Narrow Results

The paper devotes to the synthesis of local and global analysis of a discrete model describing the second-order digital filter with nonlinear signal processors. The discrete model gives rise to a two-dimensional non-invertible map, whose basins of attraction have complicated topological structures due to the intrinsic multi-stability. The influences of joint parameters on the local dynamics are presented in great details. Both theoretical and numerical results are plotted on the two-dimensional parametric planes. To show more detailed bifurcation structure, the isoclines are extended to higher periodic orbits for detecting the cusps of resonant entrainments. Invariant manifolds and critical curves are employed to illustrate the global dynamics of the model vividly. The tangency and intersections of invariant manifolds expound the process of erosions of basins of attraction. The global bifurcations of basins of attraction are deduced dynamically by critical curves.

We introduce the standard Rössler oscillator and extend it from a system defined on ℝ³ to a system defined on . We extend the system in such a way that the simple Rössler type dynamics are preserved, whilst introducing an additional symmetry to the system such that when the oscillators are linearly coupled, there exists an embedding of symmetry forced invariant manifolds. We contrast our results with a previously considered model for a linearly coupled solid-state laser system which possessed a similar embedding of invariant manifolds. We discuss whether the mechanisms for loss of synchronization are system dependent or a more generic property of systems with embedded invariant manifolds.

In this paper, generalized synchronization (GS) on a linear manifold between two algebraic nonlinear systems is considered. In particular, conditions for generalized synchronization on a linear manifold are derived and a general approach to the design of a nonlinear receiver that operates under GS conditions is presented. The coupling signal is unidirectional and depends on the state variables of the drive. Moreover, a method that ensures the occurrence of linear GS is discussed and applied to standard chaotic systems (i.e. Lorenz system and Chua's circuit).

The general Euler-Poisson problem of rigid body motion is investigated. We study the three-dimensional algebraic level surfaces of the first integrals, and their topological bifurcations. The main result of this article is an analytical and qualitatively complete description of the projections of these integral manifolds to the body-fixed space of angular velocities. We classify the possible types of these invariant sets and analyze the dependence of their topology on the parameters of the body and the constants of the first integrals. Particular emphasis is given to the enveloping surfaces of the sets of admissible angular velocities. Their pre-images in the reduced phase space induce a Heegaard splitting which lends itself for a general choice of complete Poincaré surfaces of section, irrespective of whether or not the system is integrable.

This is a survey on discretizing delay equations from a geometric-qualitative view-point. Concepts like compact attractors, hyperbolic periodic orbits, the saddle structure around hyperbolic equilibria, center-unstable manifolds of equilibria, inertial manifolds, structural stability, and Kamke monotonicity are considered. Error estimates for smooth and nonsmooth initial data in various C^j topologies are provided. The emphasis is put on Runge–Kutta methods with natural interpolants. The paper ends with a collection of the related results on retarded functional differential equations with bounded delay.

We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids.

Invariant surfaces in three-dimensional continuous piecewise linear homogeneous systems with two pieces separated by a plane are detected. The Poincaré map associated to this plane transforms half-straight lines passing through the origin into half-straight lines of the same type. The invariant half-straight lines under this map determine invariant cones for which the existence, stability and bifurcation are studied. This analysis lead us to consider some questions about the topological type and stability of the origin.

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L₁ between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L₁ and the other collinear libration points L₂, L₃ is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

The method of invariant manifolds was originally developed for hyperbolic rest points of autonomous equations. It was then extended from fixed points to arbitrary solutions and from autonomous equations to nonautonomous dynamical systems by either the Lyapunov–Perron approach or Hadamard's graph transformation.

We go one step further and study meaningful notions of hyperbolicity and stable and unstable manifolds for equations which are defined or known only for a finite time, together with matching notions of attraction and repulsion. As a consequence, hyperbolicity and invariant manifolds will describe the dynamics on the finite time interval.

We prove an analog of the Theorem of Linearized Asymptotic Stability on finite time intervals, generalize the Okubo–Weiss criterion from fluid dynamics and prove a theorem on the location of periodic orbits. Several examples are treated, including a double gyre flow and symmetric vortex merger.

The stochastic bifurcation, defined as the collision of a stochastic attractor with a stochastic saddle, of an asymmetric single-well potential Duffing oscillator subject to bounded noise excitation is studied by means of the modified digraph cell mapping method. The vivid evolution of stochastic bifurcation is described in detail. Meanwhile, the invariant manifolds of the system accompanied by the stochastic bifurcation are also obtained by this method. Furthermore, the relationship between unstable manifold and the growth direction in the shape and size of the stochastic attractor and stochastic saddle as the system parameter is varied, is given through our analysis.

In this paper, we perform a global analysis of the dynamics of the Chen system

We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four-dimensional spaces of section, we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stability to simple instability, in the neighborhood of the bifurcated simple unstable x1v2 periodic orbits, we encounter the phenomenon of stickiness as the asymptotic curves of the unstable manifold surround regions of the phase space occupied by rotational tori existing in the region. For larger energies, away from the bifurcating point, the consequents of the chaotic orbits form clouds of points with mixing of color in their 4D representations. In the case of double instability, close to x1v2 orbits, we find clouds of points in the four-dimensional spaces of section. However, in some cases of double unstable periodic orbits belonging to the z-axis family we can visualize the associated unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky to this surface for long times (of the order of a Hubble time or more). Among the orbits we studied, we found those close to the double unstable orbits of the x1v2 family having the largest diffusion speed. The sticky chaotic orbits close to the bifurcation point of the simple unstable x1v2 orbit and close to the double unstable z-axis orbit that we have examined, have comparable diffusion speeds. These speeds are much slower than of the orbits in the neighborhood of x1v2 simple unstable periodic orbits away from the bifurcating point, or of the double unstable orbits of the same family having very different eigenvalues along the corresponding unstable eigendirections.

