**§ ****5 ****The general idea of stability theory****
**

Stability theory studies how the solution changes when the right-hand side function of the differential equation changes with the initial conditions .

The concept of stability

[ Stability and instability of solutions ] Let the system of differential equations

_{}

The solution that satisfies the initial conditions is ._{}_{}

If any given *ε* > 0 , there is always a corresponding positive number *δ* = *δ* ( *t *_{0 } , *ε* ) such that as long as the initial value satisfies_{}

_{}

The corresponding solution of this system of differential equations ( *i* = 1,2, … , *n* ) is satisfied for all *t* > *t *_{0}_{} _{}

_{}

The solution is said to be stable ._{}

Simply put , a solution is said to be stable if all solutions whose initial value is close to the initial value of a solution are always close to a solution when *t* > *t *_{0} .

If for any given *ε* > 0 , no matter how small *δ* > 0 is taken, there is always something satisfying

_{}

The initial value of and *τ* > *t *_{0} , its corresponding solution does not satisfy the condition_{}_{}_{}

_{}

The solution is said to be unstable ._{}

If stable, and the initial value satisfies_{}

_{}

All solutions for satisfies_{}

_{}

is said to be asymptotically stable ._{}

[ Simplification of the problem ] Given a system of equations

_{}

In order to study the stability of its solution satisfying the initial conditions , the deviation of other solutions from it must be investigated ._{}_{}_{}

_{}

The original system of equations becomes as follows:

_{} (1)

The stability of the equations (1) is attributed to the stability of the zero solution *x ** _{i}* ≡ 0 (

The constant solution of any system of differential equations is often called its equilibrium point (singularity) . So the zero solution of (1 )

_{}

is the balance point .

If any given *ε* > 0 , there is always a corresponding positive number *δ* = *δ* ( *t *_{0 } , *ε* ) such that as long as the initial value satisfies

_{}

The corresponding solution of (1 )_{} is satisfied for all *t* > *t *_{0}

_{}

Then the equilibrium point of (1) is said to be stable ._{}

If the condition is then met

_{}

The equilibrium point is said to be asymptotically stable ._{}

If no matter how small the positive number *δ* is chosen, for a predetermined positive number *ε* , there is always a satisfaction

_{}

The initial value of and *τ* > *t *_{0} , its corresponding solution does not satisfy the condition_{}_{}_{}

_{}

Then the equilibrium point of (1) is said to be unstable ._{}

[ phase space ] Given a system of equations

_{}

The space of ( *x *_{1} , *x *_{2} , . _ *_ *_{_} _ *_* _ _ _ *_* _ A curve in phase space, called an orbit . In other cases, the solution curve is often called the integral curve .

The solution to the stability problem

[ Stability Problem of Equilibrium Points of Homogeneous Linear Differential Equations with Constant Coefficients ] For the sake of simplicity , only the equations with two unknown functions are studied

_{}

where *a *_{11} , *a *_{12} , *a *_{21} , and *a *_{22} are all real numbers, and

_{}

from the characteristic equation

_{}

Calculate the characteristic roots *λ *_{1} , *λ *_{2} , and substitute them into the following equations in turn:

_{}

Two sets of solutions _{1} , _{2} and ._{}

*At this time, the stability of the equilibrium point x* ≡ 0, *y* ≡ 0 of the linear equation system can be discussed in the following cases .

1 ° eigenroots are real numbers:

_{}

The general solution is in the form of

_{}

where *c *_{1} , *c *_{2} are arbitrary constants .

( i ) _{1} < 0 , _{2} < 0

The zero solution is asymptotically stable . The shape of the orbit is shown in Figure 13.1( *a *) ( the arrow indicates the direction of *t* increasing, the same below) . This type of equilibrium point (0,0) is called a stable node .

( ii ) _{1} > 0 , _{2} > 0

The zero solution is unstable . The orbital shape is shown in Figure 13.1( *b* ). This type of equilibrium point ( 0,0) is called an unstable node .

( iii )
_{1} > 0 , _{2} < 0

The zero solution is unstable . The orbital shape is shown in Figure 13.1( *c* ). This type of equilibrium point ( 0,0) is called a saddle point .

( *a* ) ( *b* ) ( *c* )
* *

Figure 13.1

2 ° eigenroots are complex numbers:

_{}

The general solution is in the form of

_{}

where *c *_{1} , *c *_{2} are arbitrary constants, and *c *_{1} *, *c *_{2} * are linear combinations of *c *_{1} , *c *_{2} .

