Qualitative Theory of Dynamical Systems

Papers
(The TQCC of Qualitative Theory of Dynamical Systems is 4. The table below lists those papers that are above that threshold based on CrossRef citation counts [max. 250 papers]. The publications cover those that have been published in the past four years, i.e., from 2021-09-01 to 2025-09-01.)
ArticleCitations
Approximate Controllability of a Coupled Nonlocal Partial Functional Integro-differential Equations with Impulsive Effects102
Some New Results on Itô–Doob Hadamard Fractional Stochastic Pantograph Equations in $$L^p$$ Spaces87
Abelian Integrals and Non-generic Turning Points84
Existence of Homoclinic Solutions for the Discrete p(k)-Laplacian Operator71
Dynamics of One-Dimensional Motion of a Gas Under the Influence of Monochromatic Radiation57
Exact Solutions and Dynamical Behaviors of the Raman Soliton Model with Anti-Cubic Nonlinearity52
Nonlinear Multi-term Impulsive Fractional q-Difference Equations with Closed Boundary Conditions38
Cosmic-Plasma Environment, Singular Manifold and Symbolic Computation for a Variable-Coefficient (2+1)-Dimensional Zakharov-Kuznetsov-Burgers Equation37
Ground State and Sign-Changing Solutions for Critical Schrödinger–Poisson System with Lower Order Perturbation30
Bifurcation of Limit Cycles and Isochronous Centers on Center Manifolds for a Class of Cubic Kolmogorov Systems in $${\mathbb {R}}^{3}$$27
Lie Subalgebras, Solutions and Conserved Vectors of a Nonlinear Geophysical Korteweg de Vries Equation in Ocean Physics and Nonlinear Mechanics with Power Law27
Globally Exponential Stability of Piecewise Pseudo Almost Periodic Solutions for Neutral Differential Equations with Impulses and Delays27
Coexistence of Analytic and Piecewise Analytic Limit Cycles in Planar Piecewise Quadratic Differential Systems26
A New Approach on the Approximate Controllability Results for Hilfer Fractional Stochastic Hemivariational Inequalities of Order $$1<\mu <2$$25
Modulation Analysis for Stochastic FitzHugh–Nagumo Type Equation with Time Delay25
A Note on Asymptotic Stability of Semilinear Thermoelastic System24
On the Sharp Existence of Normalized Ground State Solutions for Non-Autonomous Kirchhoff Equations24
Dynamical Behavior and Exact Explicit Traveling Wave Solutions of a Generalized Vakhnenko-Parkes Equation24
q-Pressure for Dynamical Systems22
A Study on the Approximate Controllability of Damped Elastic Systems Using Sequence Method22
Existence of Traveling Wave Solutions for Gardner Equation and Its Spatiotemporal Delay Form22
On Separable Cubic Stochastic Operators21
KAM Tori for a Two Dimensional Beam Equation with a Quintic Nonlinear Term and Quasi-periodic Forcing21
The Existence of Evolution Systems of Measures of Non-autonomous Stochastic Differential Equations with Infinite Delays20
Codimension-Two Bifurcation in a Class of a Discrete-Time Predator–Prey Interaction with Cannibalism20
Boundedness and Partial Convergence of Solution of a Class of Stochastic Differential Equations20
Response Solutions of Degenerate Quasi-Periodic Systems Under Small Perturbations20
Pontryagin Maximum Principle for Fractional Delay Differential Equations and Controlled Weakly Singular Volterra Delay Integral Equations19
Existence of Periodic Solutions for a Class of Dynamic Equations with Multiple Time Varying Delays on Time Scales19
Stability Analysis of Second Order Impulsive Differential Equations19
Periodic Wave Solution of the Generalized Burgers–Fisher Equation via Abelian Integral18
Solvability and Hyers-Ulam Stability of Quaternion Difference Equations With Two-sided Coefficients18
Lie Symmetries and Exact Solutions of KdV–Burgers Equation with Dissipation in Dusty Plasma18
Hopf Bifurcation Analysis of a Predator–Prey Model with Prey Refuge and Fear Effect Under Non-diffusion and Diffusion17
Complex Dynamics for an Immunosuppressive Infection Model with Virus Stimulation Delay and Nonlinear Immune Expansion17
