Conjoined Lorenz-like attractors coined

In contrast to most other periodically forced chaotic systems with infinitely many isolated coexisting strange attractors, little seems to be known about the ones that possess conjoined Lorenz-like attractors with potential existence of infinitely many pairs of wings/scrolls. To achieve this target, this note proposes a new periodically forced extended Lorenz-like system, which generates infinitely many singularly degenerate heteroclinic cycles or heteroclinic orbits to any two equilibria of a family of non-hyperbolic lines, the collapses of which create not only the desirable conjoined Lorenz-like attractors, but also infinitely many isolated coexisting ones. What is more, the state variable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>x</mml:mi></mml:math> of that conjoined Lorenz-like attractor presents stochastic behaviors, confirming the links between long period and chaos. This also generalizes the classical concept of boundedness of chaos, i.e., the system orbits beginning from one sub-two-scroll of that conjoined Lorenz-like attractor might tend to the ones at infinity. Apart from those, the existence of an invariant surface and a family of infinitely many pairs of symmetrical heteroclinic orbits are proved by utilizing the Lyapunov function, the definitions of both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math>-limit set and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ω</mml:mi></mml:math>-limit set.