数理解析研究所講究録
第 760 巻 1991 年 57-65$57\sim$

ON A QUASI-POTENTIAL
IN CONSTRAINED DIFFERENTIAL EQUATIONSKIYOYUKI TCHIZAWA
Department of Administration Engineering
Faculty of Science and Technology
Keio University

慶大 理工

知沢 清之

Abstract

It is well known that Lyapunov function gives a sufficient condition for the stability of differential equations. However, the problem how we can construct the function is not solved, except insome special cases.
On constrained differential equations, this paper gives a sufficient condition for the existence of a generalized ,Lyapunov function which is called a quasi-potential. We shall show that thispotential is induced from the constraint space. As a result, notonly a neccessary condition but a sufficient condition for the existence of the orbit with jump is obtained simultaneously. Furtheremore, being given a suitable cross section, a problem wherean attractor exists in this system is solved through the potential.
Hence, in constrained differential equations, we can now analyzeboth the unstability including jump phenomena and the stabilityof the systems.

1. Introduction
In differential equations, generally, we attempt to construct a multivariables scalar function on which the stability and the unstability of thesystem are represented.
But we shall soon be found that this attempt does not succeed, because there are no mutual relations between the differential equations
and the scalar function. Therefore, we restrict the equations by someconstraints. Then we shall come to deal with constrained differential
1

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equations or its (singular perturbations”. In the system, we shall try toconstruct the function on which the above properties are represented under the assumptions. And we shall show the conditions with which jump
phenomena in (1) occur.

2.Constrained differential equations and its quasi-potential
Throughout this paper, we shall consider the following constrained
differential equation, $x\in R^{n},$and $u\in R^{m}$ ,$y\in R^{l}$,th $=f(x, y)$(1)$( \dot{x}=\frac{dx}{dt})$$g(x, y, u)=0$ ,
$f$

$g$

::

$R^{n}xR^{l}arrow R^{n}$,$R^{n}\cross R^{l}xR^{m}arrow R^{l},$ $(n\leq l)$,whereare state variables and is a parameter. Then, we shall givea sufficient condition for the existence of a quasi-potential (multi variables scalar function) defined under mentioned. Now, letbe an$N(x_{0})$isolated singular point of (1) andbe a neighbourhood of.
$u$$x,$ $y$$F$$x_{0}\in R^{n}$$x_{0}$

Definition 2.1 A quasi-potentialproperties ), ), ) on $N(x_{0})$ :a) for any ,
$a$

$b$

$F$

: $R^{n}xR^{m}arrow R$ has following

$c$

$u$

$F(x(t), u)$ is smooth with respect to$t$,(2)b) for any ,$u$$\frac{\partial F(x,u)}{\partial x_{i}}=0$,$i=1,2,$$\cdots$, ,(3)$n$c) for any ,$u$$N(x_{0})=N_{u}^{0}(x_{0})\cup N_{u^{+}}\cup N_{u^{-}}$

and$N_{u}(x_{0})\neq N_{u}^{0}(x_{0})$

,

(4)

where
$N_{u}^{0}(x_{0})=\{x\in N(x_{0})|\dot{F}(x, u)=0\}$$N_{u}^{+}=\{x\in N(x_{0})|\dot{F}(x, u)>0\}$

2,,

(5)
(6)

59

and
$N_{u^{-}}=\{x\in N(x_{0})|\dot{F}(x, u)<0\}$.

(7)

The condition a) is a natural as sumption. The condition b) assertsin (1) is a singular point of , simulthat the isolated singular point
on which the value oftaneously. The condition c) asserts thatis invariant divides $N(x_{0})$ into two regions. One is monotone increasing
and the other is monotone decreasing, respectively.
$F$$x_{0}$$F$$N_{u}^{0}(x_{0})$Definition 2.2 If $N(x_{0})\neq R^{n}$ , we call this function
potential. If $N(x_{0})=R^{n}$ , we call it a global one.

$F$a local quasi-

If only the property (7) holds on $N(x_{0})$ , then it is called Lyapunov
function. Therefore, this function defined above is a generalized Lyapunov function.

3. $The$ conditions for the existence of the quasi-potential
On the parameter space
there is a divisionof$U_{1},$$U_{2}$$R^{m}$, we assume that a critical state in which$R^{m}$;$J_{u\in U_{1}}\neq\emptyset$

and

(8)

$J_{u\in U_{2}}=\emptyset$occur, where$J_{u}=${ $x\in R^{n}|f(x,$$y)$

is discontinuous}.

