Synchronization of the Lohe model on the hyperboloid under a directed graph

We study the emergent behaviors of the Lohe model on the unit hyperbolic sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {H}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under more general interconnection topologies. In previous literature, the Lohe model on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {H}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> was studied under the complete graph. Using the LaSalle invariance principle, we show that the Lohe model on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {H}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> converges to a synchronized state for directed graphs containing spanning trees if all oscillator frequencies are equal. Moreover, we show that exponential convergence is achieved by analyzing the error dynamics of the Lohe model on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {H}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also provide several numerical simulations and compare them with theoretical results .