aarjaneiro/planetmath huggingface dataset:
Bernoulli equation

Bernoulli equation

The

Bernoulli equation

has the form

d

⁢

y

d

⁢

x

+

f

⁢

(

x

)

⁢

y

=

g

⁢

(

x

)

⁢

y

k

(1)

where

f

and

g

are

continuous

real functions and

k

is a (

≠

0

,

≠

1

).  Such a nonlinear equation (

http://planetmath.org/DifferentialEquation

) is got e.g. in examining the motion of a by

y

k

.  It yields

y

-

k

⁢

d

⁢

y

d

⁢

x

+

f

⁢

(

x

)

⁢

y

-

k

+

1

=

g

⁢

(

x

)

.

(2)

The substitution

z

:=

y

-

k

+

1

(3)

transforms (2) into

d

⁢

z

d

⁢

x

+

(

-

k

+

1

)

⁢

f

⁢

(

x

)

⁢

z

=

(

-

k

+

1

)

⁢

g

⁢

(

x

)

which is a

linear differential equation of first order

.  When one has obtained its

general solution

and made in this the substitution (3), then one has solved the Bernoulli equation (1).

References

1

N. Piskunov:

Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele

.  – Kirjastus Valgus, Tallinn (1966).

Title

Bernoulli equation

Canonical name

BernoulliEquation

Date of creation

2013-03-22 15:15:03

Last modified on

2013-03-22 15:15:03

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Derivation

Classification

msc 34C05

Synonym

Bernoulli differential equation

Related topic

RiccatiEquation

Generated on Fri Feb 9 20:41:35 2018 by

LaTeXML
Clairaut’s equation

Clairaut’s equation

The

ordinary differential equation

y

=

x

⁢

d

⁢

y

d

⁢

x

+

ψ

⁢

(

d

⁢

y

d

⁢

x

)

,

(1)

where

ψ

is a given

differentiable

real function

, is called

Clairaut’s equation

.

For solving the equation we use an auxiliary variable

p

=

:

d

⁢

y

d

⁢

x

and write (1) as

y

=

p

⁢

x

+

ψ

⁢

(

p

)

.

Differentiating this equation gives

p

=

x

⁢

d

⁢

p

d

⁢

x

+

p

+

ψ

′

⁢

(

p

)

⁢

d

⁢

p

d

⁢

x

,

or

[

x

+

ψ

′

⁢

(

p

)

]

⁢

d

⁢

p

d

⁢

x

=

0

.

The zero rule of product now yields the alternatives

d

⁢

p

d

⁢

x

=

0

(2)

and

x

+

ψ

′

⁢

(

p

)

=

0

.

(3)

Integrating (2) we get

p

=

C

(), and substituting this in (1) gives the

general solution

y

=

C

⁢

x

+

ψ

⁢

(

C

)

(4)

which presents a family of straight lines.

If (3) allows to solve

p

in of

x

,

p

=

p

⁢

(

x

)

,  we can write (1) as

y

=

x

⁢

p

⁢

(

x

)

+

ψ

⁢

(

p

⁢

(

x

)

)

,

(5)

which is easy to see satisfying (1).  The solution (5) may not be gotten from (4) using any value of

C

.  It is a

singular solution

which may be obtained by eliminating the

parameter

p

from the equations

y

=

p

⁢

x

+

ψ

⁢

(

p

)

,

x

+

ψ

′

⁢

(

p

)

=

0

.

Thus the singular solution presents the envelope of the family (4).

Example.