(9)

And we as sume that $f(x, y)$ is Lipschitz continuity for any
Let $D_{y}g(x, y, u)$ denote Jacobian matrix of and $g_{y}(x, y, u)$ denotepartial derivative of with respect to .
$x\in R^{n}\backslash J_{u}$

$g$

$y$

$g$

Lemma 3.1 Changing the coordinate, in the system (1), if$rankD_{x}g(x, y, u)=n,$ $rankD_{y}g(x, y, u)\geq n$ and $rankD_{y}g(H_{1}(y, u),$)for some , then the set
in the system rewritten into (12) is notempty.
$y,$

$n$

$u$

$U_{1}$

3$u$$<$

60proof) Calculating the constraint equation in (1),$g_{x}(x, y, u)\dot{x}+g_{y}(x, y, u)\dot{y}=0$.

(10)

When choosing the appropriate coordinate which is a sub vector ofas state variables, from the first condition, there is a smooth function:
such that$z$$y$$H_{1}$$R^{l}\cross R^{m}arrow R^{n}$$x=H_{1}(y, u)$.

(11)

Substituting (11) to (10), (l)is transformed into (12),)$g_{y}(H_{1}(y, u),$ $y,$ $u$ $\dot{y}=-g_{x}(H_{1}(y, u),$ $y,$ $u$

From the second condition, 1 $D_{y}g(H_{1}(y, u),$in (12) is represented as follows:
fore$y,$ $u$) $f(H_{1}(y, u),$$y,$ $u$) $|=0$ for some).

$y,$ $u$(12), there-$J_{u}$$J_{u}=\{z\in R^{n}|rankD_{y}g(H_{1}(y, u), y, u)<n\}$.

(13)

Then,
$U_{1}=\{u\in R^{m}|rankD_{y}g(H_{1}(y, u), y)u)<n\}\neq\emptyset$.

(14)$\square$

Let $f_{2}(x, u)$ denote $f(x, y)$ constrained by (1). Then, under the twoconditions again, we get the following theorem.
Theorem 3.2 If rank $D_{y}g(x, y, u)\geq n$ and the equation (1) has theproperty of hyperblicity at $x_{0}\in R^{n}$ then there is a quasi-potential on$N(x_{0})$.

Proof) From the condition $|D_{y}g(x, y, u)|\neq 0$ , there is a smooth func:tionsuch that$H_{2}$$R^{n}\cross R^{m}arrow R^{l}$$y=H_{2}(x, u)$.

(15)

Choosing n-dimensional coordinate from (15), multi variables scalar function, $F_{k}(x, u)$ : $R^{n}xR^{m}arrow R$ is defined as follows:$F_{k}(x, u)= \sum_{i=1}^{n}k_{i}y;$4

.

(16)

61

Then, (16) represents the constraint space$\Sigma_{u}$;$\Sigma_{u}=\{x\in R^{n}|g(x, H_{2}(x, u), u)=0\}$.

(17)

On the other hand, by using (15)$f_{2}(x, u)=f(x, H_{2}(x, u))$,(18)

therefore we can rewrite the system (1) as follows:.

$\dot{x}=f_{2}(x, u)$(19)From the other condition of hyperblicity in (1), for some $u,$
has positive or negative eigen values. As the distinct eigen vectors holdorthogonal relations (22), following (20),(21) are established.
Let $\lambda_{r:}\in R^{n}(1\leq i\leq n)$ be a right-side eigen vector and$j\leq n$ ,$i+j=n$ ) be a left-side eigen vector associated with a negative,, respectively.
positive eigen value ofThen, a gradient vectorofat the stable (unstable) orbit keepsorthogonal relations;$D_{x}f_{2}(x_{0}, u)$$\mu_{l_{j}}\in R^{n}(1\leq$$D_{x}f_{2}(x_{0}, u)$$\nabla_{x}F_{k}$$F_{k}$$\nabla_{x}F_{k}(x, u)\cdot\lambda_{r}$.

$=0$ ,$1\leq i\leq n$,(20),$1\leq j\leq n$

,

(21)

$\nabla_{x}F_{k}(x, u)\cdot\mu_{l_{j}}=0$$\lambda_{r_{i}}\cdot\mu_{l_{j}}=0$,$\mu_{r_{i}}\cdot\mu_{l_{j}}=0$

and$\lambda_{r_{i}}\cdot\lambda_{l_{j}}=0$.

(22)

Therefore, $k\in R^{n}$ is determined uniquely by solving (20), (21). From(16), $for$ some(23)$u$$N_{u}^{0}(x_{0})\neq\emptyset$

Moreover, as $F(x, u)$ is smooth, there is awhich contains $x_{0}\in R^{n}$tangent space atsatisfying (3). From the assumption, the signs ofeigen values are positive or negative, (20) and (21) give the property (4).
Lipschitz continuity of $f(x, y)$ assures the existence and uniqueness ofsolutions in (1). So, (4),(5),(6),(7) are satisfied.
$x_{0}$$\square$5

62

4. $A$ jumping condition on the quasi-potential

Some properties are on the quasi-potential
does not hold local solvability at some$u\in R^{m}$ , letdenote a subset ofsuch thatwhen the system (1)in (17). For some$F$$p\in\Sigma_{u}\subset R^{n+1}$$R^{n}$$L_{u}$$L_{u}=\{x\in N_{u}^{0}(x_{0})\backslash \{x_{0}\}\}$and let$\Pi_{x}$:

,

(24)

be the natural projection defined by$\Sigma_{u}arrow R^{n}$$\square _{x}(x, u)=x$.

