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Bassam's Unified Quantum Theory

بسام موفق سلمان, Bassam Muwafaq Salman

Published Oct 22, 2026• 0 citations
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Abstract

Bassam's Unified Quantum Theory Bassam Muwafaq Salman TEACHER General Directorate of personnel (Civilian Personnel), Training and Education Development Section, Iraqi Ministry of Defense. E- mail: basam.muwafaq86@gmail.com. basam.classic86@gmail.com 0009-0009-0659-7227An earlier version of this work is available on Zenodo(DOI: 10.5281/zenodo.23328953) and on HAL (hal-05796325). The priority date for the fundamental equations of Bassam's Unified Quantum Theory, in the field of Theoretical High Energy Physics (hep-th), authored by Bassam Muwaffaq Salman Abdul Karim, is established as 13/10/2026. This specific preprint was published on Figshare on 15/10/2026(DOI: 10.6084/m9.figshare.34532949). Abstract This work presents the first publication of a novel theory, termed Bassam's Unified Quantum Theory. The theory postulates that the characteristic time of any physical system can be derived directly from Bassam's fundamental time equation, t_B=S_VN/π , (in natural units) ℏ=k_B=c=1, where time emerges from S_VN , which represents the quantum information entropy (von Neumann entropy) and is a dimensionless quantity, here π is a mathematical constant. Applying the equation to the electron in the maximally mixed state, where the von Neumann entropy (unpolarized spin) is S_VN^(e) = ln 2, yields a theoretical characteristic time for the electron of t_B^e=2.846×10^(-22) s , upon restoring SI units. This time is interpreted as a fundamental timescale of the system, representing the quantum time of the electron. The theory further yields experimentally testable predictions through the solution of the Dirac-Bassam equation, a theoretical extension of the classical Dirac equation. Foremost among the theory's predictions is the existence of new excited particles-namely, excited electrons (the Bassam electron) with a predicted mass of m_B^(e)≈ 15.1 MeV/c^2. The particle is experimentally detectable in particle colliders, particularly in high-energy physics experiments such as Belle II or LHCb, where it can be produced via the decay channel e_B^-→ e^-+ γ. The theory provides a unified framework that connects quantum mechanics, relativity, and quantum information theory.Central to this theory is a fundamental relation between the time scale t_B and the quantum information entropy (von Neumann entropy), S_VN, which is a dimensionless quantity inherent to the system. which is given by Bassam's fundamental time equation, t_B=(S_VN/π)∙t_0. (upon restoring SI units), where t_0 represents the natural time scale of the system. By applying this equation to an electron in its maximally mixed state S_VN^(e)=ln⁡2, we obtain the Bassam quantum time of the electron, given by, t_B^e=2.846×10^(-22) s. This theory provides another central relationship between the length scale L_B and the quantum information entropy S_VN it contains which is a dimensionless quantity. which is given by Bassam's funda-mental spatial equation, L_B=(S_VN/π )∙λ_0. (upon restoring SI units), where λ_0 represents the reduced Compton wavelength. We can obtain the Bassam quantum length of the electron, which represents the smallest possible distance that an electron can travel from one place to another, and which is given by the calculated value L_B^(e)≈8.52×10^(-14) m . Through this theory we obtain another central relationship between the mass spectral scale m_B,_n and the quantum information entropy S_VN it contains, which is a dimensionless quantity, as given by Bassam's fundamental mass spectral equation. m_B,_n=m(1+(2π^2/S_VN) n), New particles can be obtained, namely excited electrons (excited Bassam electrons), with a mass of m_B^(e)≈ 15.1 MeV/c^2 They are ob-tained through the (Dirac–Bassam) equation via the decay channel e_B^-→ e^-+ γ. The theory also explains the behavior of the photon, where S_VN=0 and its characteristic time must be t_B^(Ph)=0, which provides a quantum expla-nation for the relativistic freezing of time at the speed of light c This is attributed to the fact that the photon does not possess quantum informa-tion entropy therefore its speed is maximal. This theory also seeks to provide an interpretation by exploring the implications and predictions it generates for various physical scenarios. The existence of quantum time necessarily leads us to the existence of a corre-sponding quantum space, thereby paving the way toward the unification of space and time. Through this endeavor, we aim to contribute to the ongoing dialogue concerning the origin of time and the origin of space, and to address the question of where the three spatial dimensions