The Clairaut’s equation

y

=

x

⁢

d

⁢

y

d

⁢

x

+

a

⁢

d

⁢

y

d

⁢

x

1

+

(

d

⁢

y

d

⁢

x

)

2

has the general solution

y

=

C

⁢

x

+

C

⁢

a

1

+

C

2

and the singular solution

{

x

=

-

a

(

1

+

p

2

)

3

/

2

,

y

=

-

a

⁢

p

3

(

1

+

p

2

)

3

/

2

in a parametric form.  Eliminating the parametre

p

yields the form

x

2

3

+

y

2

3

=

a

2

3

,

which can be recognized to be the equation of an

astroid

.  The envelope (see “

determining envelope

(

http://planetmath.org/DeterminingEnvelope

)”) of the lines is only the left half of this curve (

x

≦

0

).  The usual parametric of the astroid is

x

=

a

⁢

cos

3

⁡

φ

,

y

=

a

⁢

sin

3

⁡

φ

(

0

≦

φ

<

2

⁢

π

).

References

1

N. Piskunov:

Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele

.  – Kirjastus Valgus, Tallinn (1966).

Title

Clairaut’s equation

Canonical name

ClairautsEquation

Date of creation

2015-02-04 11:20:02

Last modified on

2015-02-04 11:20:02

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

21

Author

pahio (2872)

Entry type

Derivation

Classification

msc 34C05

Synonym

Clairaut differential equation

Related topic

DAlembertsEquation

Related topic

FamousCurvesInThePlane

Related topic

IndexOfDifferentialEquations

Related topic

PerimeterOfAstroid

Related topic

SingularSolution

Related topic

DerivativeAsParameterForSolvingDifferentialEquations

Defines

astroid

Generated on Fri Feb 9 20:41:38 2018 by

LaTeXML
Hopf bifurcation theorem

Hopf bifurcation theorem

Consider a planar

system of ordinary differential equations

, written in such a form as to make explicit the dependence on a parameter

μ

:

x

′

=

f

1

⁢

(

x

,

y

,

μ

)

y

′

=

f

2

⁢

(

x

,

y

,

μ

)

Assume that this system has the origin as an equilibrium for all

μ

. Suppose that the

linearization

D

⁢

f

at zero has the two purely imaginary eigenvalues

λ

1

⁢

(

μ

)

and

λ

2

⁢

(

μ

)

when

μ

=

μ

c

. If the real part of the eigenvalues verify

d

d

⁢

μ

⁢

(

ℜ

⁡

(

λ

1

,

2

⁢

(

μ

)

)

)

|

μ

=

μ

c

>

0

and the origin is

asymptotically stable

at

μ

=

μ

c

, then

r

o

m

a

n

]

e

n

u

m

e

r

a

t

e

μ

c

i

s

a

b

i

f

u

r

c

a

t

i

o

n

p

o

i

n

t

;

f

o

r

s

o

m

e

μ

1

∈

ℝ

¯

s

u

c

h

t

h

a

t

μ

1

<

μ

<

μ

c

,

t

h

e

o

r

i

g

i

n

i

s

a

s

t

a

b

l

e

f

o

c

u

s

;

f

o

r

s

o

m

e

μ

2

∈

ℝ

¯

s

u

c

h

t

h

a

t

μ

c

<

μ

<

μ

2

,

t

h

e

o

r

i

g

i

n

i

s

u

n

s

t

a

b

l

e

,

s

u

r

r

o

u

n

d

e

d

b

y

a

s

t

a

b

l

e

l

i

m

i

t

c

y

c

l

e

w

h

o

s

e

s

i

z

e

i

n

c

r

e

a

s

e

s

w

i

t

h

μ

.

T

h

i

s

i

s

a

s

i

m

p

l

i

f

i

e

d

v

e

r

s

i

o

n

o

f

t

h

e

t

h

e

o

r

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m

,

c

o

r

r

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s

p

o

n

d

i

n

g

t

o

a

s

u

p

e

r

c

r

i

t

i

c

a

l

H

o

p

f

b

i

f

u

r

c

a

t

i

o

n

.

