Periodic Limit Cycles Research Articles

Characterization and dynamics of certain classes of polynomial vector fields on the torus

In this paper, we classify all polynomial vector fields in R3 of degree up to three such that their flow makes the torusT2={(x,y,z)∈R3:(x2+y2−a2)2+z2−1=0}witha∈(1,∞) invariant. We also classify cubic Kolmogorov vector fields on T2 and prove that they exhibit a rational first integral. We study ‘pseudo-type-n’ vector fields on T2 and show that any such vector field is completely integrable. We prove that the Lie bracket of any two quadratic vector fields on T2 is completely integrable. We explicitly find all cubic vector fields on T2 which achieve the sharp bounds for the number of invariant meridians and parallels. We present necessary and sufficient conditions when invariant meridians and parallels of cubic vector fields on T2 are periodic orbits or limit cycles. We discuss invariant meridians and parallels of pseudo-type-n vector fields as well. Moreover, we characterize the singular points of a class of polynomial vector fields on T2.

Dynamical Systems Characterisation and Reduced Order Modelling of Thermoacoustics in a Lean Direct Injection (LDI) Hydrogen Combustor

Abstract In this study, the thermoacoustic instabilities of partially premixed hydrogen flames in a lean direct injection (LDI) multi-cluster combustor are characterised using dynamical systems theory. The combustor is operated at a range of bulk velocities (30 - 90 m/s) and equivalence ratios(0.2-0.6), and time-resolved pressure oscillations and integrated OH* chemiluminescence measurements were taken. The thermoacoustic system reveals a variety of dynamical states in pressure such as period-1 Limit Cycle Oscillation (LCO) with a single characteristic frequency, period-2 LCO with two characteristic frequencies, intermittent, quasi-periodic and chaotic states as either bulk velocity or equivalence ratio is varied. At a bulk velocity of 30 m/s, as the equivalence ratio is gradually decreased from 0.6 to 0.2, the dynamical behaviour follows a sequence from an intermittent state to a period-1 LCO, then to a quasi-periodic state, and eventually reaches a chaotic state. As the equivalence ratio is decreased for a bulk velocity of 60 m/s, the pressure oscillations evolve from a period-2 LCO to quasi-periodic state before flame blows off. The emergence of period-2 and quasi-periodic states indicates the presence of strong non-linear interactions among the cavity acoustic modes. The emergence of period-2 and quasi-periodic states indicates strong non-linear interactions among the cavity acoustic modes. These modes and their spatial behaviour are investigated using a Helmholtz solver, showing that hydrogen flames can excite a wide range of cavity acoustic modes

Chaos via type-II intermittency in a forced globally unstable jet

We explore the transition to chaos in a prototypical hydrodynamic oscillator, namely a globally unstable low-density jet subjected to external time-periodic forcing. As the forcing strengthens at an off-resonant frequency, we find that the jet exhibits a sequence of nonlinear states: period-1 limit cycle $\rightarrow $ quasiperiodicity $\rightarrow$ intermittency $\rightarrow$ low-dimensional chaos. We show that the intermittency obeys type-II Pomeau–Manneville dynamics by analysing the first return map and the scaling properties of the quasiperiodic lifetimes between successive chaotic epochs. By providing experimental evidence of the type-II intermittency route to chaos in a globally unstable jet, this study reinforces the idea that strange attractors emerge via universal mechanisms in open self-excited flows, facilitating the development of instability control strategies based on chaos theory.

The Demographic-Wealth model for cliodynamics.

Cliodynamics is a still a relatively new research area with the purpose of investigating and modelling historical processes. One of its first important mathematical models was proposed by Turchin and called "Demographic-Fiscal Model" (DFM). This DFM was one of the first and is one of a few models that link population with state dynamics. In this work, we propose a possible alternative to the classical Turchin DFM, which contributes to further model development and comparison essential for the field of cliodynamics. Our "Demographic-Wealth Model" (DWM) aims to also model link between population and state dynamics but makes different modelling assumptions, particularly about the type of possible taxation. As an important contribution, we employ tools from nonlinear dynamics, e.g., existence theory for periodic orbits as well as analytical and numerical bifurcation analysis, to analyze the DWM. We believe that these tools can also be helpful for many other current and future models in cliodynamics. One particular focus of our analysis is the occurrence of Hopf bifurcations. Therefore, a detailed analysis is developed regarding equilibria and their possible bifurcations. Especially noticeable is the behavior of the so-called coexistence point. While changing different parameters, a variety of Hopf bifurcations occur. In addition, it is indicated, what role Hopf bifurcations may play in the interplay between population and state dynamics. There are critical values of different parameters that yield periodic behavior and limit cycles when exceeded, similar to the "paradox of enrichment" known in ecology. This means that the DWM provides one possible avenue setup to explain in a simple format the existence of secular cycles, which have been observed in historical data. In summary, our model aims to balance simplicity, linking to the underlying processes and the goal to represent secular cycles.