and the fourth, temporal dimension come from. Since this theory derives all components of the universe from information, the universe becomes a network of indivisible quantum pixels. This testable theory, which naturally es-tablishes a connection between time and quantum information, extends to addressing other problems in physics, This opens up wide-ranging prospects for future applications in research and development, as well as in both theoretical and experimental investigations, particularly in experiments at facilities such as Belle II and LHCb, within the framework of high-energy particle physics. This theory presents for the first time and in accordance with the researcher's findings, these fundamental equations and results. Bassam's fundamental time equation, t_B, Bassam's fundamental spatial equation, L_B and Bassam's fundamental mass spectral equation, m_B,_n. The quantized Bassam coordinates are (t_B∙n_t ,L_B∙n_x ,L_B∙n_y ,L_B∙n_z),which represent new physical equations and quantities that have not been proposed or discovered in any previous physical model.2. 1 BASSAM'S FUNDAMENTAL QUANTUM EQUATION OF TIME The term ∂ψ/∂t in Equation (25) represents the derivative of the quantum wave function with respect to time and is equal to -iωψ , which gives ψ(t) [47]. Therefore, the term ω_(S_VN/π) in Equation (36) becomes in the following form├ ∂ψ/∂t ┤|_(t_B=S_VN/π) The relation t_B=τ can be obtained (in natural units). The following equation can now be written. τ =S_VN/π(46) After the researcher obtained Equation (46), substituting it into Equation (45) yields Bassam's fundamental time equation. which is denoted by the symbol t_B (where we add the letter B , which indicates the researcher’s designation). and it represents Bassam's quan-tum time, associated with, which represents the quantum information entropy (von Neumann entropy) and is a dimensionless quantity, In addition,is a mathematical constant, and t_0 represents the natural time scale of the system. Upon restoring SI units [48], Bassam Muwaffaq Salman Abdul Karim was able to derive Bassam's fundamental time equation, which is given by the following equation. t_B_assam = S_VN/π∙t_0(47) whererepresents the quantum information entropy (von Neumann entropy) for any quantum system, as given by the following equation.(48) In Equation (48), where[J/K] represents the quantum information entropy (von Neumann entropy).is the Boltzmann constant. In our derivations,, a mathematical constant, appears and is eliminated from Equation (20). Upon transitioning to natural units,. Conse-quently, the entropy becomes dimensionless. When SI units are employed, preserving dimensional consistency requires the inclusion ofin Equation (47). Thus,is algebraically canceled from the numerator and the denominator, which ensures the consistency of Equation (47). whererepresents the quantum information entropy (von Neumann entropy), and it is a dimensionless quantity.represents the density matrix, which describes the quantum state of the system.represents the trace, which is the sum of the diagonal elements of the matrix.denotes the logarithm of the density matrix. Equation (47) can be written (upon restoring SI units). Withpresent in the numerator and denominator, Bassam's fundamental time equation can be written as given by the following equation. t_B= S_VN^(SI)/k_B π∙t_0 (49) whererepresents the quantum information entropy (von Neumann entropy) with dimen-sions [J/K] (upon restoring SI units). Equation (47) can be written, when using natural units with, to yield the following equation. (50) We can write the fundamental relation connect-ing length, time, and velocity, as given by the following equation. (51) whererepresents the distance,repre-sents the velocity, andrepresents the time. When the velocity is the same as the speed of light c, we can consider, and we obtain the following equation (52) Substituting Equation (52) into Equation (47) yields Bassam's fundamental spatial equation. which is denoted by the symbol L_B , representing the Bassam quantum length. The researcher was able to obtain Bassam's fundamental spatial equation, which is given by the following equation. L_B_assam = S_VN/π∙λ_0 (53) whererepresents the quantum information entropy (von Neumann entropy), and it is a dimensionless quantity. where λ_0 [m] represents the reduced Compton wavelength. and it is given by the formula. Equation (53) can be written, when using the International System of Units, as the following equation. L_B= S_VN^(SI)/k_B π∙λ_0 (54) where S_VN^(SI) represents the quantum information entropy (von Neumann entropy) with dimensions [J/K] (when using the International System of Units). A complete correspondence can be written between Equation (47), which is Bassam's quantum time equation, and Equation (53), which is Bassam's quantum length equation (in natural units,), Since time and space are the same entity, two sides of the same coin, the following equation can be written. t_B=L_B = S_VN/π∙1/m (55) Contrary to what has been established in classical and relativistic physics. where the space and time coordinates