S

o

m

e

t

i

m

e

s

t

h

e

H

o

p

f

t

h

e

o

r

e

m

i

s

c

a

l

l

e

d

Poincaré-Andronov-Hopf theorem

s

i

n

c

e

i

t

w

a

s

i

n

d

e

p

e

n

d

e

n

t

l

y

d

i

s

c

o

v

e

r

e

d

b

y

A

n

d

r

o

n

o

v

i

n

1929

a

n

d

H

o

p

f

i

n

1943

a

n

d

P

o

i

n

c

a

r

é

h

a

d

d

i

s

c

u

s

s

i

o

n

o

f

s

u

c

h

r

e

s

u

l

t

i

n

1892

.

[

HK

]

References

HK

H

a

l

e

,

J

a

c

k

H

.

&

K

o

c

a

k

,

H

ü

s

e

y

i

n

:

D

y

n

a

m

i

c

s

a

n

d

B

i

f

u

r

c

a

t

i

o

n

s

.

S

p

r

i

n

g

e

r

-

V

e

r

l

a

g

,

N

e

w

Y

o

r

k

,

1991

.

Title

Hopf bifurcation theorem

Canonical name

HopfBifurcationTheorem

Date of creation

2013-03-22 13:18:45

Last modified on

2013-03-22 13:18:45

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

8

Author

Daume (40)

Entry type

Theorem

Classification

msc 34C05

Synonym

Poincaré-Andronov-Hopf

Generated on Fri Feb 9 20:41:47 2018 by

LaTeXML
Poincare-Bendixson theorem

Poincare-Bendixson theorem

Let

M

be an

open subset

of

ℝ

2

, and

f

∈

C

1

⁢

(

M

,

ℝ

2

)

. Consider the planar

differential equation

x

′

=

f

⁢

(

x

)

Consider a fixed

x

∈

M

. Suppose that the

omega limit set

ω

⁢

(

x

)

≠

∅

is

compact

,

connected

, and contains only finitely many equilibria. Then one of the following holds:

r

o

m

a

n

]

e

n

u

m

e

r

a

t

e

ω

(

x

)

i

s

a

f

i

x

e

d

o

r

b

i

t

(

a

p

e

r

i

o

d

i

c

p

o

i

n

t

w

i

t

h

p

e

r

i

o

d

z

e

r

o

,

i

.

e

.

,

a

n

e

q

u

i

l

i

b

r

i

u

m

)

.

ω

(

x

)

i

s

a

r

e

g

u

l

a

r

p

e

r

i

o

d

i

c

o

r

b

i

t

.

ω

(

x

)

c

o

n

s

i

s

t

s

o

f

(

f

i

n

i

t

e

l

y

m

a

n

y

)

e

q

u

i

l

i

b

r

i

a

{

x

j

}

a

n

d

n

o

n

-

c

l

o

s

e

d

o

r

b

i

t

s

γ

(

y

)

s

u

c

h

t

h

a

t

ω

(

y

)

∈

{

x

j

}

a

n

d

α

(

y

)

∈

{

x

j

}

(

w

h

e

r

e

α

(

y

)

i

s

t

h

e

a

l

p

h

a

l

i

m

i

t

s

e

t

o

f

y

)

.

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h

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c

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a

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d

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d

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p

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.

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(

a

f

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r

u

s

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n

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a

D

u

l

a

c

t

y

p

e

r

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l

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.

Title

Poincare-Bendixson theorem

Canonical name

PoincareBendixsonTheorem

Date of creation

2013-03-22 13:18:40

Last modified on

2013-03-22 13:18:40

Owner

jarino (552)

Last modified by

jarino (552)

Numerical id

4

Author

jarino (552)

Entry type

Theorem

Classification

msc 34C05

Classification

msc 34D23

Generated on Fri Feb 9 20:41:49 2018 by

LaTeXML
Bamòn’s theorem

Bamòn’s theorem

Every quadratic

vector field

in

ℝ

2

has a finite number of limit cycles.

[

BR

]

Historical note

This theorem is weaker than Dulac’s theorem but at the time of the publication of

[

BR

]

there was a problem in the proof of Dulac’s theorem.