Generalized Trial Equation Scheme and Enhance Modified Extended Tanh Function Method with Nonparaxial Pulse Propagation to the Cubic–Quintic Nonlinear Helmholtz Equation17
Symbolically Computing the Shallow Water via a (2+1)-Dimensional Generalized Modified Dispersive Water-Wave System: Similarity Reductions, Scaling and Hetero-Bäcklund Transformations17
Topological Entropy of a Factor Map Under Amenable Group Actions17
Relative Controllability and Hyers–Ulam Stability of Riemann–Liouville Fractional Delay Differential System17
Hyperbolicity, Shadowing, and Bounded Orbits17
New Results for Some Damped Dirichlet Problems with Impulses17
Degenerate Transcritical Bifurcation Point can be an Attractor: A Case Study in a Slow–Fast Modified Leslie–Gower Model16
Limit Cycles in Discontinuous Planar Piecewise Differential Systems with Multiple Nonlinear Switching Curves16
pth Moment Exponential Stability of Impulsive Stochastic Functional Differential Equations16
New Variational Principles of Topological Pressures15
Hyers-Ulam Stability to Linear Nonhomogeneous Quaternion-Valued Matrix Difference Equations via Complex Representation15
Normalized Solutions for Schrödinger Equations with Local Superlinear Nonlinearities15
Disease Dynamics of a Probabilistic Model with Vaccination Under Regime Switching15
Continuous Dependence and Differentiability of Solutions of Second-Order Impulsive Differential Equations on Initial Values and Impulsive Points15
Switched Server Systems Whose Parameters are Normal Numbers in Base 415
Qualitative Investigation of Nonlinear Fractional Coupled Pantograph Impulsive Differential Equations14
On a Certain Class of IFSs and Their Attractors14
Geometric Scattering Monodromy14
Necessary Optimality Conditions for Singular Controls of Caputo Fractional Systems with Delay in Control14
Dynamics of N-Species Cooperation Models with Feedback Controls and Continuous Delays13
Rich Dynamics of Discrete Time-Delayed Moran-Ricker Model13
Expressions and Evolution of Traveling wave Solutions in a Generalized Two-Component Rotation b-Family System13
Estimates for the Number of Limit Cycles in Discontinuous Generalized Liénard Equations13
An Optimal Control Strategy for Antiretroviral Treatment of HIV Infection in Presence of Immunotherapy13
Product Type Potential on the One-Dimensional Lattice Systems: Selection of Maximizing Probability and a Large Deviation Principle13
Existence of Ground State Solutions for Generalized Quasilinear Schrödinger Equations with Asymptotically Periodic Potential13
Some Notes on Topological Entropy of Subsets on Compact Uniform Spaces13
Local Integrability and Linearizability for Three Dimensional Lotka–Volterra Cubic Systems13
Stochastic Dynamics of a Two-Species Patch-System With Ratio-Dependent Functional Response13
Existence of Solutions for p(x)-Laplacian Elliptic BVPs on a Variable Sobolev Space Via Fixed Point Theorems13
Coexistence of Limit Cycles and Homoclinic Loops in a Leslie-Gower Model with Allee Effect13
On Unbounded Motions in a Real Analytic Bouncing ball Problem12
Existence and Hyers–Ulam Stability of Jerk-Type Caputo and Hadamard Mixed Fractional Differential Equations12
Existence Results and Ulam–Hyers Stability for a Fully Coupled System of Nonlinear Sequential Hilfer Fractional Differential Equations and Integro-Multistrip-Multipoint Boundary Conditions12
Dynamics of a Stochastic Brucellosis Model with Vaccination and Environmental Pollution Transmission12
Stochastic Bifurcation and Periodic Solution in Stochastic Single-Species Models Incorporating Allee Effect and Aposematism12
Spatiotemporal Dynamics in a Diffusive Predator–Prey System with Beddington–DeAngelis Functional Response12
Infinitesimal Prolongation for Fractional Derivative $$\varPsi $$-Caputo Variable Order and Applications11
Dynamics of Kirchhoff Wave Equations Incorporating Energy Damping Effects11
Four Limit Cycles of Discontinuous Piecewise Differential Systems with Nilpotent Saddles Separated by a Straight Line11
Strong Pairs of Periodic Segments11
