Search Results (10)

Semiclassical (SC) approximations for various representations of a quantum state are constructed on a single (Lagrangian) surface in the phase space but such surface is not available for chaotic systems. An analogous evolution surface underlies SC representations of the evolution operator, albeit in a doubled phase space. Here, it is shown that corresponding to the Fourier transform on a unitary operator, represented as a Green function or spectral Wigner function, a Legendre transform generates a resolvent surface as the classical basis for SC representations of the resolvent operator in the double-phase space, independently of the integrable or chaotic nature of the system. This surface coincides with derivatives of action functions (or generating functions) depending on the choice of appropriate coordinates, and its growth departs from the energy shell following trajectories in the double-phase space. In an initial study of the resolvent surface based on its caustics, its complex nature is revealed to be analogous to a multidimensional sponge. Resummation of the trace of the resolvent in terms of linear combinations of periodic orbits, known as pseudo orbits or composite orbits, provides a cutoff to the SC sum at the Heisenberg time. Here, it is shown that the corresponding actions for higher times can be approximately included within true secondary periodic orbits, in which heteroclinic orbits join multiple windings of relatively short periodic orbits into larger circuits. Full article

We examine the modified Hill three-body problem by incorporating the oblateness of the primary body and focus on its asymptotic orbits. Specifically, we analyze and characterize homoclinic and heteroclinic connections associated with the collinear equilibrium points. By systematically varying the oblateness parameter, we determine conditions for the existence and location of these orbits. Our results confirm the presence of both homoclinic orbits, where trajectories asymptotically connect an equilibrium point to itself, and heteroclinic orbits, which establish connections between two distinct equilibrium points, via their stable and unstable invariant manifolds, which are computed both analytically and numerically. To achieve precise computations, we employ differential correction techniques and leverage the system’s inherent symmetries. Numerical calculations are carried out for orbit multiplicities up to twelve, ensuring a comprehensive exploration of the dynamical properties. Full article

Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where $\dot{x}=a(y-x)$, $\dot{y}=cx-\sqrt[3]{x}z$, $\dot{z}=-bz+\sqrt[3]{x}y$, and uncovers the following property of these systems: decreasing the powers of the nonlinear terms in a quadratic Lorenz-like system where $\dot{x}=a(y-x)$, $\dot{y}=cx-xz$, $\dot{z}=-bz+xy$, may narrow, or even eliminate the range of the parameter c for hidden attractors, but enlarge it for self-excited attractors. By combining numerical simulation, stability and bifurcation theory, most of the important dynamics of the Lorenz system family are revealed, including self-excited Lorenz-like attractors, Hopf bifurcation and generic pitchfork bifurcation at the origin, singularly degenerate heteroclinic cycles, degenerate pitchfork bifurcation at non-isolated equilibria, invariant algebraic surface, heteroclinic orbits and so on. The obtained results may verify the generalization of the second part of the celebrated Hilbert’s sixteenth problem to some degree, showing that the number and mutual disposition of attractors and repellers may depend on the degree of chaotic multidimensional dynamical systems. Full article

Dynamic systems which underlie controlled systems are expected to increase in complexity as robots, devices, and connected networks become more intelligent. While classical stable systems converge to a stable point (a sink), another type of stability is to consider a stable path rather than a single point. Such stable paths can be made of saddle points that draw in trajectories from certain regions, and then push the trajectory toward the next saddle point. These chains of saddles are called stable heteroclinic channels (SHCs) and can be used in robotic control to represent time sequences. While we have previously shown that each saddle is visualizable as a trajectory waypoint in phase space, how to increase the fidelity of the trajectory was unclear. In this paper, we hypothesized that the waypoints can be individually modified to locally vary fidelity. Specifically, we expected that increasing the saddle value (ratio of saddle eigenvalues) causes the trajectory to slow to more closely approach a particular saddle. Combined with other parameters that control speed and magnitude, a system expressed with an SHC can be modified locally, point by point, without disrupting the rest of the path, supporting their use in motion primitives. While some combinations can enable a trajectory to better reach into corners, other combinations can rotate, distort, and round the trajectory surrounding the modified saddle. Of the system parameters, the saddle value provides the most predictable tunability across 3 orders of magnitude. Full article