References

BR

R. Bamòn:

http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1986__64_/PMIHES_1986__64__111_0/PMIHES_1986__64__111_0.pdf

Quadratic vector fields in the plane have a finite number of limit cycles, Publ. I.H.E.S. 64 (1986), 111-142.

Title

Bamòn’s theorem

Canonical name

BamonsTheorem

Date of creation

2013-03-22 14:28:37

Last modified on

2013-03-22 14:28:37

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

5

Author

Daume (40)

Entry type

Theorem

Classification

msc 34C07

Generated on Fri Feb 9 20:41:51 2018 by

LaTeXML
Bautin’s theorem

Bautin’s theorem

There are at most three

limit cycles

which can appear in the following quadratic system

x

˙

=

p

⁢

(

x

,

y

)

=

∑

i

+

j

=

0

2

a

i

⁢

j

⁢

x

i

⁢

y

j

y

˙

=

q

⁢

(

x

,

y

)

=

∑

i

+

j

=

0

2

b

i

⁢

j

⁢

x

i

⁢

y

j

from a

singular point

, if its type is either a focus or a center.

In 1939 N.N. Bautin claimed the above result and in 1952 submitted the proof

[

BNN1

]

.

[

GAV

]

References

GAV

Gaiko, A., Valery: Global

Bifurcation Theory

and Hilbert’s Sixteenth Problem. Kluwer Academic Publishers, London, 2003.

BNN1

Bautin, N.N.: On the number of limit cycles appearing from an equilibrium point of the focus or center type under varying coefficients. Matem. SB., 30:181-196, 1952. (written in Russian)

BNN2

Bautin, N.N.: On the number of limit cycles appearing from an equilibrium point of the focus or center type under varying coefficients.

Translation

of the

American Mathematical Society

, 100, 1954.

Title

Bautin’s theorem

Canonical name

BautinsTheorem

Date of creation

2013-03-22 14:28:46

Last modified on

2013-03-22 14:28:46

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

5

Author

Daume (40)

Entry type

Theorem

Classification

msc 34C07

Generated on Fri Feb 9 20:41:53 2018 by

LaTeXML
Dulac’s theorem

Dulac’s theorem

Let

x

˙

=

f

⁢

(

x

)

be an analytic planar system, then in any

bounded

region of the plane there is at most a finite number of

limit cycles

. Also any polynomial planar system has at most a finite number of limit cycles.

[

PL

]

note about the proof:

The proof was given by Dulac in 1923, but an error was found in the proof. In 1988 Jean Ecalle, Jacques Martinet, Robert Moussu, Jean Pierre Ramis and independently Yulij Ilyashenko corrected the error in the proof.

[

PL

]

References

PL

Perko, Lawrence:

Differential Equations

and

Dynamical Systems

. Springer-Verlag, New York, 1991.

Title

Dulac’s theorem

Canonical name

DulacsTheorem

Date of creation

2013-03-22 14:25:21

Last modified on

2013-03-22 14:25:21

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

6

Author

Daume (40)

Entry type

Theorem

Classification

msc 34C07

Generated on Fri Feb 9 20:41:54 2018 by

LaTeXML
Hilbert’s 16th problem for quadratic vector fields

Hilbert’s 16th problem for quadratic vector fields

Find a maximum

natural number

H

⁢

(

2

)

and relative position of

limit cycles

of a vector field

x

˙

=

p

⁢

(

x

,

y

)

=

∑

i

+

j

=

0

2

a

i

⁢

j

⁢

x

i

⁢

y

j

y

˙

=

q

⁢

(

x

,

y

)

=

∑

i

+

j

=

0

2

b

i

⁢

j

⁢

x

i

⁢

y

j

[

DRR

]

.

