Question

Assuming that the trajectory corresponding to a solution \(x=\phi(t), y=\psi(t),-\infty0\) such that \(\phi\left(t_{0}+T\right)=x_{0}, \psi\left(t_{0}+T\right)=y_{0} .\) Show that \(x=\Phi(t)=\phi(t+T)\) and \(y=\Psi(t)=\psi(t+T)\) is a solution and then use the existence and uniqueness theorem to show that \(\Phi(t)=\phi(t)\) and \(\Psi(t)=\psi(t)\) for all \(t .\)

Short answer

Answer: Yes, a closed trajectory corresponds to a periodic solution.

Step-by-step solution

Finding the point \((x_0, y_0)\) and \(T\)

Since the trajectory is closed, there must be at least one point \((x_0, y_0)\) on the trajectory and a number \(T>0\) such that the system returns to \((x_0, y_0)\) after time \(T\). This means that the functions representing the solution will satisfy \(\phi(t_0) = x_0\), \(\psi(t_0) = y_0\), \(\phi(t_0 + T) = x_0\) and \(\psi(t_0 + T) = y_0\).

Defining new functions with time shift

Now, we define new functions \(\Phi(t) = \phi(t + T)\) and \(\Psi(t) = \psi(t + T)\). These functions represent another solution of the autonomous system with a time shift of \(T\).

Proving that the new functions are also solutions

To show that \(\Phi(t)\) and \(\Psi(t)\) are also solutions, we can simply plug them into the system and see if they satisfy the equations in question. If they do, we'll have proved that they represent a valid solution.

Use the existence and uniqueness theorem

Now, we can use the existence and uniqueness theorem for autonomous systems. This theorem states that if we have two solutions of an autonomous system, and they are equal at any point in time, then they must be equal for all times. Since we know from our definition of \((x_0, y_0)\) that \(\Phi(t_0) = \phi(t_0)\) and \(\Psi(t_0) = \psi(t_0)\), the existence and uniqueness theorem implies that \(\Phi(t) = \phi(t)\) and \(\Psi(t) = \psi(t)\) for all \(t\).

Conclusion

We have shown that the solution functions \(\phi(t)\) and \(\psi(t)\) are equal to the same functions with a time shift of \(T\), which implies that the solution is periodic with period \(T\). Thus, a closed trajectory corresponds to a periodic solution.

Key concepts

Closed Trajectory

In mathematics, especially in the study of autonomous systems, a closed trajectory is a path that eventually loops back on itself. This can be visualized as a figure-8 or a circle on a graph. Such trajectories suggest that the system's state returns to its starting point after some time without additional input.
This behavior is crucial in understanding the long-term behavior of systems like pendulums or planetary orbits.

• A closed trajectory implies repeating patterns.
• It helps us to predict system stability.

Recognizing a trajectory as closed forms the basis for further analysis, such as determining if the solution is periodic.

Periodic Solution

A periodic solution is essentially a repeated cycle in the context of differential equations. When a system returns to its initial state after a fixed time interval, this period, noted as \( T \), represents one full cycle.
In our exercise, recognizing that \( \phi(t + T) \) and \( \psi(t + T) \) lead back to initial conditions helped us prove periodicity.

• Periodic solutions show predictable repeating patterns.
• They arise naturally in closed trajectories as solutions repeat over time.

Understanding this connection helps in analyzing system behaviors over time efficiently.

Existence and Uniqueness Theorem

The Existence and Uniqueness Theorem is a foundational principle in differential equations. It assures us that given initial conditions, a solution to the system exists and is unique.
In our context, this theorem helps prove that if two solutions, like \( \phi(t) \) and \( \Phi(t) \), coincide at any time, they must be identical throughout.

• This confirms that solutions are consistent and predictable.
• It is crucial for validating our periodic solutions by ensuring they maintain continuity.

This theorem allows us to rely on the behavior of autonomous systems for precise predictions.

Time Shift

A time shift in the context of autonomous systems refers to moving the timeline of a solution function. By adding a time period \( T \) to the function, such as \( \phi(t + T) \), we examine the solution after it has "shifted" forward in time by \( T \).
This approach was used to show the closed trajectory's periodic nature by proving the shifted solutions still satisfy the system.

• Time shifts help analyze periodic behaviors.
• It makes it easier to show how solutions repeat over time.

This allows us to explore the dynamic stability of the system and confirm long-term behaviors.

Solution Functions

Solution functions \( \phi(t) \) and \( \psi(t) \) represent the states or behaviors of the system over time. These functions evolve based on the system's governing equations and initial conditions.
In our problem, the solution functions provided a way to understand the system's trajectory and its periodic nature.

• These functions help map the entire behavior of the system.
• They are crucial in validating if a system has periodic solutions and closed trajectories.

By studying these solutions, we can infer the overall dynamics and stability of the system, aiding in both theoretical analysis and practical applications.