Symmetry-breaking Bifurcation Research Articles

On New Symmetric Fractional Discrete-Time Systems: Chaos, Complexity, and Control

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, a theoretical analysis examines the stability of the zero equilibrium point. It proves that the map generates typical nonlinear features, including chaos, which is confirmed numerically: phase attractors are plotted in a two-dimensional (2D) and three-dimensional (3D) space, bifurcation diagrams are drawn with variations in the derivative fractional values and in the system parameters, and we calculate the Maximum Lyapunov Exponents (MLEs) associated with the bifurcation diagram. Additionally, we use the C0 algorithm and entropy approach to measure the complexity of the chaotic symmetric fractional map. Finally, nonlinear 3D controllers are revealed to stabilize the symmetric fractional order map’s states in commensurate and incommensurate cases.

Noise-induced transitions past the onset of a steady symmetry-breaking bifurcation: The case of the sudden expansion

Noise-induced transitions past the onset of a steady symmetry-breaking bifurcation: The case of the sudden expansion

Flow-induced vibration of two tandem square cylinders at low Reynolds number: transitions among vortex-induced vibration, biased oscillation and galloping

The flow-induced vibrations (FIVs) of two identical tandem square cylinders with mass ratio m* = 3.5 at Reynolds number Re = 150 are investigated through two-dimensional direct numerical simulation (DNS) and linear stability analysis over a parameter range of spacing ratio 1.5 ≤ L* ≤ 5 and reduced velocity 3 ≤ Ur ≤ 34. Three kinds of FIV responses, namely vortex-induced vibration (VIV), biased oscillation (BO) and galloping (GA), are identified. The FIVs are then further classified into the branches of initial VIV (IV), resonant VIV (RV and RV′), flutter-induced VIV (FV), desynchronized VIV (DV), VIV developing from GA (GV), transitional state between VIV and GA (TR), BO and GA based on the characteristics of the vibration responses. The transitions among different FIV branches are examined by combining the DNS with linear stability analysis, where the transition boundaries among the VIV, BO and GA branches over the concerned parameters are identified on the branch maps. The transition from IV to RV or RV′ is found to be related to the unstable wake mode, while the FV, transiting from RV or RV′, is induced by the unstable structural factor in the wake-structure mode. The structural instability is considered as the physical origin of GA, whereas the mode competition between unstable wake and structure leads to DV, GV and TR, and thus delays the appearance of GA. The transition from DV to BO with biased equilibrium position, accompanied by the even-order harmonic frequencies, is essentially induced by the symmetry breaking bifurcation.

Rayleigh–Bénard Convection With Multiple Solutions in Trapezoidal Closed Cavities

Abstract Rayleigh–Bénard convection (RBC) in symmetric trapezoidal closed cavities with cavity angle ϕ=70°−110°, filled with air, is studied using numerical simulations where inclined side walls are adiabatic. In contrast to rectangular cavities, where no flow exists below a threshold value, there is a weak convection even at a low Rayleigh number (Ra) due to the fact that there is a component of thermal gradient in the horizontal direction in these cavities. Interestingly, these cavities show sudden and significant jumps in the convection, similar to square cavities (Rac = 2585.02 for ϕ=90°), as Ra increases beyond a critical value (Rac). It is noted here that these Rac represent symmetry-breaking pitchfork bifurcations. These bifurcations are seen in both acute (Rac = 8000 for ϕ=70°) and obtuse (Rac = 2300 for ϕ=110°) angle trapezoidal cavities. Moreover, it is observed that multiple steady-state solutions (MSSS) exist as Ra is further increased. A forward and backward continuation approach for numerical simulations is used to track the co-existence of MSSS. These steady-states have co-existing one-roll and two-roll convective patterns beyond another threshold value of Ra. Here, two types of critical Ra have been identified for different cavity angles; one shows the sudden jump in the convection, and the other is the one beyond which MSSS co-exist. Furthermore, a codimension two bifurcation analysis is carried out with Ra and ϕ as two parameters. The bifurcation analysis divides the parameter space into different regions based on the multiplicity of the solutions.

Symmetry-breaking longitude bifurcation for a free boundary problem modeling the growth of tumor cord in three dimensions

Symmetry-breaking longitude bifurcation for a free boundary problem modeling the growth of tumor cord in three dimensions

Symmetry-breaking combined latitude-longitude bifurcations for a free boundary problem modeling small plaques

Symmetry-breaking combined latitude-longitude bifurcations for a free boundary problem modeling small plaques

Motility and swimming: universal description and generic trajectories.