( i ) _{1 ,2} = *p **iq* , *p* <0, *q* 0.

The zero solution is stable . The orbital shape is shown in Figure 13.2( *a* ). This type of equilibrium point ( 0,0) is called a stable focus .

( ii ) _{1 ,2} = *p **iq* , *p* >0,
*q* 0.

The zero solution is unstable . The orbital shape is shown in Figure 13.2( *b* ). This type of equilibrium point ( 0,0) is called an unstable focus .

( iii ) _{1 }_{, 2} = *iq* , *q* 0. _{}

The zero solution is stable . The orbital shape is shown in Figure 13.2( *c* ). This type of equilibrium point ( 0,0) is called the center, and the center is stable .

Figure 13.2

The 3 ° characteristic equation has multiple roots:

_{}

The general solution is in the form of

_{}

( i ) _{1} = _{2} < 0

The zero solution is asymptotically stable . The orbital shape is shown in Figure 13.3( *a* ). This type of equilibrium point ( 0,0) is called a stable degenerate node .

If the zero solution is a stable node, it is called a critical node . The shape of the orbit is shown in Figure 13.3 ( *b* )._{}

Figure 13.3

( ii ) _{1} = _{2} > 0

The zero solution is unstable . The orbital shapes are shown in Figure 13.3 ( *a *) and ( *b* ), but the arrows are in opposite directions . This type of equilibrium point (0,0) is called the unstable degenerate node and the unstable critical node .

Combining the above situations, the following conclusions can be drawn: if the roots of the characteristic equation have negative real parts, then the zero solution is stable and asymptotically stable; if the characteristic equation has a root with a positive real part, then zero The solution is unstable .

This conclusion, for the general system of homogeneous linear differential equations with constant coefficients

_{}

is also established .

Theorem if the characteristic equation of a system of homogeneous linear differential equations with constant coefficients

_{}

A zero solution is asymptotically stable if all roots have negative real parts; a zero solution is unstable if at least one of all the roots of the characteristic equation has a positive real part .

[ Determining stability by first approximation ] Consider the system of equations

_{}

where *a ** _{ij}* (

Its first-order approximate equation system is

_{}

Stability can be determined by studying various cases of the characteristic roots * _{i}* (

** **The first theorem If all the eigenvalues of the first approximation equation system have negative real parts, then the zero solution of the original equation system is asymptotically stable .

** **The second theorem If at least one of the eigenvalues of the first-order approximate equation system has a positive real part, then the zero solution of the original equation system is unstable .

These two theorems cover all stable cases (called noncritical cases) in which the zero solution of the original system of equations can be studied with a first-order approximation . As for at least one root with a zero real part, all other roots have negative In the critical case of the real part, the higher-order terms on the right-hand side of the equation system play an important role in the stability of the zero solution, so it is generally impossible to study the stability problem by the first-order approximation equation system .

[ Hurwitz Discriminant Method ] It is a method to directly use some properties of the determinant formed by the coefficients of the characteristic equation to determine the stability of the zero solution of a system of linear differential equations with constant coefficients .

Let the characteristic equation of the system of linear differential equations with constant coefficients be

_{}

Then the necessary and sufficient conditions for the zero solution of the system of constant coefficient linear differential equations to be asymptotically stable are: *a *_{0} >0 , and all Hurwitz determinants

_{}

are all positive (the final Δ * _{n}* > 0 can be replaced by the condition

If Δ * _{n}* =0 , then since Δ

The characteristic equation is the Hurwitz discriminant condition for quadratic, cubic and quartic (for the convenience of drawing, *a *_{0} =1 is taken below ):

( i ) Characteristic equation:_{}

Hurwitz conditions are *a *_{1} >0,
*a *_{2} >0 _{}_{}

The stable region is shown in Figure 13.4( *a* ).

( ii ) Characteristic equation:_{}

The Hurwitz condition is _{}

The stable region is shown in Figure 13.4( *b* ).

Figure 13.4

( iii )
Characteristic equation:_{}

The Hurwitz condition is . _{}

[ Lyapunov's second method (direct method) ] to study systems of differential equations

_{}
(1)

There is a more general method for the stability of the equilibrium point of , the so-called Lyapunov method .