Quasi-periodic Solutions for a Generalized Higher-Order Boussinesq Equation11
Sliding Cycles of Regularized Piecewise Linear Visible–Invisible Twofolds11
Expansive Measures of Nonautonomous Iterated Function Systems11
Canonical Sequences of Optimal Quantization for Condensation Measures11
Existence Results for the Solution of Abstract Neutral Impulsive Differential Problems with State-Dependent Delay11
Heat Kernel Method for Quintic and Sextic Equations in Distributions and Hyperfunctions11
Hyers–Ulam Stability for a Coupled System of Fractional Differential Equation With p-Laplacian Operator Having Integral Boundary Conditions10
Dynamic Analysis of a Nonlocal and Delayed Reaction-Diffusion Zika Virus Model with Spatial Heterogeneity10
Qualitative Analysis and Analytical Wave Solutions for a Particular Scale-Invariant Analog of the Korteweg-de Vries Equation10
Response Solutions for Nonlinear Oscillator Equation in High Order Degeneracy10
Symmetry and Integrability of $$\mathcal{P}\mathcal{T}$$-Symmetric Semi-discrete Short pulse equation: A study of Rogue, Breather, and Soliton Solutions10
Strong Resonance Bifurcations and State Feedback Control in a Discrete Prey-Predator Model with Harvesting Effect10
Approximate Controllability for Sobolev-Type Fuzzy Hilfer Fractional Neutral Integro-Differential Inclusion with Clarke Subdifferential Type10
Sufficient Conditions to have Global Centers via Branching Theory10
Bifurcation and Stability Analysis of a Discrete Predator–Prey Model with Alternative Prey10
Turing Patterns Induced by Cross-Diffusion in a Predator–Prey System with Functional Response of Holling-II Type9
Relaxation Oscillation and Canard Explosion for a SIRS Model with Nonlinear Incidence Rate9
Natural Convection Non-Newtonian EMHD Dissipative Flow Through a Microchannel Containing a Non-Darcy Porous Medium: Homotopy Perturbation Method Study9
Minkowski Dimension and Slow–Fast Polynomial Liénard Equations Near Infinity9
Boundedness of Traveling Waves in a Discrete Diffusion Model with Delay9
Multiple Families of Bounded Solutions Near Perturbed Homoclinic Orbits, Application to a Nonlinear Wave Equation9
Novel Insight into a Single-Species Metapopulation Model with Time Delays9
On the Solutions of Ordinary Differential Equations in the Form of Dulac Series9
Dynamical Analysis of a Discrete Amensalism System with the Beddington–DeAngelis Functional Response and Allee Effect for the Unaffected Species9
Several Continuities of a Pullback Random Attractor for Stochastic Non-Autonomous Zakharov Lattice Equations8
Multiplicity of Solutions for Non-homogeneous Dirichlet Problem with One-Dimension Minkowski-Curvature Operator8
Bifurcations, Permanence and Local Behavior of the Plant-Herbivore Model with Logistic Growth of Plant Biomass8
Existence and Approximate Controllability Results for the Second-Order Abstract Neutral Differential System with Damping8
Global Hopf Bifurcation of a Diffusive Modified Leslie–Gower Predator–Prey Model with Delay and Michaelis–Menten Type Prey Harvesting8
Trajectory Controllability of Impulsive Neutral Stochastic Functional Integrodifferential Equations Driven by fBm with Noncompact Semigroup via Mönch Fixed Point8
An Investigation on the Approximate Controllability of Non-instantaneous Impulsive Hilfer Sobolev-Type Fractional Stochastic System Driven by the Rosenblatt Process and Poisson Jumps8
Some Existence Results of Coupled Hilfer Fractional Differential System and Differential Inclusion on the Circular Graph8
Homological Data on the Periodic Structure of Self-Maps on Wedge Sums8
Bifurcations and Exact Peakons, Periodic Peakons, Solitary Wave, Kink Wave Solutions for the Quartic CH-Type Equation8
Blow-Up Behaviors of Minimizers for the $$\dot{H}^{\gamma _c}$$-Critical Fourth-Order Nonlinear Schrödinger Equation with the Mixed Dispersions8
A Note on the Stationary Probability Density Function and Covariance Matrix of a Stochastic Chemostat Model with Distributed Delay8