DOI
10.6084/m9.figshare.34745475

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Bassam's Unified Quantum Theory Bassam Muwafaq Salman TEACHER General Directorate of personnel (Civilian Personnel), Training and Education Development Section, Iraqi Ministry of Defense. E- mail: basam.muwafaq86@gmail.com. basam.classic86@gmail.com 0009-0009-0659-7227An earlier version of this work is available on Zenodo(DOI: 10.5281/zenodo.23328953) and on HAL (hal-05796325). The priority date for the fundamental equations of Bassam's Unified Quantum Theory, in the field of Theoretical High Energy Physics (hep-th), authored by Bassam Muwaffaq Salman Abdul Karim, is established as 13/10/2026. This specific preprint was published on Figshare on 15/10/2026(DOI: 10.6084/m9.figshare.34532949). Abstract This work presents the first publication of a novel theory, termed Bassam's Unified Quantum Theory. The theory postulates that the characteristic time of any physical system can be derived directly from Bassam's fundamental time equation, t_B=S_VN/π , (in natural units) ℏ=k_B=c=1, where time emerges from S_VN , which represents the quantum information entropy (von Neumann entropy) and is a dimensionless quantity, here π is a mathematical constant. Applying the equation to the electron in the maximally mixed state, where the von Neumann entropy (unpolarized spin) is S_VN^(e) = ln 2, yields a theoretical characteristic time for the electron of t_B^e=2.846×10^(-22) s , upon restoring SI units. This time is interpreted as a fundamental timescale of the system, representing the quantum time of the electron. The theory further yields experimentally testable predictions through the solution of the Dirac-Bassam equation, a theoretical extension of the classical Dirac equation. Foremost among the theory's predictions is the existence of new excited particles-namely, excited electrons (the Bassam electron) with a predicted mass of m_B^(e)≈ 15.1 MeV/c^2. The particle is experimentally detectable in particle colliders, particularly in high-energy physics experiments such as Belle II or LHCb, where it can be produced via the decay channel e_B^-→ e^-+ γ. The theory provides a unified framework that connects quantum mechanics, relativity, and quantum information theory.Central to this theory is a fundamental relation between the time scale t_B and the quantum information entropy (von Neumann entropy), S_VN, which is a dimensionless quantity inherent to the system. which is given by Bassam's fundamental time equation, t_B=(S_VN/π)∙t_0. (upon restoring SI units), where t_0 represents the natural time scale of the system. By applying this equation to an electron in its maximally mixed state S_VN^(e)=ln⁡2, we obtain the Bassam quantum time of the electron, given by, t_B^e=2.846×10^(-22) s. This theory provides another central relationship between the length scale L_B and the quantum information entropy S_VN it contains which is a dimensionless quantity. which is given by Bassam's funda-mental spatial equation, L_B=(S_VN/π )∙λ_0. (upon restoring SI units), where λ_0 represents the reduced Compton wavelength. We can obtain the Bassam quantum length of the electron, which represents the smallest possible distance that an electron can travel from one place to another, and which is given by the calculated value L_B^(e)≈8.52×10^(-14) m . Through this theory we obtain another central relationship between the mass spectral scale m_B,_n and the quantum information entropy S_VN it contains, which is a dimensionless quantity, as given by Bassam's fundamental mass spectral equation. m_B,n=m(1+(2π^2/S_VN) n), New particles can be obtained, namely excited electrons (excited Bassam electrons), with a mass of m_B^(e)≈ 15.1 MeV/c^2 They are ob-tained through the (Dirac–Bassam) equation via the decay channel e_B^-→ e^-+ γ. The theory also explains the behavior of the photon, where S_VN=0 and its characteristic time must be t_B^(Ph)=0, which provides a quantum expla-nation for the relativistic freezing of time at the speed of light c This is attributed to the fact that the photon does not possess quantum informa-tion entropy therefore its speed is maximal. This theory also seeks to provide an interpretation by exploring the implications and predictions it generates for various physical scenarios. The existence of quantum time necessarily leads us to the existence of a corre-sponding quantum space, thereby paving the way toward the unification of space and time. Through this