Ground-induced suppression of chaos in the self-excited flow behind a plunging airfoil

We numerically investigate the forced synchronization of the self-excited flow behind a plunging airfoil in ground effect at a Reynolds number of Re = 1000. On varying the plunging amplitude and frequency, we find a rich array of nonlinear dynamics, such as a period-1 limit cycle due to natural vortex shedding as well as two-frequency quasiperiodicity on a torus attractor (T2). For certain non-resonant plunging frequencies without a ground surface, we find that low-dimensional chaos emerges via the Ruelle–Takens–Newhouse route. However, we find that the chaos can be suppressed by introducing a ground surface, inducing a direct transition from T2 quasiperiodicity to 1:1 phase locking as the plunging amplitude rises over the boundaries of the Arnold tongue. Apart from suppressing chaos, the ground surface also causes the lift and drag coefficients to become less sensitive to the plunging motion itself. Knowledge of the critical plunging conditions required for forced synchronization and chaos could be useful in various engineering applications, such as the design of pico air vehicles.

A mathematical model for tumor-immune competitive system with multiple time delays

A mathematical model is proposed to study the complicated dynamics of tumor-immune interactions with three discrete time delays. The proposed model consists of nine coupled ordinary differential equations (ODEs) whose components are tumor cells, tumor-specific CD8+T cells, dendritic cells, macrophages, regulatory T-cells (Tregs), interleukin-10 (IL-10), IL-12, transforming growth factor-β (TGF-β) and interferon-γ (IFN-γ). By utilizing the quasi-steady-state approximation for cytokine concentrations, we obtain a system of four coupled ODEs. We introduce three discrete time delays into our deterministic system to better understand the tumor-immune dynamics. Basic properties of the system, including existence, uniqueness, positivity, boundedness, and uniform persistence are discussed. Stability analysis of both the delayed and non-delayed system has been performed and the conditions for stability and direction of Hopf bifurcation have been determined. Furthermore, we assess the length of time delay required to maintain the stability of period-1 limit cycle. Parameter estimation techniques are discussed and numerical simulations are provided to support our theoretical analysis. Notably, we observe that time delays do not significantly influence the system behavior for the existing set of parameters value.

Period addition cascade in a model of neuron-glial interaction

We analyze a system of four differential equations that describe the dynamics of a neuron-glial network using the mean field approximation [1, 2]: (τ=–E+α ln(1+e1/α(JuxE+I0)), =(1–x)/τD–uxE,(1) =U(y)–u/τF+U(y)(1–u)E, =–y/τy+βσ(x), where E(t) is the average activity, x(t) is the fraction of available neurotransmitter released into the synaptic gap with a probability u(t); y(t) is the fraction of the gliatransmitter released by the astrocyte. Sigmoidal functions U(y) and σ(x) correspond to changes in the base probability level u(t) during the release of the gliatransmitter and activation of astrocytes during neurotransmitter release, respectively. The input inhibitory current corresponds to a bifurcation parameter with a negative value of I0 0, while the remaining parameters have positive and fixed values. The rest of the parameters are positive and fixed. For a detailed description of the model, including information on the type of functions and parameter values, refer to works [1, 2]. For a constant value of U(x)=const, the first three equations in system (1) represent the Tsodyks–Markram model, which explains the short-term synaptic plasticity phenomenon [1]. The model was enhanced with a fourth equation for y in [2], incorporating the influence of astrocytes via the concept of a tripartite synapse [3]. Model (1) illustrates a wide range of dynamic behavior including quiescence, regular tonic activity, and chaotic bursting activity. These behaviors correspond to various sets in the phase space, such as stable equilibrium states, limit cycles of period 1, limit cycles of any period n∈N, and chaotic attractors. Changing the I0 l parameter causes sets to bifurcate, resulting in the loss of stability of certain attractors and the emergence of others, leading to a shift in the dynamic regime. Therefore, in terms of dynamics, the conditions for bifurcation and the characteristics of the newly formed attractors are crucial. In this presentation, we have obtained a series of numerical bifurcations in system (1) that correspond to the shift from tonic activity to burst activity, resulting in subsequent modifications to the bursts. Specifically, our findings demonstrate that an increase in the number of spikes per burst is determined by a period adding cascade where the aforementioned limit cycle of period n becomes unstable, allowing for a previously established stable cycle of period n+1 to occupy the position of the “main” attractor. This process culminates in the vanishing of the orbit with an endless period due to the saddle-node bifurcation of cycles, followed by the creation of a dependable cycle with a period of 1. The main properties of the cascade were reproduced in our model one-dimensional piecewise-smooth map z¯=1−z6, forz0,μ−1−μ(z-1)6, forz0, where z∈R1, μ is a bifurcational parameter. The map’s results suggest that an increase in current I0 i in model (1) may lead to the emergence and disappearance of quasi-strange attractors (quasi-attractors), implying chaotic behavior in connection with burst variation.

