Abstract
We report the finding of the simple nonlinear autonomous system exhibiting infinite-scroll attractor. The system is generated from the pendulum equation with complex-valued function. The proposed system is having infinitely many saddle points of index two which are responsible for the infinite-scroll attractor.
Highlights
A variety of natural systems show a chaotic behaviour
We propose a complex version of (1) given by z = −a sin (z), (2)
Using the new variables x1 = x, x2 = ẋ, x3 = y, and x4 = ẏ, the system (3) can be written as the autonomous system of first-order ordinary differential equations given by ẋ1 = x2, ẋ2 = − a sin (x1) cosh (x3), (4) ẋ3 = x4, ẋ4 = − a sinh (x3) cos (x1)
Summary
A variety of natural systems show a chaotic (aperiodic) behaviour. There are various chaotic systems such as the Lorenz system [1], the Rossler system [2], the Chen system [3], and the Lusystem [4] where the dependent variables are the real-valued functions. We propose a complex pendulum equation exhibiting infinite-scroll attractor. We propose a complex version of (1) given by z = −a sin (z) ,. Using the new variables x1 = x, x2 = ẋ, x3 = y, and x4 = ẏ, the system (3) can be written as the autonomous system of first-order ordinary differential equations given by ẋ1 = x2, ẋ2 = − a sin (x1) cosh (x3) ,. Symmetry about the x1, x2-axes (or x3, x4 axes), since (x1, x2, x3, x4) → (x1, x2, −x3, −x4) (or (x1, x2, x3, x4) → (−x1, −x2, x3, x4)) do not change the equations
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