Proving Local Uniqueness of Autonomous System x'=f(x)

This implies that the solutions x and y are equal on the entire interval (alpha, beta).In summary, the solution to the given problem can be proved by using the local Lipschitz condition of the function f and showing that if x(s)=y(s) for some s in (alpha, beta), then x(t)=y(t) for all t in (alpha, beta), which implies that the solutions x and y are equal on the entire interval (alpha, beta). This approach is similar to the one used in the link provided.
  • #1
onie mti
51
0
i have an autonomous system x'=f(x) and teh function f is loc lip on its domain, if x and y are sol of the system defined on (alpha, beta) and x(s)= y(s) for some s in (alpha, beta) then x= y on (alpha, beta)

is the solution to prove this problem similar to this one:
http://mathhelpboards.com/differential-equations-17/local-uniqueness-10043.html#post46595
 
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  • #2
Yes, the solution to this problem is similar to the one in the link you provided. The basic idea is to use the local Lipschitz condition of the function f to prove that if x(s) = y(s) for some s in (alpha, beta), then x(t) = y(t) for all t in (alpha, beta).
 

FAQ: Proving Local Uniqueness of Autonomous System x'=f(x)

What is the purpose of proving local uniqueness of an autonomous system?

The purpose of proving local uniqueness of an autonomous system is to determine if there is a unique solution for the given initial conditions. This is important in analyzing the behavior and stability of the system.

What is the definition of local uniqueness in this context?

In this context, local uniqueness refers to the existence of a unique solution for a given set of initial conditions within a specific region or interval.

How is local uniqueness of an autonomous system proven?

Local uniqueness of an autonomous system is typically proven using the Picard-Lindelöf theorem. This theorem states that if the function f(x) is continuous and satisfies certain Lipschitz conditions, then there exists a unique solution to the system for a given set of initial conditions.

Can an autonomous system have multiple solutions?

No, an autonomous system cannot have multiple solutions for a given set of initial conditions. If there are multiple solutions, then the system is not locally unique.

What are some applications of proving local uniqueness of an autonomous system?

Proving local uniqueness of an autonomous system is important in fields such as engineering, physics, and economics. It is used to analyze the stability and behavior of systems in these fields, and to predict their future states based on initial conditions.

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