Periodic occurrence of chaotic behavior of homoclinic tangles (Q964917)

The article deals with numerical simulations of some important aspects of the dynamics of the periodically perturbed homoclinic solutions for the following differential equations \[ \frac{d^2q}{dt^2} + (\lambda - \gamma q^2) \frac{dq}{dt} - q + q^2 = 0; \] in other words, the solutions to the equation \[ \frac{d^2q}{dt^2} + (\lambda - \gamma q^2) \frac{dq}{dt} - q + q^2 = \mu \sin \omega t \] with a small \(\mu\) are studied. The article is written in the descriptive manner; it presents only tables and graphs demonstrating the creation of homoclinic tangles and some accompanying effects (horseshoes, periodic sinks, attractors, stable and unstable manifolds, and so on). There are no theorems, proofs, or sound reasoning validating the corresponding effects mentioned above.