Dear colleagues at first assumed that we have one chaotic system and in this system strange attractor so How can i find basin of attraction of it.
can you introduce some sources about it?
thanks
Its all there in the reference list of nonlinear dynamics and chaos I and others had suggested in your previous post. Have a look there!
https://www.researchgate.net/post/How_can_i_calculate_Lyapunov_exponent
1. calculate the trace of Jacobian of your system.
if negative run your system to infinity!
if positive you most inverse the time of your system and run to infinity!
the trace of Jacobian of your system specify the power of volume in state space!
2. attractor and repulsive attractor it's very important. if you have repulsive attractor and negative jacobi you must solve the big problem!
you can run your program with enough random initial condition!
3. in the end, before run program everyone should analysis the system with applications : 1 Jacobian,2 Lyaponuv Ex ,3 poincare section and 4return map and extra .
To find the basin of attraction of a chaotic system with a strange attractor, one approach is to use computational methods and numerical simulations. There are several techniques and algorithms available for this purpose, depending on the specific characteristics of the system and the nature of the attractor.
One common method is the basin boundary algorithm, which iteratively explores the phase space of the system to identify regions that converge to the attractor. This algorithm typically involves starting from a grid of initial conditions and simulating the dynamics of the system forward in time. Points that converge to the attractor are considered to belong to its basin of attraction.
Another approach is to use sensitivity analysis techniques, such as Lyapunov exponents or variational equations, to identify regions of phase space that are most likely to be attracted to the strange attractor. These techniques quantify the rate of divergence of nearby trajectories, which can help identify the boundaries of the basin of attraction.
Additionally, studying the topology of the phase space and the stability properties of the attractor can provide insights into the structure of its basin of attraction. Techniques such as Poincaré sections, bifurcation diagrams, and symbolic dynamics analysis can be useful in this regard.
For more in-depth information and specific algorithms, I recommend consulting literature on dynamical systems theory and chaos theory. Some recommended sources include:
These textbooks cover various aspects of chaotic systems, including basin of attraction analysis, and provide detailed explanations along with relevant algorithms and examples.
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