Problem 30
Question
If a plane autonomous system has a periodic solution, then there must be at least one critical point inside the curve generated by the solution. In Problems 27-30, use this fact together with a numerical solver to investigate the possibility of periodic solutions. $$ \begin{aligned} &x^{\prime}=x y \\ &y^{\prime}=-1-x^{2}-y^{2} \end{aligned} $$
Step-by-Step Solution
Verified Answer
There are no real critical points; periodic solutions are unlikely without a critical point inside.
1Step 1: Identify the System Type
We need to confirm that we are dealing with a plane autonomous system. The given equations \( x' = xy \) and \( y' = -1 - x^2 - y^2 \) are both functions of \( x \) and \( y \) alone, with no explicit dependence on \( t \). Therefore, this system is autonomous.
2Step 2: Find Critical Points
Critical points occur where both derivatives are zero. Set \( x' = xy = 0 \) and \( y' = -1 - x^2 - y^2 = 0 \). From \( xy = 0 \), we find either \( x = 0 \) or \( y = 0 \). Substituting into \( y' = 0 \), if \( x = 0 \), then \( -1 - y^2 = 0 \) leads to \( y^2 = -1 \), which has no real solutions; if \( y = 0 \), then \( -1 - x^2 = 0 \) leads to \( x^2 = -1 \), which also has no real solutions. Thus, the system as presented has no real critical points.
3Step 3: Sketch or Simulate the Phase Portrait
Without real critical points, we consider simulations or sketches of trajectories to check for periodic solutions. The two equations hint at behavior influenced by circles or ellipses due to \( x^2 + y^2 \) terms. We can use numerical solvers, such as MATLAB's \( ode45 \), to simulate solution trajectories and look for closed paths that suggest periodic solutions.
4Step 4: Analyze Numerical Results
By inserting different initial conditions into a numerical solver, check trajectories for closed loops that would indicate periodic solutions. We observe the behavior in the phase plane to identify these closed orbits and find whether any appear to encircle a region.
Key Concepts
Periodic SolutionsCritical PointsPhase PortraitNumerical Solvers
Periodic Solutions
Periodic solutions are fundamental concepts in the study of dynamic systems. In a plane autonomous system, these solutions correspond to closed paths or loops formed by the trajectories in the phase plane. This means that the system's state returns to its original condition after some time, making the behavior periodic.
A critical aspect to note is that periodic solutions can't exist in isolation; there must be at least one critical point within the closed loop. These solutions often indicate stable, repeating patterns which can be observed in various natural phenomena and engineered systems. They help us understand cycles and patterns in systems, which might include populations in ecology or oscillations in electrical circuits.
A critical aspect to note is that periodic solutions can't exist in isolation; there must be at least one critical point within the closed loop. These solutions often indicate stable, repeating patterns which can be observed in various natural phenomena and engineered systems. They help us understand cycles and patterns in systems, which might include populations in ecology or oscillations in electrical circuits.
Critical Points
Critical points play a vital role in understanding the behavior of autonomous systems. A critical point occurs where the time derivatives of the system equal zero, meaning no change is happening locally at that point.
To find critical points, you set both equations of the system to zero and solve for the variables. However, not all systems have real solutions for critical points; sometimes they may not exist in the real number setup, as seen in the original exercise. Moreover, critical points can be categorized into types based on their stability: nodes, saddles, foci, or centers.
Identifying critical points and their nature provides insight into the stability and potential periodic behavior of the system, guiding to which areas might exhibit significant dynamics.
To find critical points, you set both equations of the system to zero and solve for the variables. However, not all systems have real solutions for critical points; sometimes they may not exist in the real number setup, as seen in the original exercise. Moreover, critical points can be categorized into types based on their stability: nodes, saddles, foci, or centers.
Identifying critical points and their nature provides insight into the stability and potential periodic behavior of the system, guiding to which areas might exhibit significant dynamics.
Phase Portrait
A phase portrait is a graphical representation that illustrates the trajectories of a dynamical system within a coordinate space specified by its variables, here, typically the variables are the dependent variables in the equations. It visually provides the entire picture of how a system behaves over time without resolving the system explicitly.
In the context of our exercise, sketching or simulating a phase portrait allows us to observe the flow and direction of trajectories. By examining these paths, especially looking for loops or closed curves, we can infer the presence of periodic solutions.
Utilizing a phase portrait, one can easily identify stability, oscillatory behavior, and how different initial conditions affect the trajectories of the system. This tool complements the analytical approach, offering a visual means to examine dynamic systems.
In the context of our exercise, sketching or simulating a phase portrait allows us to observe the flow and direction of trajectories. By examining these paths, especially looking for loops or closed curves, we can infer the presence of periodic solutions.
Utilizing a phase portrait, one can easily identify stability, oscillatory behavior, and how different initial conditions affect the trajectories of the system. This tool complements the analytical approach, offering a visual means to examine dynamic systems.
Numerical Solvers
Numerical solvers are essential tools in modern mathematics and engineering for solving ordinary differential equations (ODEs) that are complicated or impossible to solve analytically. These solvers, such as the widely used MATLAB's `ode45`, provide powerful computational methods that approximate solutions by using numerical integration.
In tasks where analytical solutions are challenging to find or don't exist, numerical solvers can simulate the behavior of the system by giving approximation solutions based on specified initial conditions. With these results, we can analyze potential periodic patterns or critical behaviors that might not be obvious through basic mathematical analysis.
They are especially useful for exploring autonomous systems and analyzing the presence of periodic solutions as they provide a practical approach to visualize multiple trajectories and their interactions over time.
In tasks where analytical solutions are challenging to find or don't exist, numerical solvers can simulate the behavior of the system by giving approximation solutions based on specified initial conditions. With these results, we can analyze potential periodic patterns or critical behaviors that might not be obvious through basic mathematical analysis.
They are especially useful for exploring autonomous systems and analyzing the presence of periodic solutions as they provide a practical approach to visualize multiple trajectories and their interactions over time.
Other exercises in this chapter
Problem 28
Show that the dynamical system $$ \begin{aligned} &x^{\prime}=-\alpha x+x y \\ &y^{\prime}=1-\beta y-x^{2} \end{aligned} $$ has a unique critical point when \(\
View solution Problem 29
(a) Show that the plane autonomous system $$ \begin{aligned} &x^{\prime}=-x+y-x^{3} \\ &y^{\prime}=-x-y+y^{2} \end{aligned} $$ has two critical points by sketch
View solution Problem 31
If \(z=f(x, y)\) is a function with continuous first partial derivatives in a region \(R\), then a flow \(\mathbf{V}(x, y)=(P(x, y), Q(x, y))\) in \(R\) may be
View solution Problem 31
Use the phase-plane method to show that \((0,0)\) is a center of the nonlinear second-order differential equation \(x^{\prime \prime}+2 x^{3}=0\).
View solution