Question

Does Lyapunov stability of an equilibrium position follow if every solution tends to this equilibrium position as \(t \rightarrow+\infty\) ?

Short answer

Short Answer: No, it is not guaranteed that Lyapunov stability follows if every solution tends to the equilibrium position as time tends to infinity. This was illustrated through the example of the dynamical system given by the differential equation \(\dot x=-x^3\). Even though every solution tends to the equilibrium position, it does not ensure Lyapunov stability.

Step-by-step solution

Define Lyapunov Stability and Asymptotic Stability

Lyapunov stability refers to the property where any initial conditions around the equilibrium point do not cause the system to move away from the equilibrium position. A system is asymptotically stable if it is both Lyapunov stable and has every solution converging to the equilibrium point as time tends to infinity.

Show an Example of a Function

Consider a one-dimensional dynamical system given by the following differential equation: $$\dot x = -x^3.$$ The equilibrium position is at \(x=0\). We will investigate whether Lyapunov stability follows if the solutions tend to this equilibrium position as \(t \rightarrow +\infty\).

Analyzing the Function

Analyzing the function, we can rewrite the differential equation as: $$\frac{dx}{dt}=-x^3.$$ We can integrate both sides with respect to \(x\) and \(t\): $$\int \frac{dx}{x^3}=-\int dt.$$ This results in: $$-\frac{1}{2x^2}=-t+C.$$

Determine the Initial Conditions

Let's consider the initial condition \(x(0)=x_0\). In this case, the constant of integration \(C\) can be acquired as: $$-\frac{1}{2x_0^2}=C.$$ So the integrated equation can be written as: $$-\frac{1}{2x^2}=-t-\frac{1}{2x_0^2}.$$

Analyzing the Limit of the Solution

To check if every solution tends to the equilibrium position as time tends to infinity, let's analyze the limit: $$\lim_{t\to +\infty} x(t).$$ From the derived equation: $$\lim_{t\to +\infty} \left(-\frac{1}{2x^2(t)}\right)=\lim_{t\to +\infty} (-t-\frac{1}{2x_0^2}).$$ After taking the limit, we have: $$-\frac{1}{2x^2(+\infty)}=0.$$ Thus, the solution tends to the equilibrium position but does not ensure Lyapunov stability.

Conclusion

In this example, we showed that even though every solution tends to the equilibrium position as \(t \rightarrow +\infty\), it does not guarantee Lyapunov stability.

Key concepts

Asymptotic Stability

Asymptotic stability plays a crucial role in understanding the long-term behavior of dynamical systems. It combines two essential characteristics:

• Lyapunov Stability: The system remains close to the equilibrium initially if it starts nearby.
• Attraction: Solutions eventually converge to the equilibrium as time progresses to infinity.

In essence, a system is asymptotically stable if every solution not only stays near an equilibrium point but also approaches it as time goes on.
A common method to assess asymptotic stability is to use a Lyapunov function, which is akin to an "energy" measure of the system. If this function decreases over time, the system tends to not just stabilize but do so by heading towards a particular equilibrium.

Equilibrium Position

The concept of an equilibrium position is fundamental in analyzing dynamical systems. It refers to a state where the system remains constant over time, given it is undisturbed. In mathematical terms, for a dynamical system described by \(\dot x = f(x)\), the equilibrium positions are points \(x_e\) where \(f(x_e) = 0\).
This means that if the system starts at \(x_e\), it stays there indefinitely unless perturbed. Identifying these positions helps in understanding the various possible states a system can naturally settle into over time.
For real-world applications, knowing equilibrium positions allows for designing control systems and predicting system behavior.

Dynamical Systems

Dynamical systems encompass mathematical formulations that describe how a point's position changes over time. They are typically represented by differential equations such as \(\dot x = f(x)\), which describe the rate of change of a state variable over time.

• Deterministic Systems: These have predictable outcomes based on initial conditions.
• Stochastic Systems: Outcomes may vary due to inherent randomness.

Dynamical systems are instrumental in multiple scientific fields, from physics to economics, capturing complex evolving behaviors.
The analysis of such systems involves assessing stability conditions, long-term behavior, and response to initial states. Understanding the interplay between variables and their temporal evolution is key to solving problems ranging from population dynamics to engineering stability.