Large eddy simulation of effects of oxidizer inlet temperatures on the transition routes before and after thermoacoustic instability in a subcritical hydrogen peroxide/kerosene liquid rocket engine

This paper presents the first numerical evidence of the intermittency routes that exist before and after the occurrence of thermoacoustic instability in a subcritical single-element liquid rocket engine burning liquid kerosene and decomposed hydrogen peroxide with increasing oxidizer inlet temperatures (T). Three-dimensional compressible large eddy simulation algorithms, combined with Euler–Lagrangian frameworks, are employed to model the spray turbulent combustion process in a high-pressure rocket combustor where a one-equation eddy viscosity sub-grid turbulence model and a PaSR sub-grid combustion model are used based on OpenFOAM. After verifying the numerical framework and achieving grid independence, we focus on (i) dynamical transition routes before and after the thermoacoustic regime, (ii) frequency-locking phenomena between acoustic perturbations, vortex dynamics, and combustion heat release, and (iii) the underlying physical mechanisms associated with different dynamical states. The results show that as we increase the oxidizer inlet temperature (700 K ≤ T ≤ 900 K), the system dynamics undergo a transition from a state of combustion noise to a period-1 limit cycle via intermittency. Furthermore, by further increasing the oxidizer inlet temperature (950 K ≤ T ≤ 1450 K), a second bifurcation occurs, causing a transition from a limit cycle state back to a combustion noise state also via intermittency. These bifurcation phenomena are attributed to frequency-locking interactions among pressure fluctuations, combustion heat release, and vortex dynamics. Evidence from combustion flow subsystems including mixture fraction, burning modes, and flame-induced vorticity sources provides additional insights into the complex instability mechanism.

On the quality of stability and bifurcation sets in rotors with permanent shaft bow on nonlinear supports

The paper investigates the dynamics of simple and complex geometry rotors of various configurations with permanent bow, mounted on nonlinear bearings, in regards to nonlinear stability, quality of motion, and bifurcation set. Permanent shaft bow is a fault existing in most slender rotors in thermal machines, e.g. gas turbines, steam turbines, and turbopumps, and other applications of rotating machinery like transmission shafts and geared shafts. In this work, finite element models of multi-segment rotors are implemented, including permanent bow, and a generic rotor system of a uniform rotor with three masses is studied first on its stability characteristics for various bow values, and for different slenderness. A realistic rotor of complex geometry is used as the second application. The rotors are mounted on two plain short bearings to include the nonlinearity of impedance forces with analytical formulas; this is the unique source of nonlinearity in the systems.The rotor-bearing systems are found to produce the well-known oil whirl instability through a Hopf, or a secondary Hopf bifurcation, depending the unbalance grade, in speeds higher than the first critical speed, when bow is not included. Supercritical and subcritical Neimark-Sacker and period doubling bifurcations are produced in speeds lower than critical speed when bow is included above some threshold values. Stable and unstable limit cycles are generated both for unbalanced or balanced rotors with bow, and their periodicity is investigated. A bifurcation set of the rotor-bearing systems is evaluated utilizing the pseudo-arc length continuation method with embedded collocation method, for the evaluation of periodic limit cycles (where these appear). Neimark-Sacker bifurcations are found to collide and vanish as the bow magnitude changes from high to low values. Bow phase angle with respect to unbalance force is found not to influence the respective bifurcation sets.