As of now neither part of the problem

(i.e. the bound and the positions of the limit cycles)

are solved. Although R. Bamòn in 1986 showed

[

BR

]

that a quadratic vector field has finite number of limit cycles. In 1980 Shi Songling

[

SS

]

and also independently Chen Lan-Sun and Wang Ming-Shu

[

ZTWZ

]

showed an example of a quadratic vector field which has four limit cycles

(i.e.

H

⁢

(

2

)

≥

4

)

.

Example by Shi Songling:

The following system

x

˙

=

λ

⁢

x

-

y

-

10

⁢

x

2

+

(

5

+

δ

)

⁢

x

⁢

y

+

y

2

y

˙

=

x

+

x

2

+

(

-

25

+

8

⁢

ϵ

-

9

⁢

δ

)

⁢

x

⁢

y

has four limit cycles when

0

<

-

λ

≪

-

ϵ

≪

-

δ

≪

1

.

[

ZTWZ

]

Example by Chen Lan-sun and Wang Ming-Shu:

The following system

x

˙

=

-

y

-

δ

2

⁢

x

-

3

⁢

x

2

+

(

1

-

δ

1

)

⁢

x

⁢

y

+

y

2

y

˙

=

x

⁢

(

1

+

2

3

⁢

x

-

3

⁢

y

)

has four limit cycles when

0

<

δ

2

≪

δ

1

≪

1

.

[

ZTWZ

]

References

DRR

Dumortier, F., Roussarie, R., Rousseau, C.: Hilbert’s 16th Problem for Quadratic Vector Fields. Journal of

Differential Equations

110, 86-133, 1994.

BR

R. Bamòn: Quadratic vector fields in the plane have a finite number of limit cycles, Publ. I.H.E.S. 64 (1986), 111-142.

SS

Shi Songling, A concrete example of the

existence

of four limit cycles for plane quadratic systems, Scientia Sinica 23 (1980), 154-158.

ZTWZ

Zhang Zhi-fen, Ding Tong-ren, Huang Wen-zoa, Dong Zhen-xi. Qualitative Theory of Differential Equations.

American Mathematical Society

, Providence, 1992.

Title

Hilbert’s 16th problem for quadratic vector fields

Canonical

name

Hilberts16thProblemForQuadraticVectorFields

Date of creation

2013-03-22 14:03:35

Last modified on

2013-03-22 14:03:35

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

11

Author

Daume (40)

Entry type

Conjecture

Classification

msc 34C07

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Rolle’s theorem for dynamical systems in the plane

Rolle’s theorem for dynamical systems in the plane

Let

ϕ

be a smooth

planar dynamical system

and

γ

a

continuous

curve which has a

tangent vector

at each point

(i.e.

γ

is

differentiable

)

. Then there is a

contact point

between any two points of the intersection of

γ

and a

separating solution

of

ϕ

.

[

KGA

]

References

KGA

Khovanskiĭ, A.G.: Fewnomials.

American Mathematical Society

, Providence, 1991.

Title

Rolle’s theorem for

dynamical systems

in the plane

Canonical name

RollesTheoremForDynamicalSystemsInThePlane

Date of creation

2013-03-22 14:06:40

Last modified on

2013-03-22 14:06:40

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

9

Author

Daume (40)

Entry type

Theorem

Classification

msc 34C08

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bifurcation

bifurcation

Bifurcation

refers to the splitting of

dynamical systems

. The parameter space of a dynamical system is

regular

if all points in the sufficiently small open neighborhood correspond to the dynamical systems that are

equivalent

to this one; a

parameter

point that is not regular is a bifurcation point.

For example, the

branching

of the

Feigenbaum tree

is an

instance

of bifurcation.

A

cascade

of bifurcations is a precursor to chaotic dynamics.
The topologist René Thom in his book on

catastrophe theory

in biology discusses the cusp bifurcation
as a basic example of (dynamic) ‘catastrophe’ in morphogenesis and biological development.