(25)

Next Lemma 4.1 is a well-known neccessary condition for the existence of jumping points.

is locally solvable, then is not a jumping
Lemma 4.1 Ifpoint.
proof) As a vector field is defined uniquely under this condition, isnot ajumping point.
$p\in\Sigma_{u}$

$p$

$p$

$\square$

Theorem 4.2 Let a quasi-potential on $N(x_{0})$ exist. Thenat someand then the tangent spacedoes not intersect$KerII_{x}$ , the kernel of, transversally.
$x\in L_{u}$ ,$u\inR^{m},$proof) For some$F$

$L_{u}\neq\emptyset$$p\in\Sigma_{u}$$T_{p}(\Sigma_{u})$$II_{x}$$\dot{F}(x, u)=(\frac{\partial F}{\partial x}I\cdot(\frac{dx}{dt}I=0$.

(26)

The relation (26) implies that the tangent vector along the orbit in(1) and the gradient vector ofkeep an orthogonal relation. Therefore, is not locally solvable. As is locally solvable iffintersects$KerII_{x}$ , it is concluded from Lemma 4.1.
$T_{p}(\Sigma_{u})$$p$$T_{p}(\Sigma_{u})$$p$$\square$Now, we shall reduce a sufficient condition for the existence of ajumping point. If $HF(x, u)$ , Hessian matrix of $F(x, u)$ for some $u\in R^{m}$ , ispositive (negative) definite with respect to , then $HF(x, u)$ restricts
to jumping states. Let
such that
denote a sub set of$L_{u}$$\dot{x}$$LJ_{u}$$L_{u}$$LJ_{u}=\{x\in L_{u}|\dot{x}^{t}HF(x, u)\dot{x}\neq 0\}$

6.

(27)

63

Theorem 4.3 If, then.
proof) As a point ofgives a minimal (or maximal) value onalong the orbit in (1), from Theorem4.2 and (5), (6), (7), this point isa jumping state.
$LJ_{u}\neq\emptyset$$LJ_{u}\subset J_{u}$$F$$LJ_{u}$$\square$On the other hand, this quasi-potential has a following general property. As (elementary catastrohp\’es’’ are some equivalence families of$(m\leq$$F(x, u)$ with a non-degenerate singular point $(x_{0}, u_{0})\in R^{n}\cross R^{m}$4), we can conclude the following corolary.

Corollary 4.4 If $u_{0}\in R^{m}$ is a singular point of$F(x, u)$ belongs to elementary catastroph\’es.

$F$and $m\leq 4$ , thenIn the following section, we shall take up van der Pol’s equation as atypical, simple example $(n=m=l=1)$ .

5. $Van$ der Pol’s equation
His equation is represented as follows:$\dot{x}=-y+a$$\epsilon\dot{y}=x-f(y, u)$whereby$a$,$\epsilonarrow 0$

(28)

,

is any constant. In this system, the constraint space$\Sigma_{u}=\{x\in R|x-f(y, u)=0\}$.

$\Sigma_{u}$is given(29)

By changing coordinate,$\dot{x}=\frac{\partial}{\partial y}f(u, y)\dot{y}$,(30)and substituting (30) to (28),$\frac{\partial}{\partial y}f(y, u)\dot{y}=-y+a$.

(31)

Generally, $f(y, u)$ is given as follows:$f(y, u)=\underline{y^{3}}-uy$

3
7,

(32)

64

therefore we rewrite (31) into (33),$(y^{2}-u)\dot{y}=-y+a$ .

(33)$F(y, u)=x=f(y, u)$ ,(34)

,(35)

Then,
$U_{1}=\{u\in R|u\geq 0\}$

$U_{2}=\{u\in R|u<0\}$and$J_{u}= \{x=\frac{2}{3}u\sqrt{u}, -\frac{2}{3}u\sqrt{u}\}$.

(36)

areThis function (34) is a global quasi-potential. These points injumping states by Theorem4.3. We have already investigated the cases
in which there exists a local quasi-potential $(n=2,m=1,l=12)$ and a
global quasi-potential $(n=m=2, l=16)$ on the system [1].
$J_{u}$8

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References[1] K.Tchizawa,“On a quasi-potential in nonlinear circuits”, Yokohama
Mathematical Journal, to appear (1990)[2] S.Smale, (On the mathematical foundations of electrical circuit theory”, J.Differential Geometry, 7,193-210 (1972)[3] A.Andronov,A.Vitt and S.Khaikin, (Theory of oscillator”, Pergamon
Press (1965)[4] R.Thom, (Stabiht\’e structurelle et morpheg\’en\‘ese’’, Inter Editions (1977)

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