_{a,b}$phisms of H\’enon type on the Euclidean space of an arbitrary dimension $m$ , and showthat there is an open set$(a, b)\in \mathcal{H}_{m},$$F_{a,b}$$\mathcal{H}_{m}$in the product space of parameters such that, for every

has a hyperbolic structure. The following article is a brief summary of

the results in [3].

2

Definitions and the main result

Let $m\geq 3$ be an integer, andtwo-parameter family$F_{a,b}$$\mathbb{R}^{m}$be the $m$-dimensional Euclidean space. We define theof diffeomorphisms on$\mathbb{R}^{m}$as$F_{a,b}(x_{1}, x_{2}, \ldots, x_{m})=(a+bx_{m}-X_{1}^{2}, x_{1}, x_{2}, \ldots, x_{m-1})$

where

$a$

,

and are real parameters. One shall see why we call this family a m-dimensional$b$special H\’enon family by the following fundamental properties which are similar to thoseof the 2-dimensional H\’enon map. When $b\neq 0$ , it’s inverse is$F_{a,b}^{-1}(x_{1}, x_{2}, \ldots , x_{m})=(x_{2}, x_{3}, \ldots, x_{m}, b^{-1}(-a+x_{1}+x_{2}^{2}))$,

94

and it is a bijection. Moreover,$(a-x_{1}^{2}\rangle x_{1}, \ldots , x_{m-1})$$F_{a,b}$is a diffeomorphism on

which maps all points ofvalue of the Jacobian of$F_{a,b}$is equal to$|b|$$\mathbb{R}^{m}$$\mathbb{R}^{m}$to the parabolic surface. The absolutefor every$\mathrm{x}\in \mathbb{R}^{m}$$\det(DF_{a,b})_{\mathrm{x}}=\det$$F_{a,b}$is obtained by the composed mapping of$\phi,$$\phi_{b}$. When $b=0,$ $F_{a,0}(x1, x_{2}, .. , , x_{m})=$and$\phi_{a}$, that is,$=(-1)^{m}-1b$, that is,$\phi(x_{1}, x_{2}, \cdots, x_{m})=(x_{m}, x_{1}, \cdots, x_{m-1})$$\phi_{b}(x_{1}, x_{2}, \cdots , x_{m})=(bx_{1}, x_{2}, \cdots, x_{m})$.

$F_{a,b}=\phi_{a}\circ\phi b\circ\phi$, where;;$\phi_{a}(x_{1}, x_{2}, \cdots, x_{m})=(a+x_{1}-x_{2}^{2}, x_{2}, \cdots, x_{m})$.

These above constructions of this family are similar to those of 2-dimensional H\’enonfamily. Moreover, in the following main theorem, we get an essential property of thisfamily, which corresponds to that of the 2-dimensional H\’enon family.
Main Theorem. Let$\mathit{8}et$$F_{a,b}$be the $m- dimen\mathit{8}i_{on}alspeCial$ H\’enon family. Let$\mathcal{H}_{m}$be an open

in the product set of parameter spaces such that$\mathcal{H}_{m}=\{(a, b)\in \mathbb{R}\cross \mathbb{R}$For any$(a, b)\in \mathcal{H}_{m’}$: $a> \frac{(1+|b|)^{2}}{4}(m+4+2\sqrt{m+4}),$ $b\neq 0\}$ .

the nonwandening$\mathit{8}et$of$F_{a,b}i\mathit{8}$a$h_{or\mathit{8}e}shoe$, that is, it$i\mathit{8}\mathit{8}truCturally$

stable.

We can prove the main theorem using the following two theorems. See Figure 2. The

open set$\mathcal{H}_{m}$occupies a large area in the product space of parameters meaning that thestructural stability is$\mathrm{e}\mathrm{a}s$

ily acquired by this high-dimensional family. However, numericalexperiments showed characteristic division of the area outside of$\mathcal{H}_{m}$by some typicalbifurcations. Moreover, the combinations of parameters causing the non-trivial attractors

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are located outside$\mathcal{H}_{m}$. It may be very important to make a systematic investigation of

all the types of bifurcations occurring in this family.