endeavor, we aim to contribute to the ongoing dialogue concerning the origin of time and the origin of space, and to address the question of where the three spatial dimensions and the fourth, temporal dimension come from. Since this theory derives all components of the universe from information, the universe becomes a network of indivisible quantum pixels. This testable theory, which naturally es-tablishes a connection between time and quantum information, extends to addressing other problems in physics, This opens up wide-ranging prospects for future applications in research and development, as well as in both theoretical and experimental investigations, particularly in experiments at facilities such as Belle II and LHCb, within the framework of high-energy particle physics. This theory presents for the first time and in accordance with the researcher's findings, these fundamental equations and results. Bassam's fundamental time equation, t_B, Bassam's fundamental spatial equation, L_B and Bassam's fundamental mass spectral equation, m_B,n. The quantized Bassam coordinates are (t_B∙n_t ,L_B∙n_x ,L_B∙n_y ,L_B∙n_z),which represent new physical equations and quantities that have not been proposed or discovered in any previous physical model.2. 1 BASSAM'S FUNDAMENTAL QUANTUM EQUATION OF TIME The term ∂ψ/∂t in Equation (25) represents the derivative of the quantum wave function with respect to time and is equal to -iωψ , which gives ψ(t) [47]. Therefore, the term ω(S_VN/π) in Equation (36) becomes in the following form├ ∂ψ/∂t ┤|(t_B=S_VN/π) The relation t_B=τ can be obtained (in natural units). The following equation can now be written. τ =S_VN/π(46) After the researcher obtained Equation (46), substituting it into Equation (45) yields Bassam's fundamental time equation. which is denoted by the symbol t_B (where we add the letter B , which indicates the researcher’s designation). and it represents Bassam's quan-tum time, associated with, which represents the quantum information entropy (von Neumann entropy) and is a dimensionless quantity, In addition,is a mathematical constant, and t_0 represents the natural time scale of the system. Upon restoring SI units [48], Bassam Muwaffaq Salman Abdul Karim was able to derive Bassam's fundamental time equation, which is given by the following equation. t_B_assam = S_VN/π∙t_0(47) whererepresents the quantum information entropy (von Neumann entropy) for any quantum system, as given by the following equation.(48) In Equation (48), where[J/K] represents the quantum information entropy (von Neumann entropy).is the Boltzmann constant. In our derivations,, a mathematical constant, appears and is eliminated from Equation (20). Upon transitioning to natural units,. Conse-quently, the entropy becomes dimensionless. When SI units are employed, preserving dimensional consistency requires the inclusion ofin Equation (47). Thus,is algebraically canceled from the numerator and the denominator, which ensures the consistency of Equation (47). whererepresents the quantum information entropy (von Neumann entropy), and it is a dimensionless quantity.represents the density matrix, which describes the quantum state of the system.represents the trace, which is the sum of the diagonal elements of the matrix.denotes the logarithm of the density matrix. Equation (47) can be written (upon restoring SI units). Withpresent in the numerator and denominator, Bassam's fundamental time equation can be written as given by the following equation. t_B= S_VN^(SI)/k_B π∙t_0 (49) whererepresents the quantum information entropy (von Neumann entropy) with dimen-sions [J/K] (upon restoring SI units). Equation (47) can be written, when using natural units with, to yield the following equation. (50) We can write the fundamental relation connect-ing length, time, and velocity, as given by the following equation. (51) whererepresents the distance,repre-sents the velocity, andrepresents the time. When the velocity is the same as the speed of light c, we can consider, and we obtain the following equation (52) Substituting Equation (52) into Equation (47) yields Bassam's fundamental spatial equation. which is denoted by the symbol L_B , representing the Bassam quantum length. The researcher was able to obtain Bassam's fundamental spatial equation, which is given by the following equation. L_B_assam = S_VN/π∙λ_0 (53) whererepresents the quantum information entropy (von Neumann entropy), and it is a dimensionless quantity. where λ_0 [m] represents the reduced Compton wavelength. and it is given by the formula. Equation (53) can be written, when using the International System of Units, as the following equation. L_B= S_VN^(SI)/k_B π∙λ_0 (54) where S_VN^(SI) represents the quantum information entropy (von Neumann entropy) with dimensions [J/K] (when using the International System of Units). A complete correspondence can be written between Equation (47), which is Bassam's quantum time equation, and Equation (53), which is Bassam's quantum length equation (in natural units,), Since time and space are the same entity, two sides of the same coin, the following equation can be written. t_B=L_B = S_VN/π∙1/m (55) Contrary to what has been established in classical and relativistic physics. where the space and time coordinates
