Recently, we presented three low-order Energy Balance Models (EBMs) to propose a definition of climate tipping processes consisting of a bifurcation followed by a subsequent transition. These models were formulated as first-order ordinary differential equations comprising three key components: a time-varying forcing term, a linear feedback term, and a nonlinear quadratic or cubic term. Among these systems, the cubic EBM, which admits a double-well potential structure, represents a minimal framework for capturing regime coexistence, bifurcations, and tipping behavior. In this study, we further establish explicit mathematical connections between the cubic EBM and a class of higher-order dynamical systems, including the cubic oscillator and the nondissipative Lorenz model, both of which support oscillatory solutions. These connections highlight a shared potential-based structure underlying seemingly disparate systems. We also demonstrate the physical relevance of the cubic nonlinear term by linking the cubic EBM to the ice–albedo feedback model. Through a systematic reduction from a smooth hyperbolic-tangent albedo formulation, we show that the cubic nonlinearity represents the leading-order saturation mechanism that introduces an effective nonlinear negative feedback. In addition, we examine how bistability — a defining feature of the cubic EBM — also emerges in more comprehensive climate models, such as ice-sheet models, despite their higher dimensionality and complexity. This comparison underscores the value of the cubic EBM as a normal-form representation for understanding tipping points, regime shifts, and path dependence across a broad hierarchy of climate and geophysical models. Beyond the forward reduction pathway emphasized in this study, we outline backward embedding (upscaling) strategies for extending the cubic EBM to more sophisticated yet dynamically traceable systems, thereby providing a systematic route for linking idealized EBMs with higher-order climate models.

Berezansky et al. [2010] proposed an open problem: How are the dynamic behaviors of the well-known Nicholson’s blowflies model with a delayed linear harvesting. In this paper, we mainly study the existence of Hopf bifurcation of Nicholson’s blowflies model with a delayed linear harvesting. To that end, the stability and Hopf bifurcation of a general functional differential equation with two dependent delays are investigated. We show that if the difference between the two dependent delays is constant, by using one of the delays as a bifurcation parameter, sufficient conditions of stability and Hopf bifurcation are obtained. In addition, in order to determine the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, explicit formulas are given by using the normal form theory. The main results can be applied to guarantee the existence of Hopf bifurcation in Nicholson’s blowflies model with a delayed linear harvesting. Our research partially answers the open problem proposed by Berezansky et al. [2010].

This paper reports the emergence of rare, large-amplitude events in an extremely simple autonomous jerk circuit employing a single semi-conductor diode as its only nonlinear element. The governing equation corresponds to a third-order jerk system with exponential nonlinearity. Thanks to numerical simulations and theoretical analysis, we characterize the system’s dynamics using two-parameter Lyapunov exponent plots, bifurcation diagrams, and phase portraits. Our study reveals that these rare large-amplitude events emerge particularly at an interior crisis point. Statistical analysis confirms their rarity, with events detected at thresholds ranging from 4 to 10 standard deviations above the mean. Experimental measurements from a hardware realization of the circuit corroborate the salient features predicted by the theoretical study. To the best of our knowledge, this model represents the simplest chaotic jerk circuit reported to date capable of exhibiting such pronounced, rare large-amplitude events. Its elementary design, hinging on a single diode, makes it a paradigm of interest for fundamental studies of crisis-induced dynamics and a versatile building block for investigating complex phenomena in more advanced applications, such as coupled oscillator networks.

Extensive research has solved the dynamic problems of electromagnetic bearings under 1:1 internal resonance, but the complex nonlinear behavior of a 12-pole variable stiffness system under 1:2 internal resonance has not been explored to a large extent. This paper aims to bridge this gap by providing comprehensive dynamic analysis and PD control strategies for such systems. First, the expression for the electromagnetic force in the system is derived based on electromagnetic theory, applying Newton’s second law and considering the effect of rotor gravity. The resulting differential equations governing the dynamics and control of the magnetic bearing, which include quadratic and cubic terms, are then derived. Next, the relationships between the first-order and second-order natural frequencies are analyzed, taking into account 1:2 internal resonance, primary parameter resonance, and 1/2 subharmonic resonance. A perturbation analysis of the system is performed using the method of multiple time scales, yielding the four-dimensional averaged equations in both polar and Cartesian coordinates, as well as the amplitude–frequency response equations. Finally, numerical simulations reveal distinct vibration modulation patterns: parametric excitation dominates the first-order modal response, whereas external forcing preferentially excites the second-order mode, resulting in anisotropic oscillations along orthogonal axes. Moreover, the differential gain is demonstrated to be a critical factor in the suppression of chaos and bifurcations. These results significantly advance the understanding of the nonlinear dynamic behaviors inherent in active electromagnetic bearing systems.