Figure 2: The open set

for $\mathrm{m}=3,5,10$ and $N$

Nonwandering set

3Let$\mathcal{H}_{m}$$F_{a,b}$be the $m$-dimensional special H\’enon family. We assume $b\neq 0$ . We denote thenonwandering set of$F_{a,b}$by$\Omega_{a,b}$. In this section, we detect the location of$r= \frac{1+|b|+\sqrt{(1+|b|)2+4a}}{2}$$\Omega_{a,b}$

. Let

,

which satisfies the quadratic equation $r^{2}-(1+|b|)r-a=0$ . We as sume that $(1+|b|)2+4a\geq$$0$. Using this , partition
$r$

$D$

$\mathcal{T}_{1}$$\mathcal{T}_{i}$$\mathbb{R}^{m}$into $m+1$ areas as follows: for each $i=2,3,$$=$$\{(x_{1}, x_{2}, \cdots, x_{m})\in \mathbb{R}^{m} :$=$$\{(x_{1}, x_{2}, \cdot..

$=$$\{(x_{1}, x_{2}, \cdots, x_{m})\in \mathbb{R}^{m} :$\cdots$

,

$m$

,

|x_{j}|\leq r, j=1,2, \cdots, m\}$, x_{m})\in.\mathbb{R}^{m} :|x_{1}|>r, |x_{1}|>|x_{j}|, j=2,3, \cdot..

, m\}$|x_{i}|>r, |x_{i}|\geq|x_{j}|, j=1,2, , .., m, j\neq i\}$

When $m=3$ , these partitions are shown in Figure 3.

.

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Proposition 1 Let $N$ be a subset of the direct product set of parameter spaces such that
$N=\{(a, b)\in \mathbb{R}\cross \mathbb{R}$: $a\geq-(1+|b|)^{2}/4,$ $b\neq 0\}$ .

For any $(a, b)\in N$ , the nonwandering set
Proof. See [3].

$\Omega_{a,b}$for$F_{a,b}$is contained in $D$ .

$\blacksquare$

Figure 3: The partitions of$\mathbb{R}^{3}$.

From the following lemmas 1 and 2, we get directly the proof of this theorem. For any$\mathrm{x}=(x_{1}, x_{2}, \cdot, .

, we write,x_{m})\in \mathbb{R}^{m}$$|| \mathrm{x}||=\max\{|x_{1}|, |x_{2}|, \cdots|x_{m}|\}$

Lemma 1 For $(a, b)\in N$ ,(1).

$F_{a,b}(T_{1})\subset T1f$

(2) For every$\mathrm{x}\in T_{1},$(3) $F_{a,b}(D)\subset D\cup T_{1}$ .

$||F_{a,b}^{n}(\mathrm{x})||arrow\infty$as$narrow\infty_{i}$.

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Proof. See [3].

$\blacksquare$

Lemma 2 For any $(a, b)\in N$ ,(1)

$F_{a,b}^{-1}(\tau_{2})\subset \mathcal{T}mi$(2)$F_{a,b}^{-1}(\mathcal{T}_{i})\subset \mathcal{T}_{i-1}\cup \mathcal{T}_{m}$(3) For every

for each $i=3,4,$$\mathrm{x}\in\bigcup_{i=2}^{m}\mathcal{T}_{i_{f}}||F_{a,b}^{-}n(\mathrm{X})||arrow\infty$

Proof. See [3].

4$\cdots,$as$m$$narrow\infty$.

$\blacksquare$

Hyperbolicity

Rom the previous section, we understand that the nonwandering set ofin $D$ . Therefore, in this section, we concentrate$\Lambda_{a,b}$

be the maximal invariant set of $D$ under$0\dot{\mathrm{n}}$$F_{a,b}$$F_{a,b}$is contained

the maximal invariant set of, i.e.

$\Lambda_{a,b}=\mathrm{n}i\in \mathrm{Z}Fi(a,bD)$

$D$

. Let

. The following

proposition gives a result on its hyperbolicity.

Proposition 2 Let$ber\mathit{8}uch$$F_{a,b}$be the m-dimen8ional$\mathit{8}peCial$

H\’enon family. Let$\lambda$

be a real num-that $\sqrt{m-1}\geq\lambda>1$ , and let$a_{1}(b, m)= \frac{(1+|b|)^{2}}{4}(m+4+2\sqrt{m+4})$

For any $|b|>0$ , if $a>a_{1}(b, m)$ , then the maximal invariant
Proof. See [3].

.

$\mathit{8}et\Lambda_{a},b$

is a hyperbolic$\blacksquare$

For $m\geq 3$ , let us define$\mathcal{H}_{m}=\{(a, b)\in \mathbb{R}\cross \mathbb{R}$For any point$\mathrm{x}\in\Lambda_{a,b}$: $a> \frac{(|b|+1)^{2}}{4}(m+4+2\sqrt{m+4}),$ $b\neq 0\}$ .