A Study on Fractional Differential Equations with a Generalized Exponential Kernel Derivative7
On a Rumor Propagation Model with Spatial Heterogeneity7
Existence of Positive Solutions for (N, p)-Laplace Equation with Exponential Nonlinearity Term and Convection Term in Dimension $${\mathbb {N}}$$7
On the Number of Positive Solutions for a Higher Order Elliptic System7
Innovative Insights into Wave Phenomena: Computational Exploration of Nonlinear Complex Fractional Generalized-Zakharov System7
Existence Results for Nonlinear Hilfer Pantograph Fractional Integrodifferential Equations7
Baby Mandelbrot Sets and Spines in Some One-Dimensional Subspaces of the Parameter Space for Generalized McMullen Maps7
Orbit Growth of Sofic Shifts and Periodic-Finite-Type Shifts7
Infinitely Many Nodal Solutions for Kirchhoff-Type Equations with Non-odd Nonlinearity7
Higher-Order Nonlinear Dynamical Systems and Invariant Lagrangians on a Lie Group: The Case of Nonlocal Hunter–Saxton Type Peakons7
A Polynomial System of Degree Four with an Invariant Square Containing At Least Five Limit Cycles7
Elementary Integrability of Polynomial Rayleigh–Duffing Systems7
Limiting Dynamics of Stochastic p(x)-Laplace Equation with Nonstandard Growth Driven by Pure Jump Noise7
Global Phase Portrait and Local Integrability of Holomorphic Systems7
Rotation Domains and Stable Baker Omitted Value7
On the Decay of Solutions of the Linearized Equations of Conservation Equations with Viscosity and Relaxation7
Traveling Wave Solution with the Critical Speed for a Diffusive Epidemic System with Spatio-Temporal Delay7
Riemann–Hilbert Approach and Multiple Arbitrary-Order Pole Solutions for the Lakshmanan–Porsezian–Daniel Equation with Finite Density Initial Data6
Existence of Gibbs States and Maximizing Measures on a General One-Dimensional Lattice System with Markovian Structure6
Quasilinear Coupled System in the Frame of Nonsingular ABC-Derivatives with p-Laplacian Operator at Resonance6
Fractional Evolution Equations with Nonlocal Initial Conditions and Superlinear Growth Nonlinear Terms6
Auto-Bäcklund Transformation and Exact Solutions for a New Integrable (3+1)-dimensional KdV-CBS Equation6
Non–Equivalence Between the Melnikov and the Averaging Methods for Nonsmooth Differential Systems6
Strong Chain Transitivity via Uniformity6
Two Weighted Norm Dynamic Inequalities with Applications on Second Order Half-Linear Dynamic Equations6
Specification and Shadowing Properties of Free Semigroup Actions6
Historic Behavior in Rock–Paper–Scissor Dynamics6
Structurally Unstable Quadratic Vector Fields of Codimension Two: Families Possessing One Finite Saddle-Node and a Separatrix Connection6
Bifurcations and Exact Traveling Wave Solutions for the Generalized Alexeyev’s $$A\pm $$ Equation6
Random Uniform Attractors for First Order Stochastic Non-Autonomous Lattice Systems6
Bifurcation of Piecewise Smooth Manifolds from 3D Center-Type Vector Fields6
Representation and Stability of Solutions for Impulsive Discrete Delay Systems with Linear Parts Defined by Non-Permutable Matrices6
Global Asymptotic Stability in Cubic Systems with a Nilpotent Point at the Origin and with An Invariant Straight Line6
N-Convergence and Chaotic Properties of Non-autonomous Discrete Systems6
On Multi Order Nonlinear Langevin Type of $$\mathbb {FDE}$$ Subject to Multi-Point Boundary Conditions6
On Double Homoclinic Bifurcation of Limit Cycles in Near-Hamiltonian Systems on the Cylinder6
Dynamics of Different Soliton Solutions, Phase Portraits, Bifurcation Analysis, and Chaotic Behavior in the Benney-Luke Equation Represent Surface Waves in a Shallow Channel with Surface Tension Effec6
Exponential Stabilization of a Flexible Structure: A Delayed Boundary Force Control Versus a Delayed Boundary Moment Control6
Bifurcations of a Generalized Heteroclinic Loop in a Planar Piecewise Smooth System with Periodic Perturbations6
Asymptotic Behavior of Solutions to Difference Equations of Neutral Type6
Well-Posedness of a Class of Fractional Langevin Equations6
