Our curious visitor might roam around a little more and notice a few other paradoxes in this city of ours: A remarkable number of skyscrapers and apartment blocks, but very few tenants a huge range of shopping malls, but a dearth of shoppers.Again, we revel in initial conditions
, but fail in enabling our long-term success.
Additionally, the steel in initial condition
was annealed at T = 450[degrees]C for t = 50 and 100 hours.
where h, k [greater than or equal to] 1 in the upper bound of the sigmas and the differential transform of initial condition
is as follows:
Caption: Figure 6: Attraction basins: (a) under initial conditions
(0.1, 0.1, p, [l.sub.1]) and (b) under initial condition
[l.sub.1] and system parameter [xi].
Although this time exact initial conditions
were applied for the sake of image reconstruction but, due to the added noise, the residues are not regenerated properly.
Devaney has one of the most popular and accepted definitions of chaos in which such systems must exhibit sensitive dependence to initial conditions
, topological transitivity, and dense periodic orbits .
Next, we claim that System (12) with initial condition
(9) is equivalent to
Similar to the previous section, we depict the collocation points associated with the initial conditions
and boundary conditions in each case of the diffusion and wave equations for clarity of presentation.
It can be found that this new chaotic system has two isolated chaotic attractors or two disconnected chaotic attractors (named "positive attractor" and "negative attractor" in this paper), which depend on the distance between the initial points (initial conditions
) and the unstable equilibrium points.
For a clear comparison, Figure 3(b) provides the highest Lyapunov exponent curves only under two initial conditions
. For initial condition
[[alpha].sub.1], by numerical calculation, the finite-time local Lyapunov exponents of the system at t = 15 s were [L.sub.1] = -13.01, [L.sub.2] = -13.32, [L.sub.3] = -25.42, [L.sub.4] = -26.09, [L.sub.1] < [L.sub.2] < [L.sub.3] < [L.sub.4] < 0, whereas the maximum Lyapunov exponent [L.sub.1] was below 0.
While these simulations represent some of the most accurate to date, simulations generally follow the evolution for a handful of orbits and are based upon initial conditions
of quasicircular orbits obtained in the conformally flat approximation.
The sequence induced by the logistic map with this initial condition
is given by: