What are the fixed points and stability of a non-linear system of ODEs?

In summary, the conversation discusses a non-linear system with two equations, X' = x² - ay and Y' = y² - y. The person needs help finding the fixed points of the system and studying the stability of each fixed point through linearization or analyzing the vector field. They also mention determining bifurcation values for a and drawing phase portraits for the system before, at, and after each bifurcation, and identifying the fixed points and their stable/unstable manifolds on the phase portraits. The conversation ends with a request for the person to contact their professor to verify if it is okay to receive outside help for this question.
  • #1
ZiniaDuttaGupta
3
0
I need help with the following so please help me --

Consider the following non-linear system
X’ = x² - ay
Y’ = y² - y(a) Find the fixed points of this system. (depending on a, there may be different fixed points!)

(b) Study stability of each fixed point via linearization. In the case the linearization is inconclusive, use directions of vector field analysis (or any other information contained in the equations) to show stability/instability.

(c) Use the information above to determine the bifurcation values for a, and draw the phase portraits for the system before, at, and after each bifurcation. On phase portraits identify the fixed points as well as their stable/unstable manifolds (curves) where appropriate.
 
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  • #2
I have reason to believe this is part of a graded assignment. Please contact me with your professor's contact information so that I can verify that it is okay for you to receive outside help with this question.

Best Regards,

Mark.
 

Related to What are the fixed points and stability of a non-linear system of ODEs?

1. What is a non-linear system of ODEs?

A non-linear system of ODEs (ordinary differential equations) is a set of equations that describe the behavior of a system, where the relationships between the variables are non-linear. This means that the rate of change of a variable is not directly proportional to the value of that variable, as is the case in linear systems. Non-linear systems are often more complex and difficult to solve than linear systems.

2. What are some examples of non-linear systems of ODEs?

Examples of non-linear systems of ODEs include population growth models, chemical reaction kinetics, and weather forecasting models. These systems involve multiple variables that interact with each other in complex ways, resulting in non-linear relationships between them.

3. How are non-linear systems of ODEs solved?

Unlike linear systems, there is no general method for solving non-linear systems of ODEs. Depending on the complexity of the system, different techniques such as numerical methods, series solutions, or qualitative analysis may be used to approximate solutions. In some cases, it may not be possible to find an exact solution and approximations must be used.

4. What are the applications of non-linear systems of ODEs?

Non-linear systems of ODEs are used in a wide range of fields, including physics, biology, chemistry, economics, and engineering. They are particularly useful for modeling complex systems that cannot be accurately described by linear relationships, such as chaotic systems or systems with feedback loops.

5. How do non-linear systems of ODEs differ from linear systems?

In linear systems, the relationships between variables are described by linear equations, meaning that the rate of change of a variable is directly proportional to the value of that variable. Non-linear systems, on the other hand, have non-linear relationships between variables, making their behavior more complex and often more difficult to predict or solve. Additionally, linear systems have well-defined analytical solutions, while non-linear systems may require approximations or numerical methods to find solutions.

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