Autonomous locomotion is a ubiquitous phenomenon in biology and in physics of active systems at microscopic scale. This includes prokaryotic, eukaryotic cells (crawling and swimming) and artificial swimmers. An outstanding feature is the ability of these entities to follow complex trajectories, ranging from straight, curved (circular, helical...), to random-like ones. The non-straight nature of these trajectories is often explained as a consequence of the asymmetry of the particle or the medium in which it moves, or due to the presence of bounding walls, etc... Here, we show that for a particle driven by a concentration field of an active species, straight, circular and helical trajectories emerge naturally in the absence of asymmetry of the particle or that of suspending medium. Our proof is based on general considerations, without referring to an explicit form of a model. We show that these three trajectories correspond to self-congruent solutions. Self-congruency means that the states of the system at different moments of time can be made identical by an appropriate combination of rotation and translation of the coordinate space. We show that these solutions are exhibited by spherically symmetric particles as a result of a series of pitchfork bifurcations, leading to spontaneous symmetry breaking in the concentration field driving the particle motility. Self-congruent dynamics in one and two dimensions are analyzed as well. Finally, we present a simple explicit nonlinear exactly solvable model of fully isotropic phoretic particle that shows the transitions from a non-motile state to straight motion to circular motion to helical motion as a series of spontaneous symmetry-breaking bifurcations. Whether a system exhibits or not a given trajectory only depends on the numerical values of parameters entering the model, while asymmetry of swimmer shape, or anisotropy of the suspending medium, or influence of bounding walls are not necessary.

Symmetry-breaking response of a flexible splitter plate attached to a circular cylinder in uniform flow

Static and dynamic fluid–structure interaction of a flexible plate behind a stationary cylinder in uniform flow is explored for a body-to-fluid mass ratio of 10. Steady flow-structure computations for Re = 150, based on the diameter of cylinder and free-stream speed, reveal four regimes with respect to increasing flexibility of the plate. The plate does not undergo any lateral deflection in regime 1 beyond which it undergoes a symmetry-breaking bifurcation causing it to spontaneously deflect laterally. The curvature is of the same sign along the entire length of plate in regime 2 while it changes sign along the plate in regime 3. The lateral deflection, however, is still maximum at the plate tip. The location of maximum deflection moves away from the plate tip in regime 4. The evolution of flow structures including the strength of various standing vortices, with flexibility, is studied. The role of reattachment of flow on the surface of the plate and the modification of the pressure distribution is explored. The critical Re, beyond which the splitter plate spontaneously achieves a deflected shape, decreases with increase in flexibility. It is estimated to be Re = 122.33 for the limiting case of an infinitely flexible plate. Computations for dynamic fluid–structure interaction reveal several regimes of lock-in with different natural vibration modes of the plate and related hysteresis. No lateral bias in the time-averaged deflection is found during lock-in; it occurs in the desynchronization regime that precedes the lock-in regime with second mode. For the mass ratio considered, the bias in the static and dynamic simulations start at the same flexibility.

Wall mode dynamics and transition to chaos in magnetoconvection with a vertical magnetic field

Quasistatic magnetoconvection of a fluid with low Prandtl number ( ${\textit {Pr}}=0.025$ ) with a vertical magnetic field is considered in a unit-aspect-ratio box with no-slip boundaries. At high relative magnetic field strengths, given by the Hartmann number ${\textit {Ha}}$ , the onset of convection is known to result from a sidewall instability giving rise to the wall-mode regime. Here, we carry out three-dimensional direct numerical simulations of unprecedented length to map out the parameter space at ${\textit {Ha}} = 200, 500, 1000$ , varying the Rayleigh number ( ${\textit {Ra}}$ ) over the range $6\times 10^5 \lesssim {\textit {Ra}} \lesssim 5\times 10^8$ . We track the development of stable equilibria produced by this primary instability, identifying bifurcations leading to limit cycles and eventually to chaotic dynamics. At ${\textit {Ha}}=200$ , the steady wall-mode solution undergoes a symmetry-breaking bifurcation producing a state that features a coexistence between wall modes and a large-scale roll in the centre of the domain, which persists to higher ${\textit {Ra}}$ . However, under a stronger magnetic field at ${\textit {Ha}}=1000$ , the steady wall-mode solution undergoes a Hopf bifurcation producing a limit cycle which further develops to solutions that shadow an orbit homoclinic to a saddle point. Upon a further increase in ${\textit {Ra}}$ , the system undergoes a subsequent symmetry break producing a coexistence between wall modes and a large-scale roll, although the large-scale roll exists only for a small range of ${\textit {Ra}}$ , and chaotic dynamics primarily arise from a mixture of chaotic wall-mode dynamics and arrays of cellular structures.

Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition

Dynamical systems containing heteroclinic cycles and networks can be invoked as models of intransitive competition between three or more species. When populations are assumed to be well-mixed, a system of ordinary differential equations (ODEs) describes the interaction model. Spatially extending these equations with diffusion terms creates a system of partial differential equations which captures both the spatial distribution and mobility of species. In one spatial dimension, travelling wave solutions can be observed, which correspond to periodic orbits in ODEs that describe the system in a steady-state travelling frame of reference. These new ODEs also contain a heteroclinic structure. For three species in cyclic competition, the topology of the heteroclinic cycle in the well-mixed model is preserved in the steady-state travelling frame of reference. We demonstrate that with four species, the heteroclinic cycle which exists in the well-mixed system becomes a heteroclinic network in the travelling frame of reference, with additional heteroclinic orbits connecting equilibria not connected in the original cycle. We find new types of travelling waves which are created in symmetry-breaking bifurcations and destroyed in an orbit flip bifurcation with a cycle between only two species. These new cycles explain the existence of ‘defensive alliances’ observed in previous numerical experiments. We further describe the structure of the heteroclinic network for any number of species, and we conjecture how these results may generalise to systems of any arbitrary number of species in cyclic competition.

Symmetry breaking bifurcation of membranes with boundary

We use a bifurcation theory due to Crandall and Rabinowitz to show the existence of a symmetry breaking bifurcation of a specific one parameter family of axially symmetric disc type solutions of a membrane equation with fixed boundary. In place of working directly with the fourth order membrane equation, it is replaced by a second order reduction found in Palmer and Pámpano (2022).

Symmetry-breaking bifurcations and excitations of solitons in linearly coupled NLS equations with PT-symmetric potentials

We address symmetry-breaking bifurcations (SBBs) in the ground state (GS) and dipole-mode (DM) solitons of the one-dimensional linearly coupled NLS equations, modelling the propagation of light in a dual-core planar waveguide with the Kerr nonlinearity and two types of P T -symmetric potentials. The P T -symmetric potential is employed to obtain different types of solutions. A supercritical pitchfork bifurcation occurs in families of symmetric solutions of both the GS and DM types. A novel feature of the system is interplay between breakings of the P T and inter-core symmetries. Stability of symmetric GS and DM modes and their asymmetric counterparts, produced by SBBs of both types, is explored via the linear-stability analysis and simulations. It is found that the instability of P T -symmetric solutions takes place prior to the inter-core symmetry breaking. Surprisingly, stable inter-core-symmetric GS solutions may remain stable while the P T symmetry is broken. Fully asymmetric GS and DM solitons are only partially stable. Moreover, we construct symmetric and asymmetric GS solitons under the action of a pure imaginary localized potential, for which the SBB is subcritical. These results exhibit that stable solitons can still be found in dissipative systems. Finally, excitations of symmetric and asymmetric GS solitons are investigated by making the potential’s parameters or the system’s coupling constant functions, showing that GS solitons can be converted from an asymmetric shape onto a symmetric one under certain conditions. These results may pave the way for the study of linear and nonlinear phenomena in a dual-core planar waveguide with P T potential and related experimental designs.

The Singularity of Expanded Jacobian Matrix in Incremental Harmonic Balance Method Directly Locates Bifurcation Points of Steady-State Responses

Abstract As a semi-analytical approach, the incremental harmonic balance (IHB) method is widely implemented for solving steady-state (including both periodic and quasi-periodic) responses through an iteration process. The iteration is carried out through a Jacobian matrix (JM) and a residual vector, both updated in each iteration. Though the JM is known to be singular at certain bifurcation points, the singularity is still an open question and could play a pivotal role in real applications. In this study, we define and calculate an expanded JM (EJM) by applying an expanded solution expression in the IHB iteration. The singularity of the EJM at several different bifurcation points is proved in a general manner, according to the bifurcation theory for equilibria in nonlinear dynamical systems. Given the possible bifurcation type, furthermore, the singularity is applied to locate the corresponding bifurcation point directly and precisely. Considered are the cases of the period-doubling, symmetry breaking, and Neimark-Sacker bifurcations of periodic and/or quasi-periodic responses.

