Dynamical Behavior of a New Chaotic System with One Stable Equilibrium

This paper reports a simple three-dimensional autonomous system with a single stable node equilibrium. The system has a constant controller which adjusts the dynamic of the system. It is revealed that the system exhibits both chaotic and non-chaotic dynamics. Moreover, chaotic or periodic attractors coexist with a single stable equilibrium for some control parameter based on initial conditions. The system dynamics are studied by analyzing bifurcation diagrams, Lyapunov exponents, and basins of attractions. Beyond a fixed-point analysis, a new analysis known as connecting curves is provided. These curves are one-dimensional sets of the points that are more informative than fixed points. These curves are the skeleton of the system, which shows the direction of flow evolution.

Influence of Controller’s Parameters on Static Bifurcation of Magnetic-Liquid Double Suspension Bearing

Magnetic-Liquid Double Suspension Bearing (MLDSB) is composed of an electromagnetic supporting and a hydrostatic supporting system. Due to greater supporting capacity and static stiffness, it is appropriate for occasions of middle speed, overloading, and frequent starting. Because of the complicated structure of the supporting system, the probability and degree of static bifurcation of MLDSB can be increased by the coupling of hydrostatic force and electromagnetism force, and then the supporting capacity and operation stability are reduced. As the key part of MLDSB, the controller makes an important impact on its supporting capacity, operation stability, and reliability. Firstly, the mathematical model of MLDSB is established in the paper. Secondly, the static bifurcation point of MLDSB is determined, and the influence of parameters of the controller on singular point characteristics is analyzed. Finally, the influence of parameters of the controller on phase trajectories and basin of attraction is analyzed. The result showed that the pitchfork bifurcation will occur as proportional feedback coefficient Kp increases, and the static bifurcation point is Kp = −60.55. When Kp < −60.55, the supporting system only has one stable node (0, 0). When Kp > −60.55, the supporting system has one unstable saddle (0, 0) and two stable non-null focuses or nodes. The shape of the basin of attraction changed greatly as Kp increases from −60.55 to 30, while the outline of the basin of attraction is basically fixed as Kp increases from 30 to 80. Differential feedback coefficient Kd has no effect on the static bifurcation of MLDSB. The rotor phase trajectory obtained from theoretical simulation and experimental tests are basically consistent, and the error is due to the leakage and damping effect of the hydrostatic system within the allowable range of the engineering. The research in the paper can provide theoretical reference for static bifurcation analysis of MLDSB.

Controlling effort, catch and biomass in models of the global marine fisheries

The starting point is the reduced model of global marine fisheries designated by W-3. The main variables of an ordinary differential equation are: the stock of bioresource, its net natural increase, as well as the catch value, which linearly depends on exogenous effort and nonlinearly on available biomass. In W-4, the effort became endogenous as a result of its positive feedback from biomass. In both models, there are values of control parameters in the catch equations, in which the value of the latter can be maintained for a long time at the maximum stable level, with the exception of transition sections. The principle of necessary precaution is fulfilled for small fish stocks more reliably in W-4 than in W-3, thanks to the transformation of the saddle into an unstable node with a common stable node. For these one-dimensional models, the author proposed an original generalization - the R-1 model of two nonlinear ordinary differential equations. In the latter effort, a new phase variable appears, subordinated to proportional control and derivative regulation. Biomass serves as a “prey”, and the effort appears as a “predator”. For two key control parameters, areas of change were identified for which the target stationary state is a locally asymptotically stable node or focus in R-1. A policy has been proposed for the restoration of depleted fish stocks and a transition to a long-term maximum sustainable harvest has been determined. Optimization over a wide time frame (from 40 to 400 years) allows us to calculate the values of the selected control parameters for which the integral catch volume in R-1 is higher than in W-4 for the same initial values of stock, effort and catch. Social constraints from below on the magnitude of the effort, as well as the desired nature of the transition to the target stationary state, are taken into account. The danger of biomass collapse is overcome, unlike previous models.

Load Balanced Multipath Routing using Seagull Optimizer Based Garson’s Pruned Recurrent Neural Network for Multipath Routing in MANET

In recent years, routing is considered one of the most challenging issues in MANET. The location of the stable node and the routing is based on the predicted locations that assists in establishing a routing path in MANET. The major intention of this paper is to detect the stable neighbor node in the MANET and also to establish stable multi-path routing for the various mobility patterns. Also, this paper deals with the data packet scheduling over multi-paths for balancing the load and forward the entire packets in the least broadcast time. The proposed approach elucidates four significant phases: stable node prediction, determination of stability measure, route exploration and packet dissemination. Initially, the stable node is predicted using the RMSG approach. Here, the stable neighbors are selected via Garson’s pruning based Recurrent neural network with a Modified seagull optimization algorithm (RMSG). In the route exploration phase, the path is created among the source and destination by the stable node. If any of the links fails, the route recovery process is established. Finally, the structure is formed for data packet distribution across the multipath. The proposed approach is evaluated by few performance measures such as throughput, packet delivery ratio, end-to-end delay, routing overhead energy consumption, and optimal path. This result describes that the proposed approach outperforms other state-of-art approaches.

Complex Dynamics of a Four-Dimensional Circuit System

Combining qualitative analysis and numerical technique, the present work revisits a four-dimensional circuit system in [Ma et al., 2016] and mainly reveals some of its rich dynamics not yet investigated: pitchfork bifurcation, Hopf bifurcation, singularly degenerate heteroclinic cycle, globally exponentially attractive set, invariant algebraic surface and heteroclinic orbit. The main contributions of the work are summarized as follows: Firstly, it is proved that there exists a globally exponentially attractive set with three different exponential rates by constructing a suitable Lyapunov function. Secondly, the existence of a pair of heteroclinic orbits is also proved by utilizing two different Lyapunov functions. Finally, numerical simulations not only are consistent with theoretical results, but also illustrate potential existence of hidden attractors in its Lorenz-type subsystem, singularly degenerate heteroclinic cycles with distinct geometrical structures and nearby hyperchaotic attractors in the case of small [Formula: see text], i.e. hyperchaotic attractors and nearby pseudo singularly degenerate heteroclinic cycles, i.e. a short-duration transient of singularly degenerate heteroclinic cycles approaching infinity, or the true ones consisting of normally hyperbolic saddle-foci (or saddle-nodes) and stable node-foci, giving some kind of forming mechanism of hyperchaos.