Analysis and simulations of the HANDY model with social mobility, renewables and nonrenewables

We expand the HANDY (Human And Nature DYnamics) model for the socioeconomic dynamics of a large stratified society. The basic model was introduced in Motesharrei (Dissertation 2014) and Motesharrei et al. (2016). It is a nonlinear system of ODEs for a `simple society' of Elites, Workers, Wealth, and Natural Resources. Following Ali Kadhim (Dissertation 2021), we add social mobility between the classes and split natural resources into renewables and nonrenewables. We establish the existence, boundedness and positivity of the solutions, and investigates the stability of the steady states. The model admits stable steady states, and there is numerical evidence of stable periodic solutions and limit cycles. Simulations depict the different qualitative types of model behavior: convergence to steady states, periodic oscillations, collapse.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/59/abstr.html

Erratum: “Dynamical Characterization of Thermoacoustic Oscillations in a Hydrogen-Enriched Partially Premixed Swirl-Stabilized Methane/Air Combustor” [ASME J. Eng. Gas Turbines Power, 143(12), p. 121022; DOI: 10.1115/1.4052091

Abstract In this study, we systematically analyze the effects of hydrogen enrichment in the well-known PRECCINSTA burner, a partially premixed swirl-stabilized methane/air combustor. Keeping the equivalence ratio and thermal power constant, we vary the hydrogen percentage in the fuel. Successive increments in hydrogen fuel fraction increase the adiabatic flame temperature and also shift the dominant frequencies of acoustic pressure fluctuations to higher values. Under hydrogen enrichment, we observe the emergence of periodicity in the combustor resulting from the interaction between acoustic modes. As a result of the interaction between these modes, the combustor exhibits a variety of dynamical states, including period-1 limit cycle oscillations (LCO), period-2 LCO, chaotic oscillations, and intermittency. The flame and flow behavior is found to be significantly different for each dynamical state. Analyzing the coupled behavior of the acoustic pressure and the heat release rate oscillations during the states of thermoacoustic instability, we report the occurrence of 2:1 frequency-locking during period-2 LCO, where two cycles of acoustic pressure lock with one cycle of the heat release rate. During period-1 LCO, we notice 1:1 frequency-locking, where both acoustic pressure and heat release rate repeat their behavior in every cycle.

Fluid-structure interaction analysis of nonlinear flapping dynamic behaviors of variable stiffness composite laminated plates in viscous flows

This paper is concerned with the development of a numerical model for the analysis of nonlinear fluid–structure interaction problems of large-deformable curvilinear fiber-reinforced composite laminated plates in viscous flows. A higher-order shear deformation zig-zag theory in conjunction with nonlinear von Kármán strains is employed to accommodate the large deformations of variable stiffness composite laminated plates. An implicit partitioned fluid–structure interaction method based on an arbitrary Lagrangian-Eulerian framework is adopted for dealing with the coupling of the large-deformable plate and the viscous flow. The validity of the proposed model and method is confirmed by comparing the computed results of a two-layered plate with those available solutions in the literature. Effects of material properties and fiber paths of composite plates on the periodic limit cycle oscillation and the wake-flow vortices are examined. It is found that symmetrical lay-ups of the constant stiffness composite laminated plates can produce asymmetrical flapping shapes, resulting in asymmetrical hairpin vortex structures in the wake flow. For a variable stiffness composite laminated plate, the flexibility of the leading or trailing sections can be adjusted by the curved fiber path, which in turn leads to changes in the plate flapping shapes.

Construction of a reduced-order model of an electro-osmotic micromixer and discovery of attractors for petal structure

The chaotic state of microfluidic devices such as electroosmotic micromixers has received extensive attention. Its unsteady flow and multi-physics mask low-dimensional structure and potential attractors. Based on the dynamic mode decomposition and the sparse identification of nonlinear dynamics, this study aims to construct a manifold equation with the minimum degree of freedom, reveal the mixing mechanism of micromixers, and discover the evolution of chaotic states. The attenuation degree of freedom was introduced to force the modal coefficients to be pure oscillations. The six, four, and two-dimensional minimum reduced-order models (ROMs) were constructed under different mixing conditions. The nonlinear dynamics evolves on attractors resembling a six-petal structure based on the amplitude-phase method. The attractor periodicity and decay map the evolution of the periodic oscillation and limit cycle of the active modes and are related to the appearance of the low-energy dominant non-axisymmetric modes. These results emphasize the significance of ROM technology in revealing the low-dimensional structure and attractor of the electroosmotic micromixer.