N-Soliton and Other Analytic Solutions for a ($$3 + 1$$)-Dimensional Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff Equation with the Time-Dependent Coefficients for the Shallow Water Waves6
Multiple Positive Periodic Solutions to Minkowski-Curvature Equations with a Singularity of Attractive Type6
On the Variational Principle for BS Dimension in Amenable Group Actions6
Quasi-Stationary Distribution of a Prey–Predator Model Driven by Demographic Stochasticity6
A Note on the Existence and Controllability Results for Fractional Integrodifferential Inclusions of Order $$r \in (1,2]$$ with Impulses6
Conley Index and Schwartz Index on Singular Surfaces6
Morse Predecomposition of an Invariant Set6
Isochronicity Conditions and Lagrangian Formulations of the Hirota Type Oscillator Equations6
Sufficient Conditions and Bounds Estimate of Blow-Up Time for a Fourth Order Parabolic Equation5
Approximate Controllability for Semilinear Fractional Stochastic Evolution Equations5
Stochastic Averaging of Non-linear Implicit Caputo’s Fractional Differential Equation with Multiplicative Fractional Brownian Motion5
Bilinear Bäcklund Transformations, as well as N-Soliton, Breather, Fission/Fusion and Hybrid Solutions for a (3+1)-Dimensional Integrable Wave Equation in a Fluid5
Existence and Regularity of Solutions to Damping Elastic System with State-Dependent Nonlocal Condition5
Transmission Dynamics of a High Dimensional Rabies Epidemic Model in a Markovian Random Environment5
Filling Times for Linear Flow on the Torus with Truncated Diophantine Conditions: A Brief Review and New Proof5
Second-Order Noncanonical Delay Differential Equations with Sublinear and Superlinear Terms: New Oscillation Criteria via Canonical Transform and Arithmetic–Geometric Inequality5
Generalized Fractional Differential Systems with Stieltjes Boundary Conditions5
Exactly Solvable Quadratic Differential Equation Systems Through Generalized Inversion5
An Improved Reciprocally Convex Inequality and its Application to Time-Varying Delay Systems5
Dynamic Analysis of a Toxin-Producing Phytoplankton Zooplankton Model with Allee Effect5
Hilbert Number for a Family of Piecewise Nonautonomous Equations5
A van der Pol-Duffing Oscillator with Indefinite Degree5
Phase Portraits of the Family IV of the Quadratic Polynomial Differential Systems5
Interactions among the Rogue Waves and Solitons for a (2+1)-dimensional Maccari System in Fluid Mechanics and Nonlinear Optics5
Qualitative Analysis for an HIV Infection Model with Caspase-1-Mediated Pyroptosis of the Predominance: Threshold Dynamics and Traveling Waves5
Integrable (3 + 1)-Dimensional Generalization for the Dispersionless Davey–Stewartson System5
Oscillatory Behavior of Semi-canonical Nonlinear Neutral Differential Equations of Third-Order Via Comparison Principles5
Representation of a Solution for a Neutral Type Differential Equation with Pure Delay on Fractal Sets5
The Controllability for Second-Order Semilinear Impulsive Systems5
Exploring Discrete-Time Epidemic Behavior Based on Saturated Incidence Rate and Multiple Transmission Pathways5
On the Bielecki–Hyers–Ulam Stability of Non–linear Impulsive Fractional Hammerstein and Mixed Integro–dynamic Systems on Time Scales5
Nonlinear Dynamical Analysis and Optimal Control Strategies for a New Rumor Spreading Model with Comprehensive Interventions5
Persistence of Invariant Tori in Infinite-Dimensional Hamiltonian Systems5
A Class of New Solvable Nonlinear Isochronous Systems and Their Classical Dynamics4
Dimension of a Class of Intrinsically Transversal Solenoidal Attractors in High Dimensions4
Dynamics of Predator–Prey Model Based on Fear Effect with Bifurcation Analysis and Chaos Control4
Global Dynamics of 3D Type-K Competitive Lotka-Volterra System with the Identical Intrinsic Growth Rate4
Bifurcation Analysis and Soliton Solutions to the Kuralay Equation Via Dynamic System Analysis Method and Complete Discrimination System Method4