References

1

“Bifurcations”,

http://mcasco.com/bifurcat.html

http://mcasco.com/bifurcat.html

2

“Bifurcation”,

http://spanky.triumf.ca/www/fractint/bif_type.html

http://spanky.triumf.ca/www/fractint/bif_type.html

3

“Quadratic Iteration, bifurcation, and chaos”,

http://mathforum.org/advanced/robertd/bifurcation.html

http://mathforum.org/advanced/robertd/bifurcation.html

Title

bifurcation

Canonical

name

Bifurcation

Date of creation

2013-03-22 12:34:21

Last modified on

2013-03-22 12:34:21

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 34C23

Classification

msc 35B32

Classification

msc 37H20

Related topic

DynamicalSystem

Related topic

SystemDefinitions

Generated on Fri Feb 9 20:42:00 2018 by

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Bendixson’s negative criterion

Bendixson’s negative criterion

Let

𝐱

˙

=

𝐟

⁢

(

𝐱

)

be a planar system where

𝐟

=

(

𝐗

,

𝐘

)

t

and

𝐱

=

(

x

,

y

)

t

. Furthermore

𝐟

∈

C

1

⁢

(

E

)

where

E

is a simply connected region of the plane. If

∂

⁡

𝐗

∂

⁡

x

+

∂

⁡

𝐘

∂

⁡

y

(the

divergence

of the

vector field

f

,

∇

⋅

𝐟

)

is always of the same sign but not identically zero then there are no

periodic solution

in the region

E

of the planar system.

Title

Bendixson’s negative criterion

Canonical name

BendixsonsNegativeCriterion

Date of creation

2013-03-22 13:31:01

Last modified on

2013-03-22 13:31:01

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

5

Author

Daume (40)

Entry type

Theorem

Classification

msc 34C25

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Dulac’s criteria

Dulac’s criteria

Let

𝐱

˙

=

𝐟

⁢

(

𝐱

)

be a planar system where

𝐟

=

(

𝐗

,

𝐘

)

t

and

𝐱

=

(

x

,

y

)

t

. Furthermore

𝐟

∈

C

1

⁢

(

E

)

where

E

is a simply connected region of the plane. If there exists a

function

p

⁢

(

x

,

y

)

∈

C

1

⁢

(

E

)

such that

∂

⁡

(

p

⁢

(

x

,

y

)

⁢

𝐗

)

∂

⁡

x

+

∂

⁡

(

p

⁢

(

x

,

y

)

⁢

𝐘

)

∂

⁡

y

(the divergence of the vector field

p

⁢

(

x

,

y

)

⁢

𝐟

,

∇

⋅

p

⁢

(

x

,

y

)

⁢

𝐟

)

is always of the same sign but not identically zero then there are no

periodic solution

in the region

E

of the planar system. In addition, if

A

is an

annular region

contained in

E

on which the above condition is satisfied then there exists at most one periodic solution in

A

.

Title

Dulac’s criteria

Canonical name

DulacsCriteria

Date of creation

2013-03-22 13:37:09

Last modified on

2013-03-22 13:37:09

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

5

Author

Daume (40)

Entry type

Theorem

Classification

msc 34C25

Generated on Fri Feb 9 20:42:04 2018 by

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Lissajous curves

Lissajous curves

We consider the

ordinary differential equation

d

⁢

y

d

⁢

x

=

±

k

⁢

b

2

-

y

2

a

2

-

x

2

,

(1)

i.e.

(

d

⁢

y

d

⁢

x

)

2

=

k

2

⋅

b

2

-

y

2

a

2

-

x

2

,

(2)

where

a

,

b

and

k

are

positive

.  It’s evident that the equations

x

=

±

a

,

y

=

±

b

(3)

give

singular solutions

of (1).  The lines (3) divide the

x

⁢

y

-plane into nine parts, each one of which contains a family of

integral curves

of (1).