, we define the unstable cone:$C^{u}(\mathrm{x})=\{(\xi_{1}, \xi_{2}, \cdots, \xi_{m})\in T_{\mathrm{x}}\mathbb{R}^{m}$:$\lambda\sqrt{\xi_{2}^{2}+\xi_{3}^{2}++\xi^{2}m}\leq|\xi_{1}|\}$,$\mathit{8}et$.

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and take a stable cone in the complement of$C^{s}(_{\mathrm{X})}=\{$

where$\lambda>1$

in$C^{u}(\mathrm{x})$$(\xi_{1}, \xi_{2}, \cdot.., \xi_{m})\in T_{\mathrm{x}}\mathbb{R}^{m}$:$T_{\mathrm{x}}\mathbb{R}^{m}$:$\sqrt{\xi_{2^{+}}^{2}\xi^{2}3++\xi_{m}^{2}}\geq\lambda|\xi_{1}|\mathrm{I}$,is a real number. The next lemma guarantees that these cones are invariant

and expanded exponentially byand$(DF_{a,b})_{\mathrm{x}}$$(DF_{a,b}^{-1})_{\mathrm{x}}$, respectively. Proposition 2 is given

directly from this lemma.
Lemma 3 Let
and

$b$

belong to$\mathit{8}ati\mathit{8}fying$$F_{a,b}$$\mathcal{H}_{m}$be the m-dimen8ional special H\’enon family. For each$\mathrm{x}\in\Lambda_{a,b}$, there exist the$(DF_{a,b})\mathrm{x}C^{u}(\mathrm{x})\subset C^{u}(F_{a,b(\mathrm{x})})$

and

(ii)

$(DF_{a,b}-1){}_{\mathrm{x}}C^{s}(\mathrm{X})\subset C^{s}(F_{a,b}^{-1}(\mathrm{x}))$and$|| \mathrm{v}||=\max\{|\xi_{1}|, |\xi_{2}|, \cdots , |\xi_{m}|\}$

Proof. See [3].

5and$un\mathit{8}table$that the $parameter\mathit{8}a$$\mathit{8}tablecone\mathit{8}$in$T_{\mathrm{x}}\mathbb{R}^{m}$the following conditions:

(i)

where

$\mathit{8}uch$$||(DF_{a,b})_{\mathrm{x}}\mathrm{v}||\geq\lambda||\mathrm{v}||$

for all$||(DF_{a}^{-},m+1)b\mathrm{x}\mathrm{V}||\geq\lambda||\mathrm{v}||$for$\mathrm{v}=$$(\xi_{1}, \xi_{2}, \cdots , \xi_{m})$$\mathrm{v}\in C^{u}(\mathrm{x})$for all;$\mathrm{v}\in C^{s}(\mathrm{X})f$.

$\blacksquare$

Proof for the 3-dimensional special caseIn [3], all proofs of the above statements are presented perfectly. In this last section, wegive a proof of the above lemma for the special 3-dimensional case.

Let$b=0.1$ .

$\{F_{a,b}\}$be the same family as the previous subsection. We take $m=3,$ $a=5$ andThen, we have
$r= \frac{1+0.1+\sqrt{(1+01)^{2}+45}}{2}<3$$\tilde{D}=\{(x, y, z)\in \mathrm{R}^{3} :Let A be the maximal invariant set of$\tilde{D}$.

|x|\leq 3, |y|\leq 3, |z|\leq 3\}$

under

$F$

. For any point$\mathrm{x}\in\Lambda$

unstable cone by$C^{u}(\mathrm{x})=\{(\xi, \eta,\zeta)\in\tau \mathrm{x}\mathrm{R}^{3}$:$\lambda\sqrt{\eta^{2}+\zeta^{2}}\leq|\xi|\}$.