This paper investigates the impact of combined axial through flow and radial mass flux on Taylor–Couette flow in a counter-rotating configuration, in which different branches of nontrivial solutions appear via Hopf bifurcations. Using direct numerical simulation, we elucidate flow structures, dynamics, and bifurcation behavior in qualitative and quantitative detail as a function of axial Reynolds numbers ($Re$) and radial mass flux ($\alpha$) spanning a parameter space with a very rich variety of solutions. We have determined nonlinear properties such as anharmonicity, asymmetry, flow rates (axial and radial) and torque for toroidally closed Taylor vortices and helical spiral vortices. Small to moderate radial flow $\alpha$ initially decreases the symmetry of the different flows, before for larger values, $\alpha$, the symmetry eventually increases, which appears to be congruent with the degree of anharmonicity. Enhancement in the total torque with $\alpha$ are elucidated whereby the strength varies for different flow structures, which allows for potential better selection and control. Further, depending on control parameters, heteroclinic connections (and cycles) of oscillatory type in between unstable and topological different flow structures are detected. The research results provide a theoretical basis for simple modification the conventional Taylor flow reactor with a combination of additional mass flux to enhance the mass transfer mechanism. Full article

This paper deals with a diffusive predator–prey model with two delays. First, we consider the local bifurcation and global dynamical behavior of the kinetic system, which is a predator–prey model with cooperative hunting and Allee effect. For the model with weak cooperation, we prove the existence of limit cycle, and a loop of heteroclinic orbits connecting two equilibria at a threshold of conversion rate $p=p^{\#}$, by investigating stable and unstable manifolds of saddles. When $p>p^{\#}$, both species go extinct, and when $p<p^{\#}$, there is a separatrix. The species with initial population above the separatrix finally become extinct, and the species with initial population below it can be coexisting, oscillating sustainably, or surviving of the prey only. In the case with strong cooperation, we exhibit the complex dynamics of system, including limit cycle, loop of heteroclinic orbits among three equilibria, and homoclinic cycle with the aid of theoretical analysis or numerical simulation. There may be three stable states coexisting: extinction state, coexistence or sustained oscillation, and the survival of the prey only, and the attraction basin of each state is obtained in the phase plane. Moreover, we find diffusion may induce Turing instability and Turing–Hopf bifurcation, leaving the system with spatially inhomogeneous distribution of the species, coexistence of two different spatial-temporal oscillations. Finally, we consider Hopf and double Hopf bifurcations of the diffusive system induced by two delays: mature delay of the prey and gestation delay of the predator. Normal form analysis indicates that two spatially homogeneous periodic oscillations may coexist by increasing both delays. Full article

The aim of this work is to characterize Traveling Waves (TW) solutions for a coupled system with KPP-Fisher nonlinearity and weak advection. The heterogeneous diffusion introduces certain instabilities in the TW heteroclinic connections that are explored. In addition, a weak advection reflects the existence of a critical combined TW speed for which solutions are purely monotone. This study follows purely analytical techniques together with numerical exercises used to validate or extent the contents of the analytical principles. The main concepts treated are related to positivity conditions, TW propagation speed and homotopy representations to characterize the TW asymptotic behaviour. Full article

In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee the uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is a dynamic gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections. Full article

In this paper, we consider the second order discontinuous differential equation in the real line, ${\left(a\left(t,u\right)\varphi \left(u^{\prime }\right)\right)}^{\prime }=f\left(t,u,u^{\prime }\right),a.e.t\in \mathbb{R},u(-\infty )={\nu }^{-},u(+\infty )={\nu }^{+},$ with $\varphi$ an increasing homeomorphism such that $\varphi \left(0\right)=0$ and $\varphi \left(\mathbb{R}\right)=\mathbb{R}$ , $a\in C({\mathbb{R}}^2,\mathbb{R})$ with $a(t,x)>0$ for $(t,x)\in {\mathbb{R}}^2$ , $f:{\mathbb{R}}^3\to \mathbb{R}$ a $L^1$ -Carathéodory function and ${\nu }^{-},{\nu }^{+}\in \mathbb{R}$ such that ${\nu }^{-}<{\nu }^{+}$ . The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities $\varphi$ and $f$ . To the best of our knowledge, this result is even new when $\varphi \left(y\right)=y$ , that is for equation ${\left(a\left(t,u\left(t\right)\right)u^{\prime }\left(t\right)\right)}^{\prime }=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right),a.e.t\in \mathbb{R}$ . Moreover, these results can be applied to classical and singular $\varphi$ -Laplacian equations and to the mean curvature operator. Full article

We study 4 x 4 games for which the best response dynamics contain a cycle. We give examples in which multiple Shapley polygons occur for these kinds of games. We derive conditions under which Shapley polygons exist and conditions for the stability of these polygons. It turns out that there is a very strong connection between the stability of heteroclinic cycles for the replicator equation and Shapley polygons for the best response dynamics. It is also shown that chaotic behaviour can not occur in this kind of game. Full article