Dynamic Advertising Games in Duopolies Under One-Step-Ahead Optimal Control

In discrete-time Vidale–Wolfe–Deal duopoly models and variants, two firms compete for market share, in a dynamic game setting, described by a pair of difference equations. This paper studies these dynamic games, using the natural concept of one-step-ahead optimal control, in which each firm optimizes its own performance index at the next step, and only has access to some information about its competitor. Two cases are studied: with and without stipulating target market shares for each firm, under sequential and parallel game playing procedures. It is shown that when target market shares are not specified, for the VWD model, limit cycles of large period can occur when each firm uses linear performance indices, while multiple equilibria may arise when quadratic performance indices are used. Three other proposed models result in games that lead to equilibria and do not have limit cycle behavior. When target market shares are specified, convergence to an equilibrium occurs for all the models proposed in this paper.

Heteroclinic cycling and extinction in May–Leonard models with demographic stochasticity

May and Leonard (SIAM J Appl Math 29:243–253, 1975) introduced a three-species Lotka–Volterra type population model that exhibits heteroclinic cycling. Rather than producing a periodic limit cycle, the trajectory takes longer and longer to complete each “cycle”, passing closer and closer to unstable fixed points in which one population dominates and the others approach zero. Aperiodic heteroclinic dynamics have subsequently been studied in ecological systems (side-blotched lizards; colicinogenic Escherichia coli), in the immune system, in neural information processing models (“winnerless competition”), and in models of neural central pattern generators. Yet as May and Leonard observed “Biologically, the behavior (produced by the model) is nonsense. Once it is conceded that the variables represent animals, and therefore cannot fall below unity, it is clear that the system will, after a few cycles, converge on some single population, extinguishing the other two.” Here, we explore different ways of introducing discrete stochastic dynamics based on May and Leonard’s ODE model, with application to ecological population dynamics, and to a neuromotor central pattern generator system. We study examples of several quantitatively distinct asymptotic behaviors, including total extinction of all species, extinction to a single species, and persistent cyclic dominance with finite mean cycle length.

Memristor synapse-coupled piecewise-linear simplified Hopfield neural network: Dynamics analysis and circuit implementation

Electromagnetic induction current is generated between the adjacent neurons in neural network caused by the existence of membrane potential difference. Memristor is the fourth fundamental electric element, which can mimic the behavior of neural synapses and simulate the electromagnetic induction effect. In this paper, in order to simplify practical implementation, the commonly used hyperbolic tangent activation function of Hopfield neural network (HNN) is replaced by a piecewise-linear function, and a simple bi-neuron-based memristor synapse-coupled HNN is proposed. Theoretical analysis and numerical simulation results illustrate that the memsirstive HNN possesses five equilibria including one unstable saddle-focus, two unstable saddle points, and two stable node points (or node-foci). Local attraction basins and phase plane plots show that the memristive HNN model behaves multistability of coexisting chaos, periodic limit cycles, and stable point attractors. Various system dynamical behaviors affected by parameters of the piecewise-linear activation function, memristor coupling strength, and initial conditions of the neurons are investigated by numerical simulations. Furthermore, an analog circuit of the memristive HNN model is simply designed, which is easier for hardware implementation as the piecewise-linear activation function can be implemented by simple op-amp-based module. Finally, the experimental results verify the correctness of the design and analyses.

Optimizing energy dissipation in gas foil bearings to eliminate bifurcations of limit cycles in unbalanced rotor systems

High-speed rotor systems mounted on gas foil bearings present bifurcations which change the quality of stability, and may compromise the operability of rotating systems, or increase noise level when amplitude of specific harmonics drastically increases. The paper identifies the dissipating work in the gas film to be the source of self-excited motions driving the rotor whirling close to bearing’s surface. The energy flow among the components of a rotor gas foil bearing system with unbalance is evaluated for various design sets of bump foil properties, rotor stiffness and unbalance magnitude. The paper presents a methodology to retain the dissipating work at positive values during the periodic limit cycle motions caused by unbalance. An optimization technique is embedded in the pseudo-arc length continuation of limit cycles, those evaluated (when exist) utilizing an orthogonal collocation method. The optimization scheme considers the bump foil stiffness and damping as the variables for which bifurcations do not appear in a certain speed range. It is found that secondary Hopf (Neimark–Sacker) bifurcations, which trigger large limit cycle motions, do not exist in the unbalanced rotors when bump foil properties follow the optimization pattern. Period-doubling (flip) bifurcations are possible to occur, without driving the rotor in high response amplitude. Different design sets of rotor stiffness and unbalance magnitude are investigated for the efficiency of the method to eliminate bifurcations. The quality of the optimization pattern allows optimization in real time, and gas foil bearing properties shift values during operation, eliminating bifurcations and allowing operation at higher speed margins. Compliant bump foil is found to enhance the stability of the system.

