Normal form of cibrario

The normal form of Cibrario is the normal form of a differential equation that is not resolved with respect to the derivative in the vicinity of the simplest singular point. The name was proposed by V.I. Arnold in honor of the Italian mathematician Maria Cibrario , who established this normal form for one class of equations ^[1] ^[2] ^[3] .

Content

1 Related Definitions
- 1.1 Special points
- 1.2 Raising the equation
2 Normal form theorem
3 Examples
4 Literature
5 notes

Related Definitions

Feature Points

Let the differential equation have the form

$F(x,y,p)=0,\$ ${\ displaystyle F (x, y, p) = 0, \}$ Where $p={\frac {dy}{dx}}.$ ${\ displaystyle p = {\ frac {dy} {dx}}.}$

Function $F$ ${\ displaystyle F}$ assumed real, smooth class $C^{\infty }$ ${\ displaystyle C ^ {\ infty}}$ (or analytical ) in the aggregate of all three variables. The singular points of such an equation are points of three-dimensional space with coordinates $(x,y,p)$ ${\ displaystyle (x, y, p)}$ lying on the surface given by the equation $F=0$ ${\ displaystyle F = 0}$ in which the derivative $F_{p}$ ${\ displaystyle F_ {p}}$ vanishes, i.e. design $\pi$ ${\ displaystyle \ pi}$ surface $\{F=0\}$ ${\ displaystyle \ {F = 0 \}}$ to the plane of variables $x, y$ ${\ displaystyle x, y}$ along the axis $p$ ${\ displaystyle p}$ irregularly. In the general case, many singular points form on the surface $\{F=0\}$ ${\ displaystyle \ {F = 0 \}}$ a curve called a criminant . Projection of criminants on a plane $(x,y)$ ${\ displaystyle (x, y)}$ is called the discriminant curve , its points are also often called singular points of the equation, although inaccuracy is possible: when designing $\pi$ ${\ displaystyle \ pi}$ different surface points $\{F=0\}$ ${\ displaystyle \ {F = 0 \}}$ the same point in the plane of variables can correspond $(x,y)$ ${\ displaystyle (x, y)}$ ^[1] ^[4] ^[5] .

Raising an equation

Differential ratio $p=dy/dx$ ${\ displaystyle p = dy / dx}$ sets in space $(x,y,p)$ ${\ displaystyle (x, y, p)}$ contact plane field $pdx-dy=0$ ${\ displaystyle pdx-dy = 0}$ . The intersection of contact planes with planes tangent to the surface $\{F=0\}$ ${\ displaystyle \ {F = 0 \}}$ , sets on the last field of directions (defined at all points where the contact and tangent planes do not coincide with each other). The integral curves of the field constructed in this way are 1-graphs of the solutions of the original equation, and their projections onto the plane $(x,y)$ ${\ displaystyle (x, y)}$ - decision graphs ^[4] ^[5]

Raising the equation to the surface

The described construction of the study of equations that are not resolved with respect to the derivative goes back to the third memoir of A. Poincare “On curves determined by differential equations” (1885); in modern mathematical literature, it is often called lifting an equation to the surface ^[3] .

Normal form theorem

The simplest singular points of the equation $F(x,y,p)=0$ ${\ displaystyle F (x, y, p) = 0}$ are the so-called regular singular points at which the design $\pi$ ${\ displaystyle \ pi}$ has a feature called the Whitney fold , and the contact plane does not touch the surface $F=0.$ ${\ displaystyle F = 0.}$ This is equivalent to the fulfillment of conditions at a given point:

F=0,\quad F_{p}=0,\quad F_{pp}\neq 0,\quad F_{x}+pF_{y}\neq 0.

{\ displaystyle F = 0, \ quad F_ {p} = 0, \ quad F_ {pp} \ neq 0, \ quad F_ {x} + pF_ {y} \ neq 0.}

Theorem In a neighborhood of a regular singular point, the equation $F(x,y,p)=0$ ${\ displaystyle F (x, y, p) = 0}$ with smooth (or analytic) function $F$ ${\ displaystyle F}$ smoothly (respectively, analytically) is equivalent to the equation

p^{2}-x=0,

{\ displaystyle p ^ {2} -x = 0,}

called the normal form of Cibrario ^[1] ^[4] ^[5] .

In 1932, Cibrario got this normal form, exploring the characteristics of a mixed- order second-order partial differential equation ^[2] .