This work presents the first publication of a novel theory, termed Bassam's Unified Quantum Theory. The theory postulates that the characteristic time of any physical system can be derived directly from Bassam's fundamental time equation:t_B = S_VN / πin natural units ℏ = k_B = c = 1, where time emerges from S_VN, which represents the quantum information entropy (von Neumann entropy) and is a dimensionless quantity; π is a mathematical constant. Applying the equation to the electron in the maximally mixed state, where the von Neumann entropy (unpolarized spin) isS_VN^(e) = ln 2,yields a theoretical characteristic time for the electron oft_B^(e) = 2.846 × 10^(−22) s,upon restoring SI units. This time is interpreted as a fundamental timescale of the system, representing the quantum time of the electron. The theory further yields experimentally testable predictions through the solution of the Dirac–Bassam equation, a theoretical extension of the classical Dirac equation. Foremost among the theory's predictions is the existence of new excited particles—namely, excited electrons (the Bassam electron)—with a predicted mass ofm_B^(e) ≈ 15.1 MeV/c².The particle is experimentally detectable in particle colliders, particularly in high-energy physics experiments such as Belle II or LHCb, where it can be produced via the decay channele_B⁻ → e⁻ + γ.The theory provides a unified framework that connects quantum mechanics, relativity, and quantum information theory.
This work presents the first publication of a novel theory, termed Bassam's Unified Quantum Theory. The theory postulates that the characteristic time of any physical system can be derived directly from Bassam's fundamental time equation, t_B=S_VN/π , (in natural units) ℏ=k_B=c=1, where time emerges from S_VN , which represents the quantum information entropy (von Neumann entropy) and is a dimensionless quantity, here π is a mathematical constant. Applying the equation to the electron in the maximally mixed state, where the von Neumann entropy (unpolarized spin) is S_VN^(e) = ln 2, yields a theoretical characteristic time for the electron of t_B^e=2.846×10^(-22) s , upon restoring SI units. This time is interpreted as a fundamental timescale of the system, representing the quantum time of the electron. The theory further yields experimentally testable predictions through the solution of the Dirac-Bassam equation, a theoretical extension of the classical Dirac equation. Foremost among the theory's predictions is the existence of new excited particles-namely, excited electrons (the Bassam electron) with a predicted mass of m_B^(e)≈ 15.1 MeV/c^2. The particle is experimentally detectable in particle colliders, particularly in high-energy physics experiments such as Belle II or LHCb, where it can be produced via the decay channel e_B^-→ e^-+ γ. The theory provides a unified framework that connects quantum mechanics, relativity, and quantum information theory.
This work presents the first publication of a novel theory, termed Bassam's Unified Quantum Theory. The theory postulates that the characteristic time of any physical system can be derived directly from Bassam's fundamental time equation, , (in natural units) , where time emerges from , which represents the quantum information entropy (von Neumann entropy) and is a dimensionless quantity, here is a mathematical constant. Applying the equation to the electron in the maximally mixed state, where the von Neumann entropy (unpolarized spin) is , yields a theoretical characteristic time for the electron of , upon restoring SI units. This time is interpreted as a fundamental timescale of the system, representing the quantum time of the electron. The theory further yields experimentally testable predictions through the solution of the Dirac-Bassam equation, a theoretical extension of the classical Dirac equation. Foremost among the theory's predictions is the existence of new excited particles-namely, excited electrons (the Bassam electron) with a predicted mass of . The particle is experimentally detectable in particle colliders, particularly in high-energy physics experiments such as Belle II or LHCb, where it can be produced via the decay channel . The theory provides a unified framework that connects quantum mechanics, relativity, and quantum information theory.
