kthouz said:I am new to non linear dynamics, I am reading some papers and there are terms that I am finding every which I still have no idea about their meaning. What is:
1. Inhibitory coupling
2. An attractor
Any explanation or reference is warmly welcome
Thanks
jackmell said:An attractor is the state the dynamics tends to when ever it finds itself in the attractor's basin of attraction and further, the dynamics will remain in that state until it is pushed outside that basin. Take for example the Lorenz attractor: start it with some initial state. If that initial state is outside a basin of attraction, the dynamics may fly to infinity. Start the dynamics inside the basin of some attractor and the dynamics settles into a bound state, the attractor. Purturb it a little, hit it a little, it will react. If you don't hit it much, it would jump and fall back to the attractor. Hit it hard enough and push it outside the basin, then it will either fly to infinity or find another basin to settle into. Program the Lorenz attractor in Mathematica and work with it for some time. That's the best way to understand attractors.
This is my favorite reference:
"Perspectives of nonlinear dynamics" by E. Atlee Jackson
and the quintessential reference:
"Chaos and Fractals" by Peitegen
Non-linear analysis is a mathematical approach used to study systems that do not follow a linear relationship between input and output. This means that the output of the system is not directly proportional to the input, and there may be complex interactions and relationships between various components of the system.
Some of the key terms used in non-linear analysis include bifurcation, chaos, attractors, and stability. Bifurcation refers to the emergence of new behaviors or patterns in a system as a result of changes in its parameters. Chaos refers to the unpredictable and highly sensitive behavior of some non-linear systems. Attractors are states or patterns that a system may settle into over time, and stability refers to the tendency of a system to return to a previous state after being disturbed.
The key difference between non-linear and linear analysis is that linear analysis assumes a direct and proportional relationship between input and output, while non-linear analysis takes into account the complex interactions and relationships between components in a system. Linear analysis is simpler and easier to solve, but non-linear analysis can provide a more accurate understanding of real-world systems.
Non-linear analysis is used in various fields such as physics, engineering, biology, economics, and social sciences. Some common applications include understanding weather patterns, predicting stock market trends, studying population dynamics, and analyzing the behavior of complex systems such as the human brain.
One of the main challenges in non-linear analysis is the difficulty in solving the equations that describe non-linear systems. These equations often have no analytical solution and require numerical methods for approximation. Another challenge is the sensitivity of non-linear systems to small changes in initial conditions or parameters, making it difficult to predict their long-term behavior.