Forced Symmetry-Breaking Bifurcations in Networks with Disordered Parameters

Emergent behavior in complex networks can be predicted and analyzed via the mechanism of spontaneous symmetry-breaking bifurcation, in which solutions of related bifurcation problems lose symmetry as some parameters are varied, even though the equations that such solutions satisfy retain the full symmetry of the system. A less common mechanism is that of forced symmetry-breaking, in which either a bifurcation problem has symmetry on both the state variables and the parameters, or one where the equations have less symmetry when a certain parameter is varied. In this manuscript, it is shown that in certain networks with parameter mismatches the governing equations remain unchanged when the group of symmetries acts on both the state variables and the parameter space. Based on this observation we study the existence and stability of collective patterns in symmetric networks with parameters mismatches from the point of view of forced symmetry-breaking bifurcations. Treating the parameters as state variables, we perform center manifold reductions, which allow us to understand how the disorder in parameters affects the bifurcation points as well as the stability properties of the ensuing patterns. Theoretical results are validated with numerical simulations.

Coupled responses of the flow-induced vibration and flow-induced rotation of a rigid cylinder-plate body

In this study, coupled responses of flow-induced vibration and rotation for an elastically mounted cylinder-plate body are numerically investigated at a low Reynolds number of 120. A wide vibrational reduced velocity range of Uy = 3–18 under four rotational reduced velocities Uθ = 5, 8, 12, and 18 are considered. The non-bifurcation responses, bifurcation only in rotation responses, and bifurcation in both vibration and rotation responses are identified. Typical vortex-induced vibration (VIV) responses are recognized when considering the passive rotations, different from the full interactions between VIV and galloping for the vibration-only case. As Uθ increases, the peak vibration amplitudes increase, the onset Uy of the lock-in region becomes larger, and the lock-in region is wider. The phase angles of displacements versus lift coefficients experience a jump from 0° to 180° in the lock-in region, and the larger the Uθ, the wider the Uy range of phase jump. Whether the instantaneous posture of the cylinder-plate body is streamlined or not is determined by oscillation amplitudes and phase differences between displacements versus rotation angles. Streamlined profiles can be achieved under small oscillation amplitudes or when the phase angles are nearly 90°. The 2S (two isolated vortices) vortex shedding mode dominates the initial and desynchronization branch, while the 2P (two pairs of vortices), 2S* (two isolated vortices with tendency to split), and 2T (two triplets of vortices) modes appear in the lock-in region. After the symmetry-breaking bifurcation, the reattachment behavior becomes simpler and the length of the recirculation region is significantly increased, as compared with those in non-bifurcation region. With the above study, a new method of improving energy harvesting from flow-induced vibration, by incorporating passive rotations simultaneously, is first introduced. It is found that passive rotations can enhance the vibration responses and thus lead to the increased output power and energy transfer ratio, although they make less contributions to the total power. Generally, this mechanical system presents a promising opportunity for energy harvesting through flow-induced vibration.

Pair of optical solitons in parallel planar waveguides coupled by linear photonic crystal

We propose a simple optical model, which consists of two Kerr nonlinear parallel planar waveguides sandwiching a one-dimensional photonic crystal for demonstrating the phenomenon of spontaneous symmetry breaking in optics. We study the shape and stability of soliton pairs propagating in two planar waveguides and the spontaneous symmetry breaking of the power distribution of solitons in the two waveguides. The results of numerical simulation show that there is supercritical type of spontaneous symmetry breaking in this system. More optical power P and smaller refractive index of photonic crystals [Formula: see text] make soliton solutions more prone to spontaneous symmetry breaking bifurcation. Asymmetric soliton solutions and symmetric soliton solutions correspond to different dispersion relations. Under the given total power P, for the symmetric soliton solutions, the propagation constant k increases linearly with [Formula: see text], and for asymmetric soliton solutions, the propagation constant k decreases nonlinearly with [Formula: see text]. The minimum point of [Formula: see text] is just the demarkation between the symmetric soliton solutions and the asymmetric soliton solutions. If [Formula: see text] is fixed, the minimum point of the Hamiltonian of the system, [Formula: see text], is also the critical point of the two different kinds of soliton solutions. We also study stable and unstable distributions of two kinds of soliton solutions.

Random number generation using spontaneous symmetry breaking in a Kerr resonator.