,, we define the99and take a stable cone in the complement of$C^{u}(\mathrm{x})$:$C^{s}(\mathrm{x})=\{(\xi, \eta, \zeta)\in\tau \mathrm{x}\mathrm{R}^{3}$

where$\lambda$

Lemma 4 There is awhere$\lambda_{1}>1$

such that for any$(DF){}_{\mathrm{x}}C^{u}(\mathrm{x})\subset C^{u}(F(\mathrm{x}))$and$(DF^{-1})\mathrm{x}Cs(\mathrm{x})\subset C^{S}(F^{-}1(\mathrm{x}))$$|| \mathrm{v}||=\max\{|\xi|, |\eta|, |\zeta|\}$for$\mathrm{x}\in\Lambda$$(DF^{-1})_{\mathrm{x}}$as:$||(DF)_{\mathrm{x}}\mathrm{V}||\geq\lambda||\mathrm{v}||$andthe unstable and stable cones

for all$||(DF^{-2})\mathrm{x}\mathrm{V}||\geq\lambda||\mathrm{v}||$$\mathrm{v}=(\xi, \eta, \zeta)\in C^{u}(\mathrm{x}),$$\mathrm{v}\in C^{u}(\mathrm{X})$for all$C^{s}(\mathrm{x})$;$\mathrm{v}\in C^{s}(\mathrm{x})$,.

$\mathrm{v}=(\xi, \eta, \zeta)\in C^{u}(\mathrm{x})$, we denote. The image of the vector by the derivative is given by$==$$(\xi_{1}, \eta_{1}, \zeta_{1})$We estimate$|\xi_{1}|=|-2x\xi+b\zeta|$$\geq$$\geq$$\geq$

As).

$1<\lambda\leq\lambda_{1}$Proof. Let $\mathrm{x}=(x, y, z)$ be a point in A. For any$(DF)_{\mathrm{x}}\mathrm{v}$:$\sqrt{\eta^{2}+\zeta^{2}}\geq\lambda|\xi|\}$and$(DF)_{\mathrm{x}}$satisfy the following conditions for all(ii)$T_{\mathrm{x}}\mathrm{R}^{3}$

is a positive real number. In the next lemma, we show that these cones areinvariant, moreover expanded, under

(i)

in

$\mathrm{x}=(x, y, z)\in\tilde{D}\cap F(\tilde{D}),$.

$2|x||\xi|-|b||\zeta|$$2|x||\xi|-|b|\sqrt{|\xi/\lambda|^{2}-\eta^{2}}$$(2|x|-|b||\lambda^{-1}|)|\xi|$.

$a=5,$ $b=0.\acute{1}$ and $r=3$ , we have$a+bz-x^{2}\leq r,$$i.e$. $5+0.1z-x^{2}\leq 3$ .

Then we have $x^{2}\geq 2+0.1z\geq 2-0.3$ . Thus $|x|\geq\sqrt{17}>1.3$ . We assume that $\lambda=1.5$ .

As $|x|>1.3$ , we have.

$|\xi_{1}|>(2\cdot 1.3-\mathrm{o}.1\sim$$\frac{2}{3}\mathrm{I}|\xi|>2.533|\xi|$

.

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We next estimate$\lambda\sqrt{\eta_{1^{2}}+\zeta_{1^{2}}}<$$\lambda\sqrt{\xi^{2}+|\xi/\lambda|2-\zeta 2}$$<$$\lambda\sqrt{\xi^{2}+\xi^{2}\lambda^{-2}}$$<$$(\lambda\sqrt{1+\lambda^{-2}})|\xi|<2.167|\xi|$.

Therefore,$|\xi_{1}|>2.533|\xi|>2.167|\xi|>\lambda\sqrt{\eta_{1^{2}}+\zeta_{1^{2}}}$

Then, we obtain$||(\xi_{1},$$(DF)_{\mathrm{x}}\mathrm{v}=(\xi_{1}, \eta_{1}, \zeta_{1})\in C^{u}(F(\mathrm{x}))$$\eta_{1},$

,

, and

() $||=|\xi_{1}|>2.533|\xi|>2.167|\xi|>1.5|\xi|=\lambda||(\xi,$$\eta,$$()||$.

This completes the proof of (1).
For the second claim of this lemma, for any
by$(\xi_{-1}, \eta_{-1}, \zeta_{-1})i.e.$$\mathrm{v}=(\xi, \eta, \zeta)\in C^{s}(\mathrm{x})$, we denote$(DF^{-1})_{\mathrm{x}^{\mathrm{V}}}$,$==\sim$
As before we take $b=0.1$ and $\lambda=1.5$ . As$\mathrm{x}\in\overline{D}\cap F^{-1}(\tilde{D})$, we have$|b^{-1}(-a+x+y^{2})|\leq r$ ,and then$y^{2}\geq a-x-br\geq 5-3-0.1\cdot 3=1.7$ .