1 ° Lyapunov stability theorem For the system of equations (1) , if one can find a differentiable function *V* ( *x *_{1} , *x *_{2} , … , *x ** _{n}* ) (called the Lyapunov function ) that satisfies the following conditions in the neighborhood of the origin ):

( i ) *V* ( *x *_{1} , *x *_{2} , … , *x ** _{n}* ) ≥ 0 (or ≤ 0 ), and only when

( ii ) When *t* ≥ *t *_{0} , the full derivative of *V* along the integral curve of the system of equations ( 1)

_{}( or )_{}

Then the equilibrium point *x ** _{i}* =0 (

2 ° Lyapunov Asymptotic Stability Theorem For the system of equations (1) , if one can find a differentiable function *V* ( *x *_{1} , *x *_{2} , … , *x ** _{n}* ) in the neighborhood of the origin that satisfies the following conditions (called Lyapunov Nov function):

( i ) *V* ( *x *_{1} , *x *_{2} , … , *x ** _{n}* ) ≥ 0 (or ≤ 0 ), and only when

( ii ) The total derivative along the integral curve of the system of equations ( 1)

_{}( or )_{}

And outside a suitably small *delta* neighborhood of the origin (ie ), when *t* ≥ *t *_{0} , (or ) then the equilibrium point *x ** _{i}* =0 (

Case Study of Differential Equations

_{}

Stability at the equilibrium point *x* = 0, *y* = 0 .

The characteristic equation of the first-order approximation equation for solving this system of equations has two pure imaginary roots, so it is a critical case and cannot be studied by the first-order approximation method . Now use the Lyapunov method . Take

_{}

because

( i ) _{}

( ii ) _{}

For any *δ* > 0 , when *t* > *t *_{0} , . Therefore, the equilibrium point (0,0) is asymptotically stable ._{}_{}_{}

3. Limit cycle (or limit cycle)

Only the case of *n =* 2 is discussed here .

[ periodic solution ] equation

_{}

A periodic solution with period T is a solution that satisfies x *( **t* + *T* ) = *x* ( *t* ) and *y* ( *t+T* ) = *y* ( *t* ) . The orbit corresponding to the periodic solution is a closed curve . Conversely, a closed orbit corresponds to the periodic solution .

[ Limit cycle ] The isolated periodic solution is called the limit cycle of the equation . In a complete way , it is: let *x=x* ( *t* ), *y=y* ( *t* ) be the periodic solution of the equation, and *K* is the solution drawn on the phase plane The closed curve of . If there is a positive number *ρ* , such that for any point *ζ* on the phase plane whose distance from *K* is less than *ρ* , the solution of the equation through the point *ζ* is not periodic, then *x=x* ( *t* ) , *y=y* ( *t* ) (that is, the closed orbit *K* ) is an isolated periodic solution, or a limit cycle .

example

_{}

Do coordinate transformation: *x* = *r* cos , *y* = *r* sin _{}_{}, the equations are transformed into

_{}

The general solution is

Figure 13.5 |

_{}

Where *k* , *t *_{0} are arbitrary . Take *t *_{0} =0 , then the solution of the system of equations is

_{}

When *k=* 0 , it is a circle *x *^{2} + *y *^{2} =1 ; when *k=c *^{2} ( *c>* 0) , it is a spiral, when *t* - , it tends to the origin, and when *t* , it is approached from the inner spiral Circumference *x *^{2} + *y *^{2} =1 ; when *k=* ( *c>* 0) , the orbit is a curve, when *t* log *c* +0 , it tends to infinity, and when *t* , it circles from the outside to approximate the circumference *x *^{2} + *y *^{2} =1.
^{}^{}^{
}^{}^{}_{}^{ }^{}^{}

The track distribution is shown in Figure 13.5.

Then the circumference *x *^{2} + *y *^{2} =1 is the only limit cycle of the system of equations . (0,0) is the only singular point .

[ Limit cycle existence theorem ] For the system of equations

_{}

1 ° There are two simple closed curves *C *_{1} and *C *_{2} on the *xy* plane ,
*C *_{2} is inside *C *_{1} , and the following two conditions are satisfied: _{}_{}_{}_{}

( i ) The vector field of a point on _{C1 }_{points }*from* the outside of C1 to the inside, and _{the} vector field of a point on _{C2 }*points from the **inside of **C2* to the outside ;_{}_{}_{}

( ii ) There is no singularity in the system of equations in the annular region enclosed by *C *_{1} and *C *_{2} ;

Then in the annular region enclosed by *C *_{1} and *C *_{2} , there must be a stable limit cycle (called the Poincare - Bendikesen theorem) .
_{}_{}

2 ° If in a simple connected region *G* , the sign is constant, and it is not equal to zero in any subregion *D* (* G* ) , then in *G* , the system of equations does not have any closed orbits . _{}

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