Lid-driven cavity flow-induced dynamics of a neutrally buoyant solid: Effect of Reynolds number, flexibility, and size

The present work is on Fluid flexible–Solid Interaction (FfSI), involving a recirculating flow-induced motion of a neutrally buoyant and deformable circular solid. For a Newtonian fluid flow and neo-Hookean flexible-solid deformation, a single FfSI solver—based on fully Eulerian and monolithic approaches—is used. The effect of Reynolds Number Re (20–500), volume fraction Φ (1%–12%) of the solid, and its non-dimensional shear modulus G*(0.02–1) on transient/periodic flow-induced solid-motion and the associated FfSI analysis is presented. The solid undergoes a transient spiraling motion before attaining a periodic orbit-based limit cycle. The flow also attains the periodic state after the initial transients. Time-averaged flow velocity-magnitude ⟨v*⟩ surrounding the limit cycle increases with increasing Re, increasing G*, and decreasing Φ. Equivalent radius req* of the limit cycle and time-averaged velocity-magnitude ⟨vc*⟩ of the centroid of the solid increase with increasing Re and decrease with decreasing G* (or increasing flexibility) and increasing volume fraction Φ (or size) of the solid. Also, frequency f* of the limit cycle decreases with increasing Re and remains almost constant with G* and Φ. With increasing Φ, the limit cycle undergoes a transition from the single loop to double loop beyond a critical volume fraction Φc=2%. A critical Reynolds number Rec, below which the periodic limit cycle collapses to a point, decreases with decreasing Φ. Our findings will help in the prediction and control of the motion of the solid in a bounded fluid flow involving solids of varying flexibility, which is relevant to a wide range of industrial and biological applications.

A novel algorithm to analyze the dynamics of digital chaotic maps in finite-precision domain

Chaotic maps are widely used to design pseudo-random sequence generators, chaotic ciphers, and secure communication systems. Nevertheless, the dynamic characteristics of digital chaos in finite-precision domain must be degraded in varying degrees due to the limited calculation accuracy of hardware equipment. To assess the dynamic properties of digital chaos, we design a periodic cycle location algorithm (PCLA) from a new perspective to analyze the dynamic degradation of digital chaos. The PCLA can divide the state-mapping graph of digital chaos into several connected subgraphs for the purpose of locating all fixed points and periodic limit cycles contained in a digital chaotic map. To test the versatility and availability of our proposed algorithm, the periodic distribution and security of 1-D logistic maps and 2-D Baker maps are analyzed in detail. Moreover, this algorithm is helpful to the design of anti-degradation algorithms for digital chaotic dynamics. These related studies can promote the application of chaos in engineering practice.

Nonlinear oscillations of a collapsible tube subjected to unsteady external pressure

The non-linear dynamics of an extremely thin-walled collapsible tube with internal flow subjected to a time-varying external pressure are studied experimentally and theoretically. For the constant chamber pressure case, we observe the existence of a fixed-point attractor, period-1 attractor, and quasiperiodic attractor. The period-1 limit cycle oscillations are essentially relaxation oscillations with up-down asymmetry in the time domain, and as the Reynolds number increases, the asymmetry becomes greater. With the forcing (varying chamber pressure), the system has no fixed points; its response can be period-n, quasiperiodic, or chaotic, depending upon the Reynolds number, driving amplitude, and frequency. For the forced system, at a low Reynolds number, the external forcing dominates the self-excited oscillations and symmetric oscillations are observed; at a higher Reynolds number, the reverse is true. In experiments and theory, aperiodic oscillations for the forced system are always observed in regimes beyond the Hopf bifurcation point of the unforced system. Distended and collapsed cases, under forcing, exhibit only 1:1 synchronous oscillation. These suggest that a natural oscillation timescale of the system must be present for the external forcing to induce aperiodicity. In the experiments, the forced system exhibits signs of quasiperiodic route to chaos at lower driving amplitude, while period-doubling route to chaos at higher driving amplitude. When the system is forced near its natural frequency, an aperiodic response is totally suppressed.