We mainly investigate the existence, stability and number of invariant cones in 3-dim homogeneous piecewise linear systems with two zones separated by a plane containing the 1-dim invariant manifold of each linear subsystem. By transforming the system into a proper form with the 1-dim invariant manifolds on the separation plane either coincident or perpendicular, we obtain complete results on the existence, stability and number of invariant cones and show that the maximum number of invariant cones is two. The explicit parameter relations obtained here contribute to understanding and investigating bifurcation phenomena occurring in nonsmooth dynamical systems.

Existence and number of invariant cones in general 3-dim homogeneous piecewise linear differential systems with two zones separated by a plane are investigated. Implicit parametric expressions of two proper half slope maps whose intersections determine the existence and number of invariant cones are obtained. Based on these expressions, some sufficient conditions for the existence of at most three invariant cones are provided, and it is proved that the maximum number of invariant cones for some special cases is equal to 1 plus the maximum number of limit cycles in planar piecewise linear systems with a straight line separation. Moreover, it is illustrated by a numerical example with four invariant cones that the maximum number of invariant cones is not less than four. Specially, the main results provide a method to completely solve the existence and number of invariant cones in any specific 3-dim homogeneous piecewise linear differential systems with two zones separated by a plane by using numerical method.

In this paper, we present a derivation of Chesnavich’s Hamiltonian for a model ion-molecule reaction. The model system has the basic structure of a rigid body coupled to a structureless particle. Using the form of the potential energy of interaction given by Chesnavich, we derive the equilibria, determine their stability, and construct two, two-dimensional invariant manifolds and determine their stability.

In this paper, we continue our studies of the two-dimensional caldera potential energy surface in a parametrized family that allows for a study of the effect of symmetry on the phase space structures that govern how trajectories enter, cross, and exit the region of the caldera. As a particular form of trajectory crossing, we are able to determine the effect of symmetry and phase space structure on dynamical matching. We show that there is a critical value of the symmetry parameter which controls the phase space structures responsible for the manner of crossing, interacting with the central region (including trapping in this region) and exiting the caldera. We provide an explanation for the existence of this critical value in terms of the behavior of the Hénon stability parameter for the associated periodic orbits.

Complementary to existing applications of Lagrangian descriptors as an exploratory method, we use Lagrangian descriptors to find invariant manifolds in a system where some invariant structures have already been identified. In this case, we use the parametrization of a periodic orbit to construct a Lagrangian descriptor that will be locally minimized on its invariant manifolds. The procedure is applicable (but not limited) to systems with highly unstable periodic orbits, such as the isokinetic Chesnavich CH${}_4^{+}$ model subject to a Hamiltonian isokinetic theromostat. Aside from its low computational requirements, the method enables us to study the invariant structures responsible for roaming in the isokinetic Chesnavich CH${}_4^{+}$ model.

The paper proposes a novel input–output approach to characterize the dynamical properties of a class of circuits composed by a linear time-invariant two-terminal element coupled with one of the ideal memelements (memory elements) introduced by Prof. L. O. Chua, i.e. memristors, memcapacitors, and meminductors. The developed approach permits to readily determine the conditions under which the dynamics of any circuit of the class admits a first integral. It is also shown that the circuit dynamics can be obtained by collecting the dynamical behaviors displayed by a canonical reduced-order input–output system subject to a constant input of any amplitude. Notably, the reduced-order system exactly describes the dynamics of a circuit forced by a constant generator and with a nonlinear memoryless element in place of the memelement. The relation between the proposed input–output approach and the available state space ones (e.g. Flux-Charge Analysis Method (FCAM)) is also addressed. The main result is that explicit expressions of the invariant manifolds can be directly obtained in the voltage–current state space. Finally, it is shown how the approach also applies to circuits which contain forcing generators. It is believed that the proposed input–output approach can be a useful alternative to state space methods for studying multistability and control issues.

In this paper, we report the bifurcations of mode-locked periodic orbits occurring in maps of three or higher dimensions. The “torus” is represented by a closed loop in discrete time, which contains stable and unstable cycles of the same periodicity, and the unstable manifolds of the saddle. We investigate two types of “doubling” of such loops: (a) two disjoint loops are created and the iterates toggle between them, and (b) the length of the closed invariant curve is doubled. Our work supports the conjecture of Gardini and Sushko, which says that the type of bifurcation depends on the sign of the third eigenvalue. We also report the situation arising out of Neimark–Sacker bifurcation of the stable and saddle cycles, which creates cyclic closed invariant curves. We show interesting types of saddle-node connection structures, which emerge for parameter values where the stable fixed point has bifurcated but the saddle has not, and vice versa.

The goal of the paper is to construct nonautonomous inertial manifolds via outflowing invariant manifolds. As an example, a nonautonomous, nonlocal Burgers equation is considered.