Dynamics of Travelling Waves to KdV–Burgers–Kuramoto Equation with Marangoni Effect Perturbation4
Titchmarsh–Weyl Theory for Impulsive Dynamic Dirac System4
Finite Time Attractivity and Exponentially Stable of a Multi-Stage Epidemic System with Discontinuous Incidence4
Analysis on the Controllability of Hilfer Fractional Neutral Differential Equations with Almost Sectorial Operators and Infinite Delay via Measure of Noncompactness4
Exploring the Impact of Distributed Delay and Environmental Fluctuation on Food Chain Stability in a Chemostat Model4
Normalized Solutions to Biharmonic Equation with Bounded Potential and Mixed Power Nonlinearities4
On the Stability of Hybrid Polycycles4
Existence and Multiplicity of Blow-Up Profiles for a Quasilinear Diffusion Equation with Source4
Solitons, Breathers and Modulation Instability for a Higher-Order Coupled Nonlinear Schrödinger System for the Ultrashort Optical Pulses in a Nonlinear Medium4
Breather Transitions and Their Mechanisms of a (2+1)-Dimensional Sine-Gordon Equation and a Modified Boussinesq Equation in Nonlinear Dynamics4
Analysis of Optimal Control of the Mechanical Structures Vibrating in the Viscoelastic Environment4
Hausdorff Dimension of the Exceptional Set of Interval Piecewise Affine Contractions4
An Age-Structured Diffusive Model for Epidemic Modelling: Lie Symmetries and Exact Solutions4
Periodic Solutions of a Second Order Discontinuous Nonautonomous Differential Equation4
Normalized Solutions for the Fractional Choquard Equations with Hardy–Littlewood–Sobolev Upper Critical Exponent4
Global Stability and Permanence of a Discrete-Time Two-Predators One-Prey System with Holling Type-III Response4
Qualitative Analysis, Chaotic Behavior and Optical Solitons Perturbation for the Coupled Fokas-Lenells System in Birefringent Fibers4
General Solution to a Nonlocal Linear Differential Equation of First-Order4
General Solutions of a Class of nth-Order Nonlocal Differential Equations4
Nonlinear Second Order Delay Dynamic Equations on Time Scales: New Oscillatory Criteria4
Streams and Graphs of Dynamical Systems4
Analysis of Neutral Fractional Differential Equation via the Method of Upper and Lower Solution4
Bifurcation of Limit Cycles of a Piecewise Smooth Hamiltonian System4
Power Series Solutions of Non-linear q-Difference Equations and the Newton–Puiseux Polygon4
Asymptotic and Oscillatory Behaviour of Third Order Non-linear Differential Equations with Canonical Operator and Mixed Neutral Terms4
On $$(\alpha ,\nu )$$-Relaxed Polygonal Metric Spaces and Fixed Point Results4
New Solitary Wave Solutions and Dynamical Behaviors of the Nonlinear Fractional Zakharov System4
Ground State Sign-Changing Solutions for a Schrödinger–Poisson System with Steep Potential Well and Critical Growth4
On the Menger Probabilistic Bipolar Metric Spaces: Fixed Point Theorems and Applications4
Traveling Wave Solution of Bad and Good Modified Boussinesq Equations with Conformable Fractional-Order Derivative4
Dirichlet Problems for Fractional Laplace Equations with Singular Nonlinearity4
Total Collision in a Four-Body Problem with Jacobi Potential4
Parameter Identification Problem for Semilinear Abstract Degenerate Delay Differential Equation4
Qualitative Results of Non-linear Adjoint Dynamic System with Delays and Impulses on Arbitrary Time Domains4
Riemann–Hilbert Approach and N-Soliton Solutions of a Novel n-Component Shifted Nonlocal Reverse-Time NLS Equation4
Bielecki–Ulam’s Types Stability Analysis of Hammerstein and Mixed Integro–Dynamic Systems of Non–Linear Form with Instantaneous Impulses on Time Scales4
On Generalized Two Dimensional Linear Hybrid Dynamical Systems4
Observer Design and State-Feedback Stabilization for Nonlinear Systems via Equilibrium Manifold Expansion Linearization4
Ground State Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth4
Oscillation and Nonoscillation for Nonlinear Delay Difference Equations by Phase Plane Analysis4
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