Denote by

R

the

rectangle

|

x

|

≦

a

,

|

y

|

≦

b

.  After

separation of variables

, we can write in

R

the

differential equation

as

∫

d

⁢

y

b

2

-

y

2

=

k

⁢

∫

d

⁢

x

a

2

-

x

2

,

which leads to the general integral

arcsin

⁡

y

b

=

±

k

⁢

arcsin

⁡

x

a

+

C

(4)

and therefore

y

=

±

b

⁢

sin

⁡

(

k

⁢

arcsin

⁡

x

a

+

C

)

.

(5)

The equation (5) with the

plus sign

represents a family of smooth arcs once the rectangle

R

.  Every single arc two opposite sides of

R

which are

tangent lines

of the arc.  The situation is analogical in the case of the minus sign.  One arc of both kinds

passes through

every

interior point

of

R

.

For better examining the curves (5) one may take

arcsin

x

a

=

:

t

for a

parametre

.  Then,

x

=

a

⁢

sin

⁡

t

,  and (5) may be replaced by

{

x

=

a

⁢

sin

⁡

t

,

y

=

b

⁢

sin

⁡

(

k

⁢

t

+

C

)

.

(6)

Letting

t

to change freely, for any given value of

C

the equations (6) represent a continuous curve formed by an arc from the first family and another arc from the second family.  All such curves (6) are integral curves of the equation (2) and are called

Lissajous curves

.

It may be shown that for any

rational

value of

k

, (6) is a smooth

closed curve

, except when the curve comes to a vertex of the rectangle

R

.  If the value of

k

is

irrational

, then (6) is never a closed curve, and any such curve fills the whole rectangle in the sense that it comes arbitrarily to every point of

R

.  In the former case, all integral curves of (2) are

algebraic

(

http://planetmath.org/AlgebraicFunction

).

Some figures in

http://de.wikipedia.org/wiki/Lissajous-Figur

Wiki

References

1

E. Lindelöf:

Differentiali- ja integralilasku III 1

.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).

Title

Lissajous curves

Canonical

name

LissajousCurves

Date of creation

2013-03-22 19:13:33

Last modified on

2013-03-22 19:13:33

Owner

rrogers (21140)

Last modified by

rrogers (21140)

Numerical id

13

Author

rrogers (21140)

Entry type

Example

Classification

msc 34C25

Classification

msc 34A05

Related topic

SawBladeFunction

Related topic

SequenceAccumulatingEverywhereIn11

Generated on Fri Feb 9 20:42:06 2018 by

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proof of Bendixson’s negative criterion

proof of Bendixson’s negative criterion

Suppose that there exists a periodic solution called

Γ

which has a period of

T

and lies in

E

. Let the interior of

Γ

be denoted by

D

. Then by Green’s

Theorem

we can observe that

∫

∫

D

∇

⋅

𝐟

⁢

d

x

⁢

d

y

=

∫

∫

D

∂

⁡

𝐗

∂

⁡

x

+

∂

⁡

𝐘

∂

⁡

y

⁢

d

⁢

x

⁢

d

⁢

y

=

∮

Γ

(

𝐗

⁢

d

⁢

y

-

𝐘

⁢

d

⁢

x

)

=

∫

0

T

(

𝐗

⁢

y

˙

-

𝐘

⁢

x

˙

)

⁢

d

t

=

∫

0

T

(

𝐗𝐘

-

𝐘𝐗

)

⁢

d

t

=

0

Since

∇

⋅

𝐟

is not identically zero by

hypothesis

and is of one sign, the

double integral

on the left must be non zero and of that sign. This leads to a

contradiction

since the right hand side is equal to zero. Therefore there does not exists a periodic solution in the simply connected region

E

.

Title

proof of Bendixson’s negative criterion

Canonical name

ProofOfBendixsonsNegativeCriterion

Date of creation

2013-03-22 13:31:07

Last modified on

2013-03-22 13:31:07

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

5

Author

Daume (40)

Entry type

Proof

Classification

msc 34C25

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