We demonstrate an all-optical random number generator based on spontaneous symmetry breaking in a coherently driven Kerr resonator. Random bit sequences are generated by repeatedly tuning a control parameter across a symmetry-breaking bifurcation that enacts random selection between two possible steady-states of the system. Experiments are performed in a fiber ring resonator, where the two symmetry-broken steady-states are associated with orthogonal polarization modes. Detrimental biases due to system asymmetries are suppressed by leveraging a recently discovered self-symmetrization phenomenon that ensures the symmetry-breaking dynamics act as an unbiased coin toss, with a genuinely random selection between the two available steady-states. We optically generate bits at a rate of 3 MHz without post-processing and verify their randomness using the National Institute of Standards and Technology and Dieharder statistical test suites.

Symmetry-breaking transitions in quiescent and moving solitons in fractional couplers.

We consider phase transitions, in the form of spontaneous symmetry breaking (SSB) bifurcations of solitons, in dual-core couplers with fractional diffraction and cubic self-focusing acting in each core, characterized by Lévy index α. The system represents linearly coupled optical waveguides with the fractional paraxial diffraction or group-velocity dispersion (the latter system was used in a recent experiment [Nat. Commun. 14, 222 (2023)10.1038/s41467-023-35892-8], which demonstrated the first observation of the wave propagation in an effectively fractional setup). By dint of numerical computations and variational approximation, we identify the SSB in the fractional coupler as the bifurcation of the subcritical type (i.e., the symmetry-breaking phase transition of the first kind), whose subcriticality becomes stronger with the increase of fractionality 2-α, in comparison with very weak subcriticality in the case of the nonfractional diffraction, α=2. In the Cauchy limit of α→1, it carries over into the extreme subcritical bifurcation, manifesting backward-going branches of asymmetric solitons which never turn forward. The analysis of the SSB bifurcation is extended for moving (tilted) solitons, which is a nontrivial problem because the fractional diffraction does not admit Galilean invariance. Collisions between moving solitons are studied too, featuring a two-soliton symmetry-breaking effect and merger of the solitons.

Spontaneous symmetry breaking, stability and adiabatic changes of 2D quantum droplets in amended Gross–Pitaevskii equation with multi-well potential

In this paper, we report the spontaneous symmetry breaking (SSB) of two-dimensional (2D) quantum droplets (QDs) for the amended Gross–Pitaevskii equation with Lee–Huang–Yang corrections and a Gaussian quadruple-well potential. The complete pitchfork symmetry breaking bifurcation diagrams are presented for the possible stationary states with four modes, which involve twelve different real solution branches and one complex solution branch, where the continuous asymmetric QDs branches bifurcate from the fundamental symmetric or antisymmetric one as the strength of nonlinearity exceeds a threshold value. Meanwhile the stability of the 2D QDs is explored by means of linear stability analysis and direct numerical simulations. Especially, for the asymmetric QDs, the SSB effect and the trend for getting the droplet asymmetry are discussed by changing the chemical potential. Finally, the adiabatic changes of stable stationary QDs are explored by modulating the position, depth, and width of the potential well as functions of time, and the strong robustness of QDs is found. Our results provide some theoretical basis for the SSB phenomena and related physical experiments on 2D asymmetric QDs.

Two-dimensional fractional [formula omitted]-symmetric cubic–quintic NLS equation: Double-loop symmetry breaking bifurcations, ghost states and dynamics

In this paper, we address the double-loop symmetry breaking bifurcations of optical vortex solitons in the two-dimensional (2D) fractional cubic–quintic nonlinear Schrödinger (FCQNLS) equation with an imprinted partially parity-time (PPT) symmetric complex potential. The continuous branches of asymmetric solitons bifurcate from the fundamental one as the power exceeds some critical value. It is found that the symmetry breaking point is exactly the point at which the fundamental symmetric soliton is destabilized. Intriguingly, the branches of asymmetry solitons (alias ghost states) are existing with complex conjugate propagation constants, which is solely in fractional media. Moreover, the branch of asymmetric solitons crosses the branch of fundamental symmetric ones when it reaches the second critical value, and eventually merges back into it at the third critical power point. The imaginary part of the propagation constant also exhibits a circular shape. And the stability of the fundamental symmetric soliton is restored. In addition, the stability of two branches of asymmetric solitons differs from each other. The reason generating this phenomenon is analyzed in detail. Besides, the branch of dipole solitons for the first excited state is also discussed numerically, which is always stable in the process of symmetry breaking. The stability and dynamics of symmetric (fundamental and the first excited ones) and asymmetric solitons are explored via the linear stability analysis, direct simulation, modulation as well as adiabatic excitation. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related experiments in fractional media with PPT-symmetric potentials.