Thus $|y|\geq\sqrt{17}>1.3$ . When the vector$\mathrm{v}$satisfies$|\zeta|\geq\lambda|\eta|$, that is,$|\zeta|\geq 1.5|\eta|$

estimate$\sqrt{\eta_{-1^{2}}+\zeta_{-1}2}=\sqrt{\zeta^{2}+b^{-2}(\xi+2y\eta)2}\geq|\zeta|\geq 1.5|\eta|=\lambda|\xi_{-1}|$

Similarly, when the vector$\sqrt{\eta_{-1^{2}}+(-12}=$$\mathrm{v}$satisfies$\lambda|\eta|\geq|\zeta|$.

, we get$\sqrt{\zeta^{2}+b^{-2}(\xi+2y\eta)2}\geq|b^{-1}||\xi+2y\eta|$$\geq$$|b^{-1}|(2|y||\eta|-|\xi|)$$\geq$$|b^{-1}|(2|y||\eta|-\lambda-1\sqrt{\eta^{2}+\lambda^{2}\eta^{2}}\mathrm{I}$$\geq$$|b^{-1}|(2|y|-\sqrt{\lambda^{-2}+1})|\eta|\geq 13.98|\eta|>1.5|\eta|=\lambda|\xi_{-1}|$

.

, we

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Therefore, we obtain$(\xi_{-1}, \eta_{-1}, \zeta_{-1})\in c^{s}(F^{-}1(_{\mathrm{X}}))$

We now estimate

vector$\mathrm{v}$$||(DF^{-n})\mathrm{x}^{\mathrm{V}|}|$

for every$\mathrm{v}\in C^{s}(\mathrm{x})$.

. We divide the proof which theis expanded by the derivative of the inverse into three parts. When$\max\{|\xi|, |\eta|, |\zeta|\}=|\xi|$, that is,and$|\xi|\geq|\eta|$$|\xi|\geq|\zeta|$, we estimate$|b^{-1}(\xi+2y\eta)|\geq|b^{-1}|(2|y||\eta|-|\xi|)$as$\sqrt{\eta^{2}+\zeta^{2}}\geq\lambda|\xi|$

Therefore we haveand$|\xi|\geq|\zeta|$$||\mathrm{v}||=$,, we have$||(DF^{-1})\mathrm{x}\mathrm{V}||=|b^{-1}(\xi+2y\eta)|\geq\lambda||\mathrm{v}||$. When$||\mathrm{v}||=|\eta|$, we similarly

estimate$||(DF^{-1})_{\mathrm{x}}\mathrm{v}||$

Let

$=$

$|b^{-1}(\xi+2y\eta)|\geq|b^{-1}|(2|y||\eta|-|\xi|)$$\geq$$|b^{-1}|(2|y|-1)|\eta|>16|\eta|>\lambda||\mathrm{v}||$

.

$==$

$||\mathrm{v}||=|\zeta|$. We denote$(DF^{-2})_{\mathrm{x}^{\mathrm{V}}}$as$(\xi_{-2}, \eta_{-2}, \zeta_{-2})$

which is given by,,where $(x_{-1}, y_{-1}, z_{-1})=F^{-1}(x, y, z)$ . As similarly we have$||(DF^{-2})\mathrm{x}\mathrm{V}||$

Therefore we get$=$$|b^{-1}(\eta+2y_{-1}\zeta)|\geq|b^{-1}|(2|y_{-1}||\zeta|-|\eta|)$$\geq$$|b^{-1}|(2|y-1|-1)|\zeta|>16|\zeta|>\lambda||\mathrm{v}||$$||(DF^{-2})\mathrm{x}\mathrm{V}||\geq\lambda||\mathrm{v}||$lemma.

for all$\mathrm{v}\in C^{s}(\mathrm{x})$.

. This completes the proof of the$\blacksquare$Acknowledgments
We would like to thank Professor Atsuro Sannami for his helpful suggestions.

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Math. Phys., 67, 137-146 (1979)[2] H\’enon, M.: A two-dimensional mapping with a strange attractor. Commun. Math.

Phys., 50, 69-77 (1976)[3] Kiriki, S., Nakajima, M.: Hyperbolicity of a multi-dimensional diffeomorphism of
H\’enon type. to appear Hokkaido Mathematical Joumal (1998)[4] Smale, S.: Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967)

747-817.
[5] Viana, M.: Strange attractors in higher dimensions. Bol. Soc. Bras. Mat.24 No. 1,

13-62 (1993)[6] Yang, Z.: H\’enon-like maps before or at the first tangency. preprint of PhD theise8(1996)