This work presents the first publication of a novel theory, termed Bassam's Unified Quantum Theory. The theory postulates that the characteristic time of any physical system can be derived directly from Bassam's fundamental time equation, , (in natural units) , where time emerges from , which represents the quantum information entropy (von Neumann entropy) and is a dimensionless quantity, here is a mathematical constant. Applying the equation to the electron in the maximally mixed state, where the von Neumann entropy (unpolarized spin) is , yields a theoretical characteristic time for the electron of , upon restoring SI units. This time is interpreted as a fundamental timescale of the system, representing the quantum time of the electron. The theory further yields experimentally testable predictions through the solution of the Dirac-Bassam equation, a theoretical extension of the classical Dirac equation. Foremost among the theory's predictions is the existence of new excited particles-namely, excited electrons (the Bassam electron) with a predicted mass of . The particle is experimentally detectable in particle colliders, particularly in high-energy physics experiments such as Belle II or LHCb, where it can be produced via the decay channel . The theory provides a unified framework that connects quantum mechanics, relativity, and quantum information theory.
This paper presents a complete and verifiable theoretical framework that unifies quantum mechanics and classical physics/general relativity under a single mathematical function: H(S,θ,K)=Fclassical(S)⋅Q(θ)+∇K⋅(ΠK[δS])H(S,θ,K)=Fclassical(S)⋅Q(θ)+∇K⋅(ΠK[δS]) We propose that quantum mechanics is not a fundamental separate theory, but rather a structural regulatory mechanism activated by specific boundary conditions within a fundamental continuous space-time. The theory introduces three core components: S: A unified continuous structural field representing both space-time and matter Q(θ): A context-dependent switching function that transitions between classical (Q=1) and quantum (Q=0) regimes K: An inherent geometric framework that governs all structural deformations The theory operates through three mathematically rigorous conditions: Pure Quantum Space (Q=0): Geometric deformations dominate via K Classical Physics + GR (Q=1): Reduces exactly to Einstein/Newton equations Local Quantum Effects (0<Q<1): Hybrid regime preserving fundamental constants Key innovations include: Resolution of the measurement problem through natural θ-transitions Preservation of all classical laws (c, G, ħ) as global constants Elimination of the quantum-gravity coupling problem Falsifiable predictions through approximate self-similarity analysis We propose three mathematical candidates for K (E₈ symmetry, Hodge-Laplacian, Spin Network) with specific experimental predictions in atomic spectra, optomechanics, and black hole entropy. The theory provides a concrete research program for 2026-2030, transitioning physics from fragmented descriptions to a unified dynamical system. This work represents not just an equation, but an ecosystem for unified physics—a testable speculative paradigm that enriches theoretical discourse regardless of its ultimate verification status.
The nonlinear equation of motion that accomplishes a self-consistent unification of quantum mechanics (QM) and thermodynamics conceptually different from the (von Neumann) foundations of quantum statistical mechanics (QSM) and (Jaynes) quantum information theory (QIT), but which reduces to the same mathematics for the thermodynamic equilibrium (TE) states, and contains standard QM in that it reduces to the time-dependent Schrodinger equation for zero entropy states is discussed in full mathematical detail. By restricting the discussion to a strictly isolated system (noninteracting, disentangled and uncorrelated), we show how the theory departs from the conventional QSM/QIT rationalization of the second law of thermodynamics, which instead emerges in QT (quantum thermodynamics) as a theorem of existence and uniqueness of a stable equilibrium state for each set of mean values of the energy and the number of constituent particles. To achieve this, the theory assumes Trρ lnρ B −k for the physical entropy and is designed to implement two fundamental ansatzs: (1) that in addition to the standard QM states described by idempotent density operators (zero entropy), a strictly isolated and uncorrelated system admits also states that must be described by non-idempotent density operators (nonzero entropy); (2) that for such additional states the law of causal evolution is determined by the simultaneous action of a Schrodinger-von Neumann-type Hamiltonian generator and a nonlinear dissipative generator which conserves the mean values of the energy and the number of constituent particles, and in forward time drives the density operator in the 'direction' of steepest entropy ascent (maximal entropy generation). The resulting dynamics is well defined for all non-equilibrium states, no matter how far from TE. Existence and uniqueness of solutions of the Cauchy initial value problem for all density operators implies that the equation of motion can be solved not only in forward time, to describe relaxation towards TE, but also in backward time, to reconstruct the 'ancestral' or primordial lowest entropy state or limit cycle from which the system originates. Zero entropy states as well as a well defined family of non-dissipative states evolve unitarily according to pure Hamiltonian dynamics and can be viewed as unstable limit cycles of the general nonlinear dynamics.