Examples

The normal form of Cibrario is the characteristic equation for the Tricomi equation

$u_{xx}-xu_{yy}=0$ ${\ displaystyle u_ {xx} -xu_ {yy} = 0}$ ,

elliptical in the half-plane $x<0$ ${\ displaystyle x <0}$ and to hyperbolic - in the half-plane $x>0$ ${\ displaystyle x> 0}$ .

Cibrario normal form solution family

The equation $p^{2}-x=0$ ${\ displaystyle p ^ {2} -x = 0}$ easy to integrate: graphs of its solutions form a family of semi-cubic parabolas ^[4] ^[5]

$y=\pm {\frac {2}{3}}x^{\frac {3}{2}}+{\rm {const,}}$ ${\ displaystyle y = \ pm {\ frac {2} {3}} x ^ {\ frac {3} {2}} + {\ rm {const,}}}$

filling half-plane $x>0$ ${\ displaystyle x> 0}$ whose return points lie on the discriminant curve - the axis $y$ ${\ displaystyle y}$ .

The asymptotic lines of a two-dimensional surface in Euclidean space in a neighborhood of a typical parabolic point look similarly. The normal form of Cibrario also corresponds to the simplest features of the field of slow motion in fast-slow dynamic systems ^[6] .

Literature

Arnold V. I. Additional chapters of the theory of ordinary differential equations, - Any publication.
Arnold V.I. Geometric methods in the theory of ordinary differential equations, - Any publication.
Arnold V.I., Ilyashenko Yu. S. Ordinary differential equations, - Itogi Nauki i Tekhniki. Ser. Modern prob. mat. Fundam. Direction., 1985, Volume 1.
Arnold V.I., Afraimovich V.S., Ilyashenko Yu.S., Shilnikov L.P. The theory of bifurcations, - Itogi Nauki i Tekhniki. Ser. Modern prob. mat. Fundam. Direction., 1986, Volume 5.
Cibrario M. Sulla reduzione a forma canonica delle equazioni lineari alle derivative parzialy di secondo ordine di tipo misto, - Rend. Lombardo 65 (1932), pp. 889–906.

Notes

↑ ¹ ² ³ Arnold V.I., Ilyashenko Yu. S. Ordinary differential equations, - Itogi Nauki i Tekhniki. Ser. Modern prob. mat. Fundam. Direction., 1985, Volume 1. - Ch. 1, par. 7.
↑ ¹ ² Cibrario M. Sulla reduzione a forma canonica delle equazioni lineari alle derivative parzialy di secondo ordine di tipo misto, - Rend. Lombardo 65 (1932), pp. 889–906.
↑ ¹ ² Remizov A.O. The multidimensional Poincare construction and features of raised fields for implicit differential equations, - CMFD, 19 (2006), 131–170.
↑ ¹ ² ³ ⁴ Arnold V.I. Additional chapters of the theory of ordinary differential equations. - ch. 1, par. four.
↑ ¹ ² ³ ⁴ Arnold V.I. Geometric methods in the theory of ordinary differential equations. - ch. 1, par. four.
↑ Arnold V.I., Afraimovich V.S., Ilyashenko Yu.S., Shilnikov L.P. The theory of bifurcations, - Itogi Nauki i Tekhniki. Ser. Modern prob. mat. Fundam. Direction, 1986, Volume 5

[autogenerated4-1] ¹ ² ³ Arnold V.I., Ilyashenko Yu. S. Ordinary differential equations, - Itogi Nauki i Tekhniki. Ser. Modern prob. mat. Fundam. Direction., 1985, Volume 1. - Ch. 1, par. 7.

[autogenerated3-2] ¹ ² Cibrario M. Sulla reduzione a forma canonica delle equazioni lineari alle derivative parzialy di secondo ordine di tipo misto, - Rend. Lombardo 65 (1932), pp. 889–906.

[autogenerated5-3] ¹ ² Remizov A.O. The multidimensional Poincare construction and features of raised fields for implicit differential equations, - CMFD, 19 (2006), 131–170.

[autogenerated1-4] ¹ ² ³ ⁴ Arnold V.I. Additional chapters of the theory of ordinary differential equations. - ch. 1, par. four.

[autogenerated2-5] ¹ ² ³ ⁴ Arnold V.I. Geometric methods in the theory of ordinary differential equations. - ch. 1, par. four.

[6] Arnold V.I., Afraimovich V.S., Ilyashenko Yu.S., Shilnikov L.P. The theory of bifurcations, - Itogi Nauki i Tekhniki. Ser. Modern prob. mat. Fundam. Direction, 1986, Volume 5