This paper introduces a competition in the Cournot duopoly game where the players adopt uncertain price functions. In contrast with other studies found in the literature, the aim of each player in this game is to maximize its expected profit along with minimizing its variance. The optimization objective functions in this game are linear combinations of the expected profit and its variance for each firm. The firms are assumed to adjust their outputs adaptively based on bounded rationality, leading to a discrete-time nonlinear system. The game’s map’s dynamic characteristics are analyzed in detail, including supporting the obtained results with numerical evidence such as contact bifurcations, basins of attraction and absorbing areas. The obtained results show that increasing the price uncertainty parameter or a firms’ adjustment speed can destabilize the Cournot–Nash equilibrium and drive the system toward cyclical or chaotic fluctuations. These results provide us a more profound understanding of how uncertainty and adaptive behavior work together to produce unexpected changes in prices and production in real markets.

The periodic orbits are central to the transition state theory and the calculation of chemical reaction rates. Since this computation is often highly challenging, we introduce two methods based on Particle Swarm Optimization (PSO), an algorithm that belongs to the field of swarm intelligence (a subfield of artificial intelligence), to efficiently locate these objects and overcome the associated difficulties.

As critical damping components of vehicles, suspension systems play an essential role in maintaining vehicle stability and enhancing ride comfort. This paper studies the dynamic behaviors and reliability of the suspension system. First, based on Newton’s second law, a single-degree-of-freedom suspension system model is established through simulating the rough road fluctuations as a combination of typical cosinusoidal road excitation and Gaussian white noise. Then, considering the linear damping and nonlinear damping, respectively, the dynamic evolution and first-passage failure of the system under primary resonance and 1/3 subharmonic resonance conditions are examined by the path integral method. The influence mechanisms of dampings, road surface excitation amplitude and noise intensity on the dynamics of suspension systems are explored. The results demonstrate that reduced damping, increased road excitation amplitude and higher noise intensity collectively impair system stability. Crucially, the system’s response to these parameters is governed by the resonance type. Within a certain range, under primary resonance, road amplitude predominantly affects displacement, whereas under 1/3 subharmonic resonance, it significantly alters both displacement and velocity distributions, even inducing stochastic P-bifurcation. These findings provide valuable insights into the design and optimization of vehicle suspension systems for improving performance and reliability.

A stochastic dynamic model is proposed for a Z-shaped folding wing with random excitations. The method of multiple scales is employed to derive amplitude equations. Bistable phenomenon on the deterministic system is revealed through analyses on bifurcation and basin of attraction. Under the effect of random loads, the noise-induced transition behavior is observed, and the noise-enhanced stabilization phenomenon is examined in detail. Reliability analysis is conducted using a proposed reliability index. Reliability region is presented versus different parameters including noise intensity. The results of this study can provide guidance for the design and reliability analysis of folding wings.

In this paper, we introduce a chaos indicator derivable from Lagrangian Descriptors (LDs), defined as the difference in LD values between two neighboring trajectories. This difference LD is remarkably easy to implement and interpret, offering a direct and intuitive measure of dynamical behavior. We provide a heuristic argument linking its growth to the regularity or chaoticity due to the underlying initial condition, considering the arclength-based formulation of LDs. To evaluate its effectiveness, we benchmark it against more elaborate LD-based chaos indicators and the Smaller Alignment Index (SALI) using two prototypical systems: the Hénon–Heiles system and the Chirikov Standard Map. Our results show that, despite its simplicity, the difference LD matches the accuracy of more sophisticated methods, making it a robust and accessible tool for chaos detection in dynamical systems.

Plotting basins of attraction reveals the global behavior of nonlinear dynamical systems, specifically in the case of multistability when multiple attractors coexist for a fixed set of parameters. However, calculation of high-resolution diagrams of attraction basins is computationally demanding due to the need for extensive numerical simulation across a large grid of initial conditions. To address this problem, we present a novel GPU-accelerated framework for efficient and scalable computation of basins of attraction using the CUDA toolkit. Our approach combines parallelized trajectory calculation with a robust feature-based clustering strategy that leverages mean peak amplitudes and mean inter-peak intervals with evaluation of fixed-point, unbound, and oscillatory regimes. In addition, we introduce an augmented data representation preserving full distributions of dynamical features, enabling subsequent analysis and parameter adjustment without additional simulation. The proposed method is validated across a diverse set of multistable systems, including cases of finite multistability, megastability, Matryoshka multistability, and extreme multistability. All designed software and obtained data have been published in public repositories to support reproducibility of the study.