Perform a study of one of the presented below systems according to the following schema:

a) Plot several realizations of

x(t) for parameter values associated with periodic and chaotic regimes.b) Demonstrate sensitivity to initial conditions in chaotic regimes. For this slightly change the initial state and select value of the variation to observe coinciding realizations on the initial part of the observing realizations and divergency on the final part.

c) Demonstrate sensitivity to initial conditions in chaotic regimes for phase trajectories in projection onto the plane ().

d) Compose a program depicting projection of attractor onto the plane () and observe transformation of the attractor under gradual parameter varying.

e) Demonstrate a possibility of coexistence of distinct attractors at some properly selected parameters.

f) Compose a program that depict not only the phase trajectory projections but additionally marks the points in the stroboscopic Poincaré cross-sections (time interval must be selected equal to a period of external force; the remark is important if you are going to regard frequency as a parameter and vary it). Observe the arising patterns in different dynamical regimes.

g) Using the program for constructing the Poincaré cross-sections obtain a chart of dynamical regimes on a parameter plane (usually use the plane of amplitude and frequency of the external force).

Non-autonomous systems to study:

a) Ueda oscillator: .

b) Forced Duffing oscillator: .

c) Forced van der Pol oscillator: .

d) Brusselator , (set

A=0.4).

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Saratov groupof theoretical nonlinear

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