Zero-Hopf dynamics in a quadratic–cubic jerk system: a complement to quadratic classification

Abstract In this paper, we study a three-dimensional jerk system $$ \dddot{x} + a\,x + \dot{x} + b\,\ddot{x} + x\,\dot{x} - d\,\ddot{x}^{\,2} = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>⃛</mml:mo> </mml:mover> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:mspace/> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mspace/> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>¨</mml:mo> </mml:mover> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mspace/> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>-</mml:mo> <mml:mi>d</mml:mi> <mml:mspace/> <mml:msup> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>¨</mml:mo> </mml:mover> <mml:mrow> <mml:mspace/> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , with $$a,b,d \ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , introduced by Li et al. [1] in the context of hidden chaotic dynamics. While previous studies focused on a Hopf bifurcation at $$a=b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:math> and $$d=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , we show that this system undergoes a zero-Hopf bifurcation at the origin when $$a=b=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , where the linearization has a simple zero eigenvalue and a pair of purely imaginary eigenvalues. By applying second-order averaging, we prove the existence and orbital stability of a small-amplitude periodic orbit that bifurcates from the zero-Hopf equilibrium under small parameter perturbations. In contrast to the general classification of zero-Hopf bifurcations in quadratic polynomial jerk systems by Llibre and Makhlouf [2], the system considered here is structurally minimal, containing a single nonlinear term and lying outside their framework. Our results therefore provide a complementary contribution by identifying the simplest quadratic–cubic jerk system